# 1.4: Properties of Whole Numbers - Mathematics

1.4: Properties of Whole Numbers - Mathematics

## Properties of Whole Numbers Class 6 Notes | EduRev

Whole numbers include natural numbers (that begin from 1 onwards), along with 0. Whole numbers are part of real numbers including all the positive integers and zero, but not the fractions, decimals, or negative numbers. Counting numbers are also considered as whole numbers. In this lesson, we will learn whole numbers and related concepts. In mathematics, the number system consists of all types of numbers, including natural numbers and whole numbers, prime numbers and composite numbers, integers, real numbers, and imaginary numbers, etc., which are all used to perform various calculations.

Predecessor and Successor

• A successor of any number is the next number to it, which is obtained by adding 1.
• A predecessor of any number is the previous number to it, which is obtained by subtracting 1.
• For example, the predecessor and the successor of the number 12 is 12 – 1 and 12 + 1 which is 11 and 13

We see numbers everywhere around the world, for counting objects, for representing or exchanging money, for measuring the temperature, telling time, etc. There is almost nothing that doesn't involve numbers, be it match scores, for players not scoring any run, we say 0 runs, be it cooking recipes, counting on objects, etc. Whole Numbers is a set of numbers formed, including all positive integers and 0.

What are Whole Numbers?

Natural numbers refer to a set of positive integers and on the other hand, natural numbers along with zero(0) form a set, referred to as whole numbers. However, zero is an undefined identity that represents a null set or no result at all.
The whole numbers are a set of numbers without fractions, decimals, or even negative integers. It is a collection of positive integers and zero. The primary difference between natural and whole numbers is zero.

Whole Number Definition

Whole Numbers are the set of natural numbers along with the number 0. The set of whole numbers in Mathematics is the set <0, 1,2,3. >.This set of whole numbers is denoted by the symbol W.
W = <0,1,2,3,4…>

• All natural numbers are whole numbers.
• All counting numbers are whole numbers.
• All positive integers including zero are whole numbers.
• All whole numbers are real numbers.

Whole Number Symbol

The symbol to represent whole numbers is the alphabet ‘W’ in capital letters, such as W = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…

Smallest Whole Number

Whole numbers start from 0 (from the definition of whole numbers). Thus, 0 is the smallest whole number. The concept of zero was first defined by a Hindu astronomer and mathematician Brahmagupta in 628. In simple language, zero is a number that lies between the positive and negative numbers on a number line. As such zero carries no value, though it is used as a placeholder. So zero is neither a positive number nor a negative number, but it is an even number.

Whole Numbers Vs Natural Numbers

From the above definitions, we can understand that every whole number other than 0 is a natural number. Also, every natural number is a whole number. So, the set of natural numbers is a part of the set of whole numbers or a subset of whole numbers. Set of Whole Numbers

Difference Between Whole Numbers and Natural Numbers

Let's understand the difference between whole numbers and natural numbers through the table given below: The Number Line

The set of natural numbers and the set of whole numbers can be shown on the number line as given below. All the positive integers or the integers on the right-hand side of 0, represent the natural numbers, whereas all the positive integers and zero, altogether represent the whole numbers. Both sets of numbers can be represented on the number line as follows: Properties of Whole Numbers

Operations on whole numbers: addition, subtraction, multiplication, and division, lead to four main properties of whole numbers that are listed below:

1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property

Closure Property

The sum and product of two whole numbers is always a whole number. The closure property of W is stated as follows: For all a,b∈W: a+b∈W and a×b∈W

Division by Zero
Division of a whole number by o is not defined, i.e., if x is a whole number then x/0 is not defined.

Multiplication by Zero
When a whole number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0.

Associative Property

The sum or product of any three whole numbers remains the same though the grouping of numbers is changed.The associative property of W is stated as follows: For all a,b,c∈W: a+=+c and a×=×c. For example, 10 + (7 + 12) = (10 + 7) + 12 = (10 + 12) + 7 = 29.

Commutative Property

The sum and the product of two whole numbers remain the same even after interchanging the order of the numbers. The commutative property of W is stated as follows: For all a,b∈W: a+b=b+a and a×b=b×a. This property states that change in the order of addition does not change the value of the sum. Let a and b be two whole numbers, commutative property states that a + b = b + a. For example, a = 10 and b = 19 ⇒ 10 + 19 = 29 = 19 + 10. It means that the whole numbers are closed under addition. This property also holds true for multiplication, but not for subtraction or division. For example: 7 x 9 = 63 or 9 x 7 = 63

Distributive Property

This property states that the multiplication of a whole number is distributed over the sum of the whole numbers. It means that when two numbers, take for example a and b are multiplied with the same number c and are then added, then the sum of a and b can be multiplied by c to get the same answer. This situation can be represented as: a × (b + c) = (a × b) + (a × c). Let a = 10, b = 20 and c = 7 ⇒ 10 × (20 + 7) = 270 and (10 × 20) + (10 × 7) = 200 + 70 = 270. The same is true for subtraction as well. For e.g we have a × (b − c) = (a × b) − (a × c). Let a = 10, b = 20 and c = 7 ⇒ 10 × (20 − 7) = 130 and (10 × 20) − (10 × 7) = 200 − 70 = 130.The distributive property of multiplication over addition is a×(b+c)=a×b+a×c. The distributive property of multiplication over subtraction is a×(b-c)=a×b-a×c.

When a whole number is added to 0, its value remains unchanged, i.e., if x is a whole number then x + 0 = 0 + x = x

Multiplicative Identity

When a whole number is multiplied by 1, its value remains unchanged, i.e., if x is a whole number then x.1 = x = 1.x

Patterns are used for easy verbal calculations and to understand the numbers better.

We can arrange the numbers using dots in elementary shapes like triangle, square, rectangle and line.

## 1.4: Properties of Whole Numbers - Mathematics

Properties and Laws of Whole Numbers

· Simplify by using the addition property of 0.

· Simplify by using the multiplication property of 1.

· Identify and use the commutative law of addition.

· Identify and use the commutative law of multiplication.

· Identify and use the associative law of addition.

· Identify and use the associative law of multiplication.

Mathematics often involves simplifying numerical expressions. When doing so, you can use laws and properties that apply to particular operations. The multiplication property of 1 states that any number multiplied by 1 equals the same number, and the addition property of zero states that any number added to zero is the same number.

Two important laws are the commutative laws, which state that the order in which you add two numbers or multiply two numbers does not affect the answer. You can remember this because if you commute to work you go the same distance driving to work and driving home as you do driving home and driving to work. You can move numbers around in addition and multiplication expressions because the order in these expressions does not matter.

You will also learn how to simplify addition and multiplication expressions using the associative laws. As with the commutative laws, there are associative laws for addition and multiplication. Just like people may associate with people in different groups, a number may associate with other numbers in one group or another. The associative laws allow you to place numbers in different groups using parentheses.

Addition and Multiplication Properties of 0 and 1

The addition property of 0 states that for any number being added to 0, the sum equals that number. Remember that you do not end up with zero as an answer – that only happens when you multiply. Your answer is simply the same as your original number.

## Notable Properties of Specific Numbers

These are some numbers with notable properties. (Most of the less notable properties are listed here.) Other people have compiled similar lists, but this is my list — it includes the numbers that I think are important (-:

A few rules I used in this list:

Everything can be understood by a typical undergraduate college student.

If multiple numbers have a shared property, that property is described under one "representative" number with that property. I try to choose the smallest representative that is not also cited for another property.

When a given number has more than one type of property, the properties are listed in this order:

1. Purely mathematical properties unrelated to the use of base 10 (example: 137 is prime.)

2. Base-10-specific mathematical properties (example: 137 is prime remove the "1": 37 is also prime remove the "3": 7 is also prime)

3. Things related to the physical world but outside human culture (example: 137 is close to the reciprocal of the fine-structure constant, once thought to be exact but later found to be closer to 137.036. )

4. All other properties (example: 137 has often been given a somewhat mystical significance due to its proximity to the fine-structure constant, most famously by Eddington)

Due to blatant personal bias, I only give one entry each to complex, imaginary, negative numbers and zero, devoting all the rest (27 pages) to positive real numbers. I also have a bit of an integer bias but that hasn't had such a severe effect. A little more about complex numbers, quaternions and so on, is here.

This page is meant to counteract the forces of Munafo's Laws of Mathematics. If you see room for improvement, let me know!

One of the square roots of i .

When I was about 12 years old, my step-brother gave me a question to pass the time: If i is the square root of -1, what is the square root of i ? . I had already seen a drawing of the complex plane, so I used it to look for useful patterns and noticed pretty quickly that the powers of i go in a circle. I estimated the square root of i to be about 0.7 + 0.7 i .

I can't remember why I didn't get the exact answer: either I didn't know trigonometry or the Pythagorean theorem, or how to solve multivariable equations, or perhaps was just tired of doing maths (I had clearly hit on Euler's formula and there's a good chance that contemplating the powers of 1+ i would have led me all the way through base- i logarithms and De Moivre's formula to the complex exponential function).

But you don't need that to find the square root of i . All you need to do is treat i as some kind of unknown value with the special property that any i 2 can be changed into a -1. You also need the idea of solving equations with coefficients and variables, and the square root of i is something of the form "a+b i ". Then you can find the square root of i by solving the equation:

Expand the (a+b i ) 2 in the normal way to get a 2 + 2ab i + b 2 i 2 , and then change the i 2 to -1:

Then just put the real parts together:

Since the real coordinate of the left side has to be equal to the real coordinate of the right, and likewise for the imaginary coordinates, we have two simultaneous equations in two variables:

From the first equation a 2 -b 2 = 0, we get a=b substituting this into the other equation we get 2a 2 = 1, and a=۫/√ 2 and this is also the value of b. Thus, the original desired square root of i is a+b i = (1+ i )/√ 2 (or the negative of this).

(This is the only complex number with its own entry in this collection, mainly because it's the only one I've had much interest in see the "blatant personal bias" note above :-).

The unit of imaginary numbers, and one of the square roots of -1.

(This is the only imaginary number with its own entry in this collection, mainly because it stands out way above the rest in notability. In addition, non-real numbers don't seem to interest me much. )

-1 is the "first" negative number, unless you define "first" to be "lowest".

In "two's complement" representation used in computers to store integers (within a fixed range), numbers are stored in base 2 (binary) with separate base-2 digits in different "bits" of a register. Negative numbers have a 1 in the highest position of the register. The value of -1 is represented by 1's in all positions, which is the same as what you'd get if you wrote a program to compute

and let it go long enough to overflow.

As it turns out, that series sum can be treated as an example of the general series sum

As discussed in the entry for 1/2, the sum is equal to 1/(1- x ), but that is valid only when | x | x =2 and use the formula anyway, we get 1/(1- x ) = 1/(1-2) = -1, which is the same as the two's complement interpretation.

(I do not have many entries for negative numbers, as they do not interest so much. Perhaps I still relate to numbers in terms of counting things like "the 27 sheep on that hill" or "the 40320 permutations of the Loughborough tower bells".)

The (in)famous sum of the positive integers:

1 + 2 + 3 + 4 + 5 + 6 + 7 + . = 1 /12

In the 19 th century, new techniques (Cesaro, Abel) were developed to tame some of the infinite series sums that do not converge normally. Examples are shown in the entries for 1/4 and 1/2. But these techniques alone are not enough to handle the infinite series sum:

C = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + .

This sum diverges monotonically (increases towards infinity, without ever taking a step in the negative direction) and Cesaro/Abel will not work.

Euler had to deal with it when performing analytic continuation on what is now called the Riemann zeta function:

Zeta( s ) = 1 - s + 2 - s + 3 - s + 4 - s + .

Euler had s = -1, which gives Zeta( s ) = 1 + 2 + 3 + 4 + . Euler's approach was to express it as a linear combination of itself with an existing Cesaro- or Abel-summable series, namely the 1-2+3-4+. =1/4 series, but by Euler's considerably easier diffentiation method:

C = 1 + 2 + 3 + 4 + 5 + 6 + 7 + .
= 1 + (4-2) + 3 + (8-4) + 5 + (12-6) + 7 + .
= 1 - 2 + 3 - 4 + 5 - 6 + 7 + . + 4 (1 + 2 + 3 + . )

Cesaro/Abel and Euler's method both give a sum 1/(1+1) 2 = 1 /4 for the first part so we have

The value of the Riemann Zeta function with argument of -1 is -1/12. As described by John Baez 100 :

The numbers 12 and 24 play a central role in mathematics thanks to a series of "coincidences" that is just beginning to be understood. One of the first hints of this fact was Euler's bizarre "proof" that

which he obtained before Abel declared that "divergent series are the invention of the devil". Euler's formula can now be understood rigorously in terms of the Riemann zeta function, and in physics it explains why bosonic strings work best in 26=24+2 dimensions.

Baez, at the end of his "24" lecture, indicates that the significance of 24 is connected to the fact that there are two ways to construct a lattice on the plane with rotational symmetry: one with 4-fold rotational symmetry and another with 6-fold rotational symmetry — and 4×6=24. A connection between zeta(-1)=-1/12 and symmetry of the plane makes more sense in light of how the Zeta function is computed for general complex arguments. Also, the least common multiple of 4 and 6 is 12.

Srinivasa Ramanujan also explained 1 + 2 + 3 + 4 + . = -1 /12, but in a more general way than Euler. He used a new analytic continuation of the Riemann zeta function.

In Ramanujan's 1913 letter to G.H. Hardy, the as-yet-undiscovered Indian mathematicical genius listed many of his discoveries and derivations. In section XI he stated:

I have got theorems on divergent series, theorems to calculate the convergent values corresponding to the divergent series, viz.

1 3 + 2 3 + 3 3 + 4 3 + . = 1 /120,

Theorems to calculate such values for any given series (say: 1 - 1 2 + 2 2 - 3 2 + 4 2 - 5 2 + . ), and the meaning of such values.

In modern notation we append (ℜ) to the end of such a series sum, to signify Ramanujan summation:

The Ramanujan sum defines a function f ( x ) whose values for integer x are the terms in the series being summed. Then

1 + 2 + 3 + 4 + . n (ℜ)
= Sigma k =1 n f ( k )
= Integral x =0 n f ( x ) + Sigma k =1 ∞ Bk / k ! ( f ( k -1) ( n )- f ( k -1) (0)) + R

where " f ( k -1) " is the ( k -1) th derivative of f (). Hardy and Ramanjuan considered just the parts of this that do not depend on n :

For a converging series, f ( x ) would approach a limit as x approaches infinity, and this would give a value that is equal to the sum of the infinite series. In our case the f ( x ) diverges, and the series sum is infinite, but this Hardy-Ramanujan sum is not. f (0) is 0, and the 1 st derivative is constant f '( x ) = 1, and all higher derivatives are zero, so it reduces to just

B 2 is the second Bernoulli number which is 1/6, so we get -1/12.

The word "zero" is the only number name in English that can be traced back to Arabic (صِفر ʂifr "nothing", "cipher" which became zefiro in Italian, later contracted by removing the fi ). The word came with the symbol, at around the same time the western Arabic numerals came to Europe. 44 , 105

The practice of using a symbol to hold the place of another digit when there is no value in that place (such as the 0 in 107 indicating there are no 10's) goes back to 5 th -century India, where it was called shunya or Śūnyatā 107 .

(This is the only zero number with its own entry in this collection, mainly because a field can have only one additive identity.)

This is the Planck time in seconds it is related to quantum mechanics. According to the Wikipedia article Planck time, "Within the framework of the laws of physics as we understand them today, for times less than one Planck time apart, we can neither measure nor detect any change" . One could think of it as "the shortest measurable period of time", and for any purpose within the real world (if one believes in Quantum mechanics), any two events that are separated by less than this amount of time can be considered simultaneous.

It takes light (traveling at the speed of light) this long to travel one Planck length unit, which itself is much smaller than a proton, electron or any particle whose size is known.

This is the Planck length in meters it is related to quantum mechanics. The best interpretation for most people is that the Planck length is the smallest measurable length, or the smallest length that has any relevance to events that we can observe. This uses the CODATA 2014 value 50 . See also 5.390×10 -44 and 299792458.

The "reduced" Planck constant in joule-seconds, from CODATA 2014 values 50 .

This is the Planck constant in joule-seconds, from CODATA 2014 values 50 . This gives the proportion between the energy of a photon and its wavelength.

As of 1 st May 2019, the Planck constant (in joule-seconds) is defined to be exactly this value, in order to define the kilogram in terms of observable properties of nature. The definition reads:

The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10 -34 when expressed in the unit J s, which is equal to kg m 2 s -1 , where the metre and the second are defined in terms of c and δνCs.

where c is the speed of light by the existing (since 1967) definition (see 299792458) and δνCs is the unperturbed ground-state hyperfine transition frequency of Caesium-133 (see 9192631770).

The mass of an electron in kilograms, from CODATA 2014 values 50 . See also 206.786.

The mass of a proton in kilograms, from CODATA 2014 values 50 .

The mass of a neutron in kilograms, from CODATA 2014 values 50 .

The approximate time (in seconds) it takes light to traverse the width of a proton.

Value of the Boltzmann constant by the old (pre-2019) definition, as given in CODATA 2014 50 . This value was based on experimental observations and also upon the definition of the Kelvin, which was determined by measuring the temperature of the triple point of water and defining the Kelvin so that the triple point temperature comes out to 273.16 K. For the current (2019 and later) definition see 1.380649×10 -23 .

The Boltzmann constant (in Joules per degree Kelvin) by the 2019 redefinition, which reads:

The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380649×10 -23 when expressed in the unit J K -1 , which is equal to kg m 2 s -2 K -1 , where the kilogram, metre and second are defined in terms of h , c and δνCs.

where h is the Planck constant by the new definition (see 6.62607015×10 -34 ), c is the speed of light by the existing (since 1967) definition (see 299792458) and δνCs is the unperturbed ground-state hyperfine transition frequency of Caesium-133 (see 9192631770).

The quantum of electric charge in coulombs (one third of the electron charge), based on from CODATA 2014 values 50 . Protons, electrons and quarks all have charges that are a (positive or negative) integer multiple of this value.

The elementary charge or "unit charge", the charge of an electron in coulombs, from CODATA 2014 values 50 . This is no longer considered the smallest quantum of charge, now that matter is known to be composed largely of quarks which have charges in multiples of a quantum that is exactly 1/3 this value.

As of the 1 st May 2019, the elementary charge is not measured in terms of coulombs instead it is defined to be exactly 1.602176634×10 -19 coulombs in other words, the coulomb is now defined in terms of the elementary charge.

The reciprocal value of the coulomb in units of the elementary charge, by the new (2019) definition.

In 2019 the International System of Units (SI) was updated to define its seven base units in a way that defines all seven of them in terms of observable properties of nature, which are given arbitrary numerical values in terms of the base units:

The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634×10 -19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of δνCs.

where δνCs is the unperturbed ground-state hyperfine transition frequency of Caesium-133 (see 9192631770).

Approximate "size" of a proton 71 , in meters (based on its "charge radius" of 0.875 femtometers). "Size" is a pretty vague concept for particles, and different definitions are needed for different problems. See 10 40 .

The vacuum permittivity constant in farads per meter, using the old (pre-2019) definitions of the vacuum permeability (see 4π/10 7 ) and the (still current) definition of the speed of light (see 299792458). In older times this was called the "permittivity of free space". Due to a combination of standard definitions, notably the exact definition of the speed of light, this constant is exactly equal to 10 7 /(4 π 299792458 2 ) = 625000/22468879468420441π farads per metre.

In 2019 and later, the vacuum permittivity is something needing to be computed based on measurement. The greatest uncertainty contributing to its value is the measurement of the fine structure constant.

The gravitational constant in cubic meters per kilogram second squared, from CODATA 2014 values 50 . This is one of the most important physical constants in physics, notably cosmology and efforts towards unifying relativity with quantum mechanics. It is also one of the most difficult constants to measure. See also 1.32712442099(10)×10 20 .

The Planck mass in kilograms, using CODATA 2014 values. The Planck mass is related to the speed of light, the Planck constant, and the gravitational constant by the formula Mp = √ hc /2π G .

The constant 4π/10 7 that appears in the old (pre-2019) definition of the "magnetic constant" or vacuum permeability. It is related to the old definition of ampere, which stated that if exactly one ampere of current flows in two straight parallel conductors of infinite length 1 meter apart, the force produced would be 2×10 -7 newton per meter of length. This derives from an older definition stating that a similar setup with the wires one centimetre apart would produce a force of 2 dynes per centimetre of length (one dyne is 10 -5 newtons).

The fine-structure constant, as given by CODATA 2014 (see 50 ). The "(17)" is the error range. See the 137.035. page for history and details.

There are a few "coincidences" regarding multiples of 1/127:

e /π = 0.865255. &asymp 110/127 = 0.866141.
√ 3 = 1.732050. &asymp 220/127 = 1.732283.
π = 3.141592. &asymp 399/127 = 3.141732.
√ 62 = 7.874007. &asymp 1000/127 = 7.874015.
e π = 23.140692. &asymp 2939/127 = 23.141732.

There are a few more for 1/7. The √ 62 coincidence is discussed in the √ 62 entry, and the π and e π ones go together (see e π ).

This is the eccentricity of the orbit of the Earth-Moon barycentre at epoch J2000 the value is currently decreasing at a rate of about 0.00000044 per year, mostly due to the influence of other planets. The Moon is massive enough and far enough to shift the Earth itself a few thousand km away from the barycentre. See also 0.054900.

The version of the Gaussian gravitational constant computed by Simon Newcomb in 1895.

The "Gaussian gravitational constant" k , as originally calculated by Gauss, related to the Gaussian year Δ t by the formula Δ t = 2π/ k . The value was later replaced by the Newcomb value 0.01720209814, but in 1938 (and again in 1976) the IAU adopted the original Gauss value.

Mean eccentricity of the Moon's orbit — the average variation in the distance of the Moon at perigee (closest point to the Earth) and apogee. Due to the influence of the Sun's gravity the actual eccentricity varies a large amount, going as low as about 0.047 and as high as about 0.070 also the ellipse precesses a full circle every 9 years (see 27.554549878). The eccentricity is greatest when the perigee and apogee coincide with new and full moon. At such times the Moon's distance varies by a total of 14%, and its apparent size (area in sky) varies by 30% when the size at apogee is compared to the size at perigee. This means that the brightness of the full moon varies by 30% over the course of the year. In 2004 the brightest full moon was the one on July 2 nd due to the orbit's precession the brightest full moon in 2006 was a couple months later, Oct 6 th .

This change in size is a little too small for people to notice from casual observation (except in solar eclipses, when the Moon sometimes covers the whole sun but at other times produces an annular eclipse). But the eccentricity is large enough to cause major differences in the Moon's speed moving through the sky from one day to the next. When the Moon is near perigee it can move as much as 16.5 degrees in a day when near apogee it moves only 12 degrees the mean is 13.2. The cumulative effect of this is that the moon can appear as much as 22 degrees to the east or west of where it would be if the orbit were circular, enough to cause the phases to happen as much as 1.6 days ahead of or behind the prediction made from an ideal circular orbit. It also affects the libration (the apparent "wobbling" of the Moon that enables us to see a little bit of the far side of the moon depending on when you look).

This is the lowest value of z for which the infinite power tower

converges to a finite value. (The highest such value is e (1/ e ) = 1.444667. see that entry for more).

This is the Kepler–Bouwkamp constant, related to a geometrical construction of concentric inscribed circles and polygons. Start with a unit circle (a circle with radius 1). Inscribe an equilateral triangle inside the circle, then inscribe a circle inside the triangle. The radius of the smaller circle will be cos(π/3) = 1/2. Now inscribe a square inside that circle, and a circle inside the square this even smaller circle has radius cos(π/3)×cos(π/4) = √ 1/8 . Continue inscribing with a pentagon, hexagon, and every successive regular polygon. The circles get smaller but they do not go all the way down to zero the limit is this number, about 10/87.

The fraction 1/7 is the simplest example of a fraction with a repeating decimal that has an interesting pattern. See the 7 article for some of its interesting properties.

Reader C. Lucian points out that many of the well-known constants can be approximated by multiples of 1/7:

gamma = 0.5772156. &asymp 4/7 = 0.571428.
e /π = 0.865255. &asymp 6/7 = 0.857142.
√ 2 = 1.414213. &asymp 10/7 = 1.428571.
√ 3 = 1.732050. &asymp 12/7 = 1.714285.
e = 2.7182818. &asymp 19/7 = 2.714285.
π = 3.1415926. &asymp 22/7 = 3.142857.
e π = 23.140692. &asymp 162/7 = 23.142857.

These are mostly all coincidences without any other explanation, except as noted in the entries for √ 2 and e π . See also 1/127.

This is the integral of sin(1/*x), from 0 to 1. Mathematica or Wolfram Alpha will give more digits: 0.5040670619­0692837198­9856117741­1482296249­8502821263­9170871433­1675557800­7436618361­6051791560­4457297012.

A reader suggested to me the idea that some people might define "zillion" as "a 1 followed by a zillion zeros". This is kind of like the definition of googolplex but contradicts itself, in that no matter what value you pick for X , 10 X is bigger than X .

However, this is actually only true if we limit X to be an integer (or a real number). If X is allowed to be a complex number, then the equation 10 X = X has infinitely many solutions.

Using Wolfram Alpha, put in "10^x=x" and you will get:

x &asymp -0.434294481903251827651 Wn (-2.30258509299404568402)

with a note describing Wk as the "product log function", which is related to the Lambert W function (see 2.50618. ). This function is also available in Wolfram Alpha (or in Mathematica) using the name " ProductLog [ k , x ]" where k is any integer and x is the argument. So if we put in "-0.434294481903251827651 * ProductLog[1, -2.30258509299404568402]", we get:

0.529480508259063653364. - 3.34271620208278281864. i

Finally, put in "10^(0.529480508259063653364 - 3.34271620208278281864 * i)" and get:

0.52948050825906365335. - 3.3427162020827828186. i

If we used -2 as the initial argument of ProductLog[] , we get 0.5294805+3.342716 i , and in general all the solutions occur as complex conjugate pairs. Other solutions include x =-0.119194. ۪.750583. i and x =0.787783. ۰.083768. i .

In light of the fact that the -illion numbers are all powers of 1000, another reader suggested that one should do the above starting with 10 (3 X +3) = X . This leads to similar results, with one of the first roots being:

-0.88063650680345718868. - 2.10395020077170002545. i

The first fraction in Conway's FRACTRAN program ( page 147) that finds all the prime numbers. The complete program is 17 /91, 78 /85, 19 /51, 23 /38, 29 /33, 77 /29, 95 /23, 77 /19, 1 /17, 11 /13, 13 /11, 15 /2, 1 /7, 55 /1. To "run" the program: starting with X =2, find the first fraction N / D in the sequence for which XN / D is an integer. Use this value NX / D as the new value of X , then repeat. Every time X is set to a power of 2, you've found a prime number, and they will occur in sequence: 2 2 , 2 3 , 2 5 , 2 7 , 2 11 and so on. It's not very efficient though — it takes 19 steps to find the first prime, 69 for the second, then 281, 710, 2375 . (Sloane's A7547).

This is e -π/2 , which is also equal to i i . (Because e ix = cos( x ) + i sin( x ), e i π/2 = i , and therefore i i = ( e i π/2 ) i = e i 2 π/2 = e -π/2 .)

The Cesaro sum of the alternating-diverging infinite series sum:

which can be used to derive the Euler/Ramanujan "infamous" sum 1 + 2 + 3 + 4 + . = -1/12.

The first-order Cesaro method is illustrated in the entry for 1/2. Here we'll apply the method twice. We start with the terms of the infinite series:

This has the partial sums:

These diverge and are unbounded both above and below. The sum of the first n terms of that series is:

The average of the first n terms of A o ( n ) is A'( n )/ n :

( C ,1)-sum = A'( n )/ n : 1, 0, 2/3, 0, 3/5, 0, 4/7, .

This is not converging but offers hope in that (like 1-1+1-. ) it manages to at least remain bounded from above as well as below. The even terms are all 0 while the odd terms approach 1/2.

Let's take successive averages of this sequence: the Cesaro sum of the Cesaro sum. The sum of the first n terms of the above "( C ,1)-sum" is

1, 1, 5/3, 5/3, 34/15, 34/15, 298/105, .

and successive averages are just these over n :

1, 1/2, 5/9, 5/12, 34/75, 34/90, 298/735, .

which converge on 1/4, though it may be a bit tough to see here. This isn't actually how Cesaro defined the 2 rd order method. Instead, he put the sum of the first n terms of A'( n ) in the numerator:

and the binomial coefficients n C 2 (the triangular numbers), called "E''( n )", in the denominator:

E''( n ) : 1, 3, 6, 10, 15, 21, 28, 35, .

The second-order averages by Cesaro's method are:

( C ,2)-sum = A''( n )/ C ( n ,2) : 1, 1/3, 3/6, 3/10, 6/15, 6/21, 10/28, .

and these also converge on 1 /4. Adding this way makes it easier to see because e.g. for even n we can let h be n /2 and we get:

A''( n )/ C ( n ,2) = h C 2 / 2 h C 2
= ( h ( h -1)/2) / (2 h (2 h -1)/2)
= ( h 2 - h )/(4 h 2 -2 h )
= (1/2) (2 h 2 - h - h 2 )/(2 h 2 - h )
= (1/2) ((2 h 2 - h )/(2 h 2 - h ) - h 2 /(2 h 2 - h ))
= 1/2 - 1/2 ( h 2 /(2 h 2 - h ))

The part " h 2 /(2 h 2 - h )" clearly converges on 1/2, so the whole thing converges to 1/2 - 1/4.

This sum of 1 /4 appears as "1/(1+1) 2 " in Ramanujan's notebook. That can be derived by noting that 1-1+1-1+1-1+. has the 1 st -order Cesaro sum 1/2, and then doing this:

(1 - 1 + 1 - 1 + 1 - . ) 2
= (1 - 1 + 1 - 1 + 1 - . )×(1 - 1 + 1 - 1 + 1 - . )
= 1 + (-1×1 + 1×-1) + (1×1 + -1×-1 + 1×1) + (-1×1 + 1×-1 + -1×1 + 1×-1) + .
= 1 - 2 + 3 - 4 + .

So the sum of 1 - 2 + 3 - 4 + 5 - 6 + 7 - . must be the square of the sum of 1 - 1 + 1 - 1 + 1 - . which is the square of 1/2, which is 1/4.

There is another, perhaps easier, way to get the same answer. Start with this infinite series sum and assume it has a value, here called C :

Subtract the second from the first:

C - Cx = 1
C (1- x ) = 1
C = 1/(1- x )

If x is something like 1/2, it's easy to see that the sum 1 + 1/2 + 1/4 + 1/8 + . is 2, and 1/(1- x ) = 1/(1-1/2) is also 2, so the derivation is valid. But if x were, say, -1, then we'd get 1 - 1 + 1 - 1 + 1 - . = 1/2, whichis discussed in the entry for 1/2. Euler didn't worry about strict convergence and just went ahead with:

1 + x + x 2 + x 3 + x 4 + . = 1/(1- x )

Let's differentiate both sides!

1 + 2 x + 3 x 2 + 4 x 3 + . = 1/(1- x ) 2

If x =-1 we have the desired sum:

and again the answer is 1/4.

This is an infinite product of (1-2 - N ) for all N . This is also the product of (1- x N ) with x =1/2. Euler showed that in the general case, this infinite product can be reduced to the much easier-to-calculate infinite sum 1 - x - x 2 + x 5 + x 7 - x 12 - x 15 + x 22 + x 26 - x 35 - x 40 + . where the exponents are the pentagonal numbers N (3 N -1)/2 (for both positive and negative N ), Sloane's A1318. 30

This is Gottfried Helms' Lucas-Lehmer constant " LucLeh " see 1.38991066352414. for more.

1/3 is the simplest non-dyadic rational, and the simplest with a non-terminating decimal in base 10.

1/3 is the "Ramanujan sum" of the non-converging infinite series sum of -2 n :

Even though we're not allowed to, we could try to apply the series sum formula:

which converges the normal way only when -1 x x = -2 and the sum would be 1/(1-(-2)) = 1/3.

This is "Artin's Constant", the product (1-1/2)(1-1/6)(1-1/20). (1-1/( p ( p -1))) for all prime p . It relates to the conjecture regarding the "density" of primes p for which 1/ p has a as a primitive root, where a meets the conditions of OEIS sequence A85397. This includes 10, meaning that about 30% of primes have a reciprocal with a decimal expansion that repeats every p -1 digits the first two are 7 and 17.

has a value that is very close to, but not exactly, π/8. From Bernard Mares, Jr. via Bailey et al.  more on MathWorld at Infinite Cosine Product Integral.

If you take a string of 1's and 0's and follow it by its complement (the same string with 1's switched to 0's and vice versa) you get a string twice as long. If you repeat the process forever (starting with 0 as the initial string) you get the sequence

and if you make this a binary fraction 0.0110100110010110. 2 the equivalent in base 10 is 0.41245403364. and is called the Thue-Morse constant or the parity constant . Its value is given by a ratio of infinite products:

4 K = 2 - PRODUCT[2 2 n -1] / PRODUCT[2 2 n ]
= 2 - (1 × 3 × 15 × 255 × 65535 × . )/(2 × 4 × 16 × 256 × 65536 × . )

The Cesaro sum of the simplest Cesaro-summable infinite series sum:

The Cesaro sum technique is a generalisation of the definition of an infinite series sum as the limit of its partial sums. To illustrate the principle, let's consider an infinite sum that actually does converge in the normal way:

this has the partial sums:

which can easily be seen (and proven, by mathematical induction) to converge to 2. Cesaro considered the series of averages (arithmetic means) of the first N partial sums:

1, (1 + 3/2)/2, (1 + 3/2 + 7/4)/3, (1 + 3/2 + 7/4 + 15/8)/4, ..

1, 5/4, 17/12, 49/32, 129/80, 321/128, 769/448, .

which also converges on 2, though more slowly. This technique of averaging the first n partial sums can yield an answer for infinite series whose partial sums taken individually so not converge. Start with:

This doesn't converge, but let's take the average of the first n of these. The sums of the first n of these (for n =1, 2, 3, . ) are:

So the average of the first n partial sums are:

which converges on 1/2. See 1/4 for an example of 2 rd order Cesaro summation, and -1/12 to see Ramanujan's extension.

Ramanujan's notebook, when discussing the -1/12 series, uses "1/(1+1) 2 ", which suggests that he viewed the sum 1 - 1 + 1 - 1 + 1 - 1 + 1 - . to be "1/(1+1)". This can be derived from the generalisation of the series sum:

which converges the normal way only when | x | x = -1 we'd get "1 - 1 + 1 - 1 + 1 - 1 + 1 - . = 1/(1- x ) = 1/(1+1)". So the value 1/2 can be "justified" in two ways.

The odds of losing a game of chance. Flip a coin: if you get heads , your score increases by π, if you get tails , your score diminishes by 1. Repeat as many times as you wish — but if your score ever goes negative, you lose. Assuming the player keeps playing indefinitely (motivated by the temptation of getting an ever-higher score), what are the odds of losing?

The answer is given by a series sum: 1/2 + 1/2 5 + 4/2 9 + 22/2 13 + 140/2 17 + 969/2 21 + 7084/2 25 + 53820/2 29 + 420732/2 34 + . (numerators in Sloane's A181784) which adds up to 0.5436433121.

A more sophisticated analysis using rational numbers like 355/113 converges on the answer more quickly, giving 0.54364331210052407755147385529445. (see ).

This is the Omega constant, which satisfies each of these simple equations (all equivalent):

Thus it is sort of like the golden ratio. In the above equations, if e is replaced with any number bigger than 1 (and "ln" by the corresponding logarithm) and you get another "Omega" constant. For example:

if 2 x =1/ x , then x =0.6411857445.
if π x =1/ x , then x =0.5393434988.
if 4 x =1/ x , then x =1/2
if 10 x =1/ x , then x =0.3990129782.
if 27 x =1/ x , then x =1/3
if 10000000000 x =1/ x , then x =1/10

(the Euler-Mascheroni constant )

This is the Euler-Mascheroni constant, commonly designated by the Greek letter gamma . It is defined in the following way. Consider the sum:

S n = 1 + 1/2 + 1/3 + 1/4 + 1/5 + . + 1/ n

The sequence starts 1, 1.5, 1.833333. 2.083333. etc. As n approaches infinity, the sum approaches ln( n ) + gamma . Numberphile has a video about this constant: The mystery of 0.577.

Here are some not-particularly-significant approximations to gamma :

1/(√ π - 1/25) = 0.5772159526.
gamma = 0.5772156649.
1/(1+ 1/√ 10 ) 2 = 0.5772153925.

One of the infinite sums in Ramanujan's 1913 letter to G.H. Hardy, section XI:

see -1/12 for a simpler example.

this sum diverges, but a partial sum can be contemplated:

0 + 1 + 2 + . + n
= SUM i in [0.. n ] f ( i )
(where f ( x ) = x )

= - f (0)/2 + i INTEGRAL0..∞ ( f ( it )- f (- it )) / ( e 2π t - 1) dt

in this specific example we get

Value of the infinite series sum

It is (1-√ 2 ) times the Riemann zeta function of 1/2. More digits: 0.604898643421630370247265914. (Sloane's sequence A113024). Oddly, though the series sum converges to a reasonably small finite value, if you square the series sum:

and sum the terms in the needed order:

the magnitudes of the parenthesised parts keep growing, so the series sum diverges. However, there clearly is a sum, and technuqies such as Cesaro summation (see the entry for 1/4) can be used to evaluate it and get the proper answer, It is for sums like this that Cesaro summation is really needed. (The case for 1/4 is a bit harder to argue.)

The golden ratio (reciprocal form): see 1.618033. .

The Buffon's needle problem involves estimating the probability that a randomly-placed line segment of some given length will cross one of a a set of parallel lines spaced some fixed distance apart. If the length of the line segment is the same as the spacing between lines, the probability is 2/π.

This is the lowest point in the function y = x x . See also 1.444667. .

The natural logarithm of 2, written "ln(2)". See 69.3147. and 72.

ln(2) is the value of this infinite series sum:

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 + .
= 1/2 + 1/12 + 1/30 + 1/56 + .

This is called a "conditionally convergent series" because the series converges if added up in the way shown above, but if you rearrange the terms:

1 + 1/3 + 1/5 + 1/7 + . - (1/2 + 1/4 + 1/6 + 1/8 + . )

then you have two series that do not converge and an undefined "infinity minus infinity".

You can create a long string of 1's and 0's by using "substitution rules" and iterating from a small starting string like 0 or 1. If you use the rule:

and start with 0, you get 1, 10, 101, 10110, 10110101, 1011010110110, . where each string is the previous one followed by the one before that (Sloane's A36299 or A61107). The limit of this is an infinite string of 1's and 0's which you can make this into a binary fraction: 0.1011010110110. 2, you get this constant (0.709803. in base 10) which is called the Rabbit Constant . It has some special relationships to the Fibonacci sequence:

• In the iteration described above, the number of digits in each string is the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, .
• Expressed as a continued fraction, the constant is 0 + 1/(2 0 + 1/(2 1 + 1/(2 1 + 1/(2 2 + 1/(2 3 + 1/(2 5 + 1/(2 8 + . ))))))) where the exponents of 2 are the Fibonacci numbers.
• If you take all the multiples of the Golden Ratio 0.618033 and round them down to integers, you get 1, 3, 4, 6, 8, 9, 11, 12, . These numbers tell you where the 1's in the binary fraction are.

If you leave off the first two binary digits (10) you get 110101101101011010110110101. the bit pattern generated by a Turing machine at the end of the Turing machine Google Doodle. As a fraction (0.1101011. ) it is 0.8392137714451.

Value of x such that x =cos( x ), using radians as the unit of angle. You can find the value with a scientific calculator just by putting in any reasonably close number and hitting the cosine key over and over again. Here are a few more digits: 0.7390851332151606416553120876738734040134117589007574649656. 26

This is 3 - √ 5 , and is related to a sequence of Grafting numbers found by Matt Parker. With more precision, it is: 0.76393­20225­00210­30359­08263­31268­72376­45593­81640­38847.

Take an odd number of digits after the decimal point, add 1, and you get a Grafting number. For example, 76393+1 = 76394. The sequence of numbers derived this way starts: 8, 764, 76394, 7639321, 763932023, 76393202251, 7639320225003, .

A fiendishly engaging approximation to the answer to the "infinite resistor network" problem in xkcd 356, which introduced the world to the sport of "nerd sniping". See ries and 0.773239. .

The answer to a fiendishly engaging "infinite resistor network" problem in xkcd 356, which introduced the world to the sport of "nerd sniping" 90 . See also 0.636619. and 0.772453. .

This number, on a early calculator with 7-segment display, says "hello" when seen upside-down:     &rarr     This is INTEGRAL0..1 x x d x , which is curiously equal to - SIGMAi..inf (- n ) - n , which was proven by Bernoulli. With more digits, it is 0.78343051071213440705926438652697546940768199014. It shares (with 1.291285. the nickname "sophomore's dream".

This is 0.1101011011010110101101101011011010110101101101011010110110. in binary, and is the slightly different version of the Rabbit constant generated by a Turing machine Google Doodle from June 2012. More digits: 0.8392137714451652585671495977783023880500088230714420678280105786051.

Decimal value of the "regular paperfolding sequence" 1 1 0 1 100 1 1100100 1 110110001100100 1 1101100111001000110110001100100 . converted to a binary fraction. This sequence of 1's and 0's gives the left and right turns as one walks along a dragon curve. It is the sum of 8 2 k /(2 2 k +2 -1) for all k &ge0, a series sum that gives twice as many digits with each additional term.

The minimum value of the Gamma function with positive real arguments. The Gamma function is the continuous analogue of the factorial function. This is Gamma(1.461632144968. ). (For more digits of both, see OEIS sequences A30171 and A30169.)

This is 1/2 of the square root of π. It is Gamma (3/2), and is sometimes also called (1/2)!, the factorial of 1/2.

This is Gamma (5/4), or "the factorial of 1/4". While some Gamma function values, like 0.886226. and 1.329340. , have simple formulas involving just π to a rational power, this one is a lot more complicated. It is π to the power of 3/4, divided by (√ 2 + 4 √ 2 ), times the sum of an infinite series for an elliptic function.

This is (4+4√ 2 )/(5+4√ 2 ), and is the best density achievable by packing equal-sized regular octagons in the plane. Notably, it is a bit smaller than 0.906899. , the density achievable with circles.

This is π/12, the density achievable by packing equal-sized circles in a plane. See also 0.906163. .

Catalan's constant, which can be defined by:

G = 1 - 1/3 2 + 1/5 2 - 1/7 2 + 1/9 2 - .

If you have a 2 n × 2 n checkerboard and a supply of 2 n 2 dominoes that are just large enough to cover two squares of the checkerboard, how many ways are there to cover the whole board with the dominoes? For large n , the answer is closely approximated by

This is the cube root of ( 5 √ 27 - 5 √ 2 ). Bill Gosper discovered the following identity, which is remarkable because the left side only has powers of 2 and 3, but the right side has a power of 5 in the denominator 108 :

( 5 √ 27 - 5 √ 2 ) (1/3) = ( 5 √ 8 5 √ 9 + 5 √ 4 - 5 √ 2 5 √ 27 + 5 √ 3 ) / 3 √ 25

(3 (3/5) -2 (1/5) ) (1/3) = (- 2 (1/5) 3 (3/5) + 2 (3/5) 3 (2/5) + 3 (1/5) + 2 (2/5) ) / 5 (2/3)

This page was written in the "embarrassingly readable" markup language RHTF, and some sections were last updated on 2020 Mar 26. s.11

## NCERT Solutions for Class 6 Math Chapter 2 - Whole Numbers

Write the three whole numbers occurring just before 10001.

3 whole numbers just before 10001 are

#### Question 3:

Which is the smallest whole number?

The smallest whole number is 0.

#### Question 4:

How many whole numbers are there between 32 and 53?

Whole numbers between 32 and 53 = 20 (53 &minus 32 &minus 1 = 20)

(33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52)

#### Question 7:

In each of the following pairs of numbers, state which whole number is on the left of the other number on the number line. Also write them with the appropriate sign (>, <) between them.

(c) 98765, 56789 (d) 9830415, 10023001

503 is on the left side of 530 on the number line.

307 is on the left side of 370 on the number line.

56789 is on the left side of 98765 on the number line.

Since 98, 30, 415 < 1, 00, 23, 001,

98,30,415 is on the left side of 1,00,23,001 on the number line.

#### Question 8:

Which of the following statements are true (T) and which are false (F)?

(a) Zero is the smallest natural number.

(b) 400 is the predecessor of 399.

(c) Zero is the smallest whole number.

(d) 600 is the successor of 599.

(e) All natural numbers are whole numbers.

(f) All whole numbers are natural numbers.

(g) The predecessor of a two digit number is never a single digit number.

(h) 1 is the smallest whole number.

(i) The natural number 1 has no predecessor.

(j) The whole number 1 has no predecessor.

(k) The whole number 13 lies between 11 and 12.

(l) The whole number 0 has no predecessor.

(m) The successor of a two digit number is always a two digit number.

(a) False, 0 is not a natural number.

(b) False, as predecessor of 399 is 398 (399 &minus 1 = 398).

(f) False, as 0 is a whole number but it is not a natural number.

(g) False, as predecessor of 10 is 9.

(h) False, 0 is the smallest whole number.

(i) True, as 0 is the predecessor of 1 but it is not a natural number.

(j) False, as 0 is the predecessor of 1 and it is a whole number.

(k) False, 13 does not lie in between 11 and 12.

(l) True, predecessor of 0 is &minus1, which is not a whole number.

(m) False, as successor of 99 is 100.

#### Question 1:

Find the sum by suitable rearrangement:

(a) 837 + 208 + 363 (b) 1962 + 453 + 1538 + 647

(a) 837 + 208 + 363 = (837 + 363) + 208

(b) 1962 + 453 + 1538 + 647 = (1962 + 1538) + (453 + 647)

#### Question 2:

Find the product by suitable rearrangement:

(c) 8 × 291 × 125 (d) 625 × 279 × 16

(d) 625 × 279 × 16 = 625 × 16 × 279

(e) 285 × 5 × 60 = 285 × 300 = 85500

#### Question 3:

Find the value of the following:

(a) 297 × 17 + 297 × 3 (b) 54279 × 92 + 8 × 54279

(c) 81265 × 169 &minus 81265 × 69 (d) 3845 × 5 × 782 + 769 × 25 × 218

(a) 297 × 17 + 297 × 3 = 297 × (17 + 3)

(b) 54279 × 92 + 8 × 54279 = 54279 × 92 + 54279 × 8

(c) 81265 × 169 &minus 81265 × 69 = 81265 × (169 &minus 69)

(d) 3845 × 5 × 782 + 769 × 25 × 218

##### Video Solution for whole numbers (Page: 40 , Q.No.: 3)

NCERT Solution for Class 6 math - whole numbers 40 , Question 3

#### Question 4:

Find the product using suitable properties.

= 738 × 100 + 738 × 3 (Distributive property)

= 854 × 100 + 854 × 2 (Distributive property)

(c) 258 × 1008 = 258 × (1000 + 8)

= 258 × 1000 + 258 × 8 (Distributive property)

(d) 1005 × 168 = (1000 + 5) × 168

= 1000 × 168 + 5 × 168 (Distributive property)

#### Question 5:

A taxi driver filled his car petrol tank with 40 litres of petrol on Monday. The next day, he filled the tank with 50 litres of petrol. If the petrol costs Rs 44 per litre, how much did he spend in all on petrol?

Quantity of petrol filled on Monday = 40 l

Quantity of petrol filled on Tuesday = 50 l

Total quantity filled = (40 + 50) l

Cost of petrol (per l) = Rs 44

Total money spent = 44 × (40 + 50)

##### Video Solution for whole numbers (Page: 40 , Q.No.: 5)

NCERT Solution for Class 6 math - whole numbers 40 , Question 5

#### Question 6:

A vendor supplies 32 litres of milk to a hotel in the morning and 68 litres of milk in the evening. If the milk costs Rs 15 per litre, how much money is due to the vendor per day?

Quantity of milk supplied in the morning = 32 l

Quantity of milk supplied in the evening = 68 l

Total of milk per litre = (32 + 68) l

Cost of milk per litre = Rs 15

Total cost per day = 15 × (32 + 68)

##### Video Solution for whole numbers (Page: 41 , Q.No.: 6)

NCERT Solution for Class 6 math - whole numbers 41 , Question 6

#### Question 7:

(i) 425 × 136 = 425 × (6 + 30 + 100)

(a) Commutativity under multiplication

(iii) 80 + 2005 + 20 = 80 + 20 + 2005

(c) Distributivity of multiplication over addition

(i) 425 × 136 = 425 × (6 + 30 + 100) [Distributivity of multiplication over addition]

(ii) 2 × 49 × 50 = 2 × 50 × 49 [Commutativity under multiplication]

(iii) 80 + 2005 + 20 = 80 + 20 + 2005 [Commutativity under addition]

#### Question 1:

Which of the following will not represent zero?

It does not represent zero.

#### Question 2:

If the product of two whole numbers is zero, can we say that one or both of them will be zero? Justify through examples.

If the product of 2 whole numbers is zero, then one of them is definitely zero.

For example, 0 × 2 = 0 and 17 × 0 = 0

If the product of 2 whole numbers is zero, then both of them may be zero.

(Since numbers to be multiplied are not equal to zero, the result of the product will also be non-zero.)

#### Question 3:

If the product of two whole numbers is 1, can we say that one of both of them will be 1? Justify through examples.

If the product of 2 numbers is 1, then both the numbers have to be equal to 1.

Clearly, the product of two whole numbers will be 1 in the situation when both numbers to be multiplied are 1.

#### Question 4:

Find using distributive property:

(b) 5437 × 1001 = 5437 × (1000 + 1)

(d) 4275 × 125 = (4000 + 200 + 100 &minus 25) × 125

= 4000 × 125 + 200 × 125 + 100 × 125 &minus 25 × 125

= 500000 + 25000 + 12500 &minus 3125

#### Question 5:

1 × 8 + 1 = 9 1234 × 8 + 4 = 9876

12 × 8 + 2 = 98 12345 × 8 + 5 = 98765

Write the next two steps. Can you say how the pattern works?

(Hint: 12345 = 11111 + 1111 + 111 + 11 + 1).

123456 × 8 + 6 = 987648 + 6 = 987654

1234567 × 8 + 7 = 9876536 + 7 = 9876543

As 123456 = 111111 + 11111 + 1111 + 111 + 11 + 1,

123456 × 8 = (111111 + 11111 + 1111 + 111 + 11 + 1) × 8

= 111111 × 8 + 11111 × 8 + 1111 × 8 + 111 × 8 + 11 × 8 + 1 × 8

## 1.4: Properties of Whole Numbers - Mathematics

Properties and Laws of Whole Numbers

· Simplify by using the addition property of 0.

· Simplify by using the multiplication property of 1.

· Identify and use the commutative law of addition.

· Identify and use the commutative law of multiplication.

· Identify and use the associative law of addition.

· Identify and use the associative law of multiplication.

Mathematics often involves simplifying numerical expressions. When doing so, you can use laws and properties that apply to particular operations. The multiplication property of 1 states that any number multiplied by 1 equals the same number, and the addition property of zero states that any number added to zero is the same number.

Two important laws are the commutative laws, which state that the order in which you add two numbers or multiply two numbers does not affect the answer. You can remember this because if you commute to work you go the same distance driving to work and driving home as you do driving home and driving to work. You can move numbers around in addition and multiplication expressions because the order in these expressions does not matter.

You will also learn how to simplify addition and multiplication expressions using the associative laws. As with the commutative laws, there are associative laws for addition and multiplication. Just like people may associate with people in different groups, a number may associate with other numbers in one group or another. The associative laws allow you to place numbers in different groups using parentheses.

Addition and Multiplication Properties of 0 and 1

The addition property of 0 states that for any number being added to 0, the sum equals that number. Remember that you do not end up with zero as an answer – that only happens when you multiply. Your answer is simply the same as your original number.

## Whole Numbers Class 6 Notes Maths Chapter 2

Natural numbers: The counting numbers 1,2, 3,4, are called natural numbers.

Predecessor: If we subtract 1 from a natural number, what we get is its predecessor. For example, the predecessor of 10 is 10 – 1 = 9.

Successor: If we add 1 to a natural number, what we get is its successor. For example, the successor of 9 is 9 + 1 = 10.

The natural number 1 has no predecessor in natural numbers.

There is no largest natural number

If we add the number 0 to the collection of whole numbers. Thus, the numbers 0, 1, 2, 3,… form the collection of whole numbers, natural numbers, what we get is the collection

We regard 0 as the predecessor of 1 in the collection of whole numbers.

Every whole number has a successor.

Every whole number except zero has a predecessor.

All the natural numbers are whole numbers but all the whole numbers are not a natural number. [0 is a whole number but not a natural number]

The Number Line
Draw a line. Mark a point on it and label it 0. Mark a second point to the right of 0 at the certain proper distance and label it 1. Then, the distance between the points labeled as 0 and 1 is called the unit distance. Now, mark another point on this line to the right of 1 at a unit distance from 1 and label it 2. Proceeding in this manner, we may find consecutive points and label them as 3, 4, 5,… in order. Thus, we can go to any whole number. This line is called the number line for whole numbers.

Addition on the number line: Let us add 2 and 3. We start from 2 on the number line and make 3 jumps to the right by unit distance each. We reach 5. So, 2 + 3 = 5.

Subtraction on the number line: Let us find 5-3. We start from 5 on the number line and make 3 jumps to the left by unit distance each. We reach 2. So, 5 – 3 = 2.

Multiplication on the number line: Let us find 2ࡩ. We start from 0 on the number line and move 2 units to the right at a time. We make 3 such moves. We reach 6. So. 2 × 3 = 6.

Properties of Whole Numbers
The result of the addition of two whole numbers is always a whole number. We say that the collection of whole numbers is closed under addition.

The result of the multiplication of two whole numbers is always a whole number. We say that the collection of whole numbers is closed p under multiplication.

The result of the subtraction of two whole numbers is not always a whole number. For example: 5 – 2 = 3 is a whole number but 2 – 4 = -2 is not a whole number. We say that the collection of whole numbers is not closed under subtraction.

The result of the division of two whole numbers is not always a whole number. For example: 6 ÷ 2 = 3 is a whole number but 2 ÷ 5 = (frac < 2 >< 5 >) is not a whole number. We say that the collection of whole numbers is not closed under division.

Division of a whole number by 0 is not defined.

We can add two whole numbers in any order. For example: 1 + 2 = 2 + 1 = 3. We say that addition is commutative for the collection of whole numbers.

We can multiply two whole numbers in any order.
For example: 2 × 3 = 3 × 2 = 6. We say that multiplication is commutative for the collection of whole numbers.

Addition is associative for whole numbers. For example: 1 + (2 + 3) = 1 + 5 = 6
(1 + 2) + 3 = 3 + 3 = 6 So, 1 + (2 + 3) = (1 + 2) + 3.

Multiplication is associative for whole numbers.
For example: 2 × (3 × 5) = 2 × 15 = 30 (2 × 3) × 5 = 6 × 5 = 30 So, 2 × (3 × 5) = (2 × 3) × 5.

Multiplication is distributive over addition for whole numbers. For example:
3 × (4 + 5) = 3 × 9 = 27
3 × 4 + 3 × 5 = 12 + 15 = 27
So, 3 × (4 + 5) = 3 × 4 + 3 × 5.
This is known as distributivity of multiplication over addition.
Note: The properties of commutativity, associativity, and distributivity of whole numbers are useful in simplifying calculations and we use them without being aware of them.

The result of the addition of zero to any whole number is the same whole number. We say that zero is the identity for the addition of whole numbers or additive identity for whole numbers.

The whole number zero has a special role in multiplication too. Any number, when multiplied by zero, becomes zero.

The result of the multiplication of 1 to any whole number is the same whole number. We say that 1 is the identity for multiplication of whole numbers or multiplicative identity for whole numbers.

Patterns in Whole Numbers
Some numbers can be arranged in elementary shapes a line, a rectangle, a square and a triangle only made up of dots.
Every number can be arranged as a line.

Some numbers like 6 can be arranged as a rectangle. Note that the number of rows should be smaller than the number of columns. Also, the rectangle should have more than 1 row.

Some numbers like 4, 9 can be arranged as a square. Note that every square number is also a rectangular number.

Some numbers like 3, 6 can be arranged as a triangle. Note that the triangle should be right-angled and its two sides must be equal. The number of dots in the rows starting from the bottom row should be like 4, 3,2, 1. The top row should always have 1 dot.

The patterns with numbers are not only interesting but also useful especially for mental calculations and help us understand the properties of numbers better.

## Adding It Up: Helping Children Learn Mathematics (2001)

Whole numbers are the easiest numbers to understand and use. As we described in the previous chapter, most children learn to count at a young age and understand many of the principles of number on which counting is based. Even if children begin school with an unusually limited facility with number, intensive instructional activities can be designed to help them reach similar levels as their peers. 1 Children&rsquos facility with counting provides a basis for them to solve simple addition, subtraction, multiplication, and division problems with whole numbers. Although there still is much for them to work out during the first few years of school, children begin with substantial knowledge on which they can build.

In this chapter, we examine the development of proficiency with whole numbers. We show that students move from methods of solving numerical problems that are intuitive, concrete, and based on modeling the problem situation directly to methods that are more problem independent, mathematically sophisticated, and reliant on standard symbolic notation. Some form of this progression is seen in each operation for both single-digit and multidigit numbers.

We focus on computation with whole numbers because learning to compute can provide young children the opportunity to work through many number concepts and to integrate the five strands of mathematical proficiency. This learning can provide the foundation for their later mathematical development. Computation with whole numbers occupies much of the curriculum in the early grades, and appropriate learning experiences in these grades improve children&rsquos chances for later success.

Whole number computation also provides an instructive example of how routine-appearing procedural skills can be intertwined with the other strands of proficiency to increase the fluency with which the skills are used. For years, learning to compute has been viewed as a matter of following the teacher&rsquos directions and practicing until speedy execution is achieved. Changes in career demands and the tasks of daily life, as well as the availability of new computing tools, mean that more is now demanded from the study of computation. More than just a means to produce answers, computation is increasingly seen as a window on the deep structure of the number system. Fortunately, research is demonstrating that both skilled performance and conceptual understanding are generated by the same kinds of activities. No tradeoffs are needed. As we detail below, the activities that provide this powerful result are those that integrate the strands of proficiency.

### Operations with Single-Digit Whole Numbers

As students begin school, much of their number activity is designed to help them become proficient with single-digit arithmetic. By single-digit arithmetic, we mean the sums and products of single-digit numbers and their companion differences and quotients (e.g., 5+7=12, 12&ndash5=7, 12&ndash7=5 and 5×7=35, 35÷5=7, 35÷7=5). For most of a century, learning single-digit arithmetic has been characterized in the United States as &ldquolearning basic facts,&rdquo and the emphasis has been on memorizing those facts. We use the term basic number combinations to emphasize that the knowledge is relational and need not be memorized mechanically. Adults and &ldquoexpert&rdquo children use a variety of strategies, including automatic or semiautomatic rules and reasoning processes to efficiently produce the basic number combinations. 2 Relational knowledge, such as knowledge of commutativity, not only promotes learning the basic number combinations but also may underlie or affect the mental representation of this basic knowledge. 3

The domain of early number, including children&rsquos initial learning of single-digit arithmetic, is undoubtedly the most thoroughly investigated area of school mathematics. A large body of research now exists about how children in many countries actually learn single-digit operations with whole numbers. Although some educators once believed that children memorize their &ldquobasic facts&rdquo as conditioned responses, research shows that children do not move from knowing nothing about the sums and differences of numbers to having the basic number combinations memorized. Instead, they move through a series of progressively more advanced and abstract methods for working out the answers

to simple arithmetic problems. Furthermore, as children get older, they use the procedures more and more efficiently. 4 Recent evidence indicates children can use such procedures quite quickly. 5 Not all children follow the same path, but all children develop some intermediate and temporary procedures.

Most children continue to use those procedures occasionally and for some computations. Recall eventually becomes the predominant method for some children, but current research methods cannot adequately distinguish between answers produced by recall and those generated by fast (nonrecall) procedures. This chapter describes the complex processes by which children learn to compute with whole numbers. Because the research on whole numbers reveals how much can be understood about children&rsquos mathematical development through sustained and interdisciplinary inquiry, we give more details in this chapter than in subsequent chapters.

#### Word Problems: A Meaningful Context

One of the most meaningful contexts in which young children begin to develop proficiency with whole numbers is provided by so-called word problems. This assertion probably comes as a surprise to many, especially mathematics teachers in middle and secondary school whose students have special difficulties with such problems. But extensive research shows that if children can count, they can begin to use their counting skills to solve simple word problems. Furthermore, they can advance those counting skills as they solve more problems. 6 In fact, it is in solving word problems that young children have opportunities to display their most advanced levels of counting performance and to build a repertoire of procedures for computation.

Most children entering school can count to solve word problems that involve adding, subtracting, multiplying, and dividing. 7 Their performance increases if the problems are phrased simply, use small numbers, and are accompanied by physical counters for the children to use. The exact procedures children are likely to use have been well documented. Consider the following problems:

Sally had 6 toy cars. She gave 4 to Bill. How many did she have left?

Sally had 4 toy cars. How many more does she need to have 6?

Most young children solve the first problem by counting a set of 6, removing 4, and counting the remaining cars to find the answer. In contrast,

they solve the second problem by counting a set of 4, adding in more as they count &ldquofive, six,&rdquo and then counting those added in to find the answer.

Children solve these problems by &ldquoacting out&rdquo the situation&mdashthat is, by modeling it. They invent a procedure that mirrors the actions or relationships described in the problem. This simple but powerful approach keeps procedural fluency closely connected to conceptual understanding and strategic competence. Children initially solve only those problems that they understand, that they can represent or model using physical objects, and that involve numbers within their counting range. Although this approach limits the kinds of problems with which children are successful, it also enables them to solve a remarkable range of problems, including those that involve multiplying and dividing.

Since children intuitively solve word problems by modeling the actions and relations described in them, it is important to distinguish among the different types of problems that can be represented by adding or subtracting, and among those represented by multiplying or dividing. One useful way of classifying problems is to heed the children&rsquos approach and examine the actions and relations described. This examination produces a taxonomy of problem types distinguished by the solution method children use and provides a framework to explain the relative difficulty of problems.

Four basic classes of addition and subtraction problems can be identified: problems involving (a) joining, (b) separating, (c) part-part-whole relations, and (d) comparison relations. Problems within a class involve the same type of action or relation, but within each class several distinct types of problems can be identified depending on which quantity is the unknown (see Table 6&ndash1). Students&rsquo procedures for solving the entire array of addition and subtraction problems and the relative difficulty of the problems have been well documented. 8

For multiplication and division, the simplest kinds of problems are grouping situations that involve three components: the number of sets, the number in each set, and the total number. For example:

Jose made 4 piles of marbles with 3 marbles in each pile. How many marbles did Jose have?

In this problem, the number and size of the sets is known and the total is unknown. There are two types of corresponding division situations depending on whether one must find the number of sets or the number in each set. For example:

Connie had 5 marbles. Juan gave her 8 more marbles. How many marbles does Connie have altogether?

Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?

Connie had some marbles. Juan gave her 5 more. Now she has 13 marbles. How many marbles did Connie have to start with?

Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left?

Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan?

Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with?

Connie has 5 red marbles and 8 blue marbles. How many marbles does she have altogether?

Connie has 13 marbles: 5 are red and the rest are blue. How many blue marbles does Connie have?

Connie has 13 marbles. Juan has 5 marbles. How many more marbles does Connie have than Juan?

Juan has 5 marbles. Connie has 8 more than Juan. How many marbles does Connie have?

Connie has 13 marbles. She has 5 more marbles than Juan. How many marbles does Juan have?

SOURCE: Carpenter, Fennema, Franke, Levi, and Empson, 1999, p. 12. Used by permission of Heinemann. All rights reserved.

Jose has 12 marbles and puts them into piles of 3. How many piles does he have?

Jose has 12 marbles and divides them equally into 3 piles. How many marbles are in each pile?

Additional types of multiplication and division problems are introduced later in the curriculum. These include rate problems, multiplicative comparison problems, array and area problems, and Cartesian products. 9

As with addition and subtraction problems, children initially solve multiplication and division problems by modeling directly the action and relations in the problems. 10 For the above multiplication problem with marbles, they form four piles with three in each and count the total to find the answer. For the first division problem, they make groups of the specified size of three and count the number of groups to find the answer. For the other problem, they make the three groups by dealing out (as in cards) and count the number in one of the groups. Although adults may recognize both problems as 12 divided by 3, children initially think of them in terms of the actions or relations portrayed. Over time, these direct modeling procedures are replaced by more efficient methods based on counting, repeated adding or subtracting, or deriving an answer from a known number combination. 11

The observation that children use different methods to solve problems that describe different situations has important implications. On the one hand, directly modeling the action in the problem is a highly sensible approach. On the other hand, as numbers in problems get larger, it becomes inefficient to carry out direct modeling procedures that involve counting all of the objects.

Children&rsquos proficiency gradually develops in two significant directions. One is from having a different solution method for each type of problem to developing a single general method that can be used for classes of problems with a similar mathematical structure. Another direction is toward more efficient calculation procedures. Direct-modeling procedures evolve into the more advanced counting procedures described in the next section. For word problems, these procedures are essentially abstractions of direct modeling that continue to reflect the actions and relations in the problems.

The method children might use to solve a class of problems is not necessarily the method traditionally taught. For example, many children come to solve the &ldquosubtraction&rdquo problems described above by counting, adding up, or thinking of a related addition combination because any of these methods is easier and more accurate than counting backwards. The method traditionally presented in textbooks, however, is to solve both of these problems by

subtracting, which moves students toward the more difficult and error-prone procedure of counting down. Ultimately, most children begin to use recall or a rapid mental procedure to solve these problems, and they come to recognize that the same general method can be used to solve a variety of problems.

Children come to understand the meaning of addition in the context of word problems. As we noted in the previous section, children move from counting to more general methods to solve different classes of problems. As they do, they also develop greater fluency with each specific method. We call these specific counting methods procedures. Although educators have long recognized that children use a variety of procedures to solve single-digit addition problems, 12 substantial research from all over the world now indicates that children move through a progression of different procedures to find the sum of single-digit numbers. 13

This progression is depicted in Box 6&ndash1. First, children count out objects for the first addend, count out objects for the second addend, and count all of the objects (count all). This general counting-all procedure then becomes abbreviated, internalized, and abstracted as children become more experienced with it. Next, they notice that they do not have to count the objects for the first addend but can start with the number in the first or the larger addend and count on the objects in the other addend (count on). As children count

Box 6&ndash1 Learning Progression for Single-Digit Addition

on with objects, they begin to use the counting words themselves as countable objects and keep track of how many words have been counted on by using fingers or auditory patterns. The counting list has become a representational tool. With time, children recompose numbers into other numbers (4 is recomposed into 3+1) and use thinking strategies in which they turn an addition combination they do not know into one they do know (3+4 becomes 3+3+1). In the United States, these strategies for derived number combinations often use a so-called double (2+2, 3+3, etc.). These doubles are learned very quickly.

As Box 6&ndash1 shows, throughout this learning progression, specific sums move into the category of being rapidly recalled rather than solved in one of the other ways described above. Children vary in the sums they first recall readily, though doubles, adding one (the sum is the next counting word), and small totals are the most readily recalled. Several procedures for single-digit addition typically coexist for several years they are used for different numbers and in different problem situations. Experience with figuring out the answer to addition problems provides the basis both for understanding what it means to say &ldquo5+3=8&rdquo and for eventually recalling that sum without the use of any conscious strategy.

Children in many countries often follow this progression of procedures, a natural progression of embedding and abbreviating. Some of these procedures can be taught, which accelerates their use, 14 although direct teaching of these strategies must be done conceptually rather than simply by using imitation and repetition. 15 In some countries, children learn a general procedure known as &ldquomake a 10&rdquo (see Box 6&ndash2). 16 In this procedure the solver makes a 10 out of one addend by taking a number from the other addend. Educators in some countries that use this approach believe this first instance of regrouping by making a 10 provides a crucial foundation for later multidigit arithmetic. In some Asian countries this procedure is presumably facilitated by the number words. 17 It has also been taught in some European countries in which the number names are more similar to those of English, suggesting that the procedure can be used with a variety of number-naming systems. The procedure is now beginning to appear in U.S. textbooks, 18 although so little space may be devoted to it that some children may not have adequate time and opportunity to understand and learn it well.

There is notable variation in the procedures children use to solve simple addition problems. 19 Confronted with that variation, teachers can take various steps to support children&rsquos movement toward more advanced procedures. One technique is to talk about slightly more advanced procedures and why

Box 6&ndash2 Make a Ten: B+6=?

they work. 20 The teacher can stimulate class discussion about the procedures that various students are using. Students can be given opportunities to present their procedures and discuss them. Others can then be encouraged to try the procedure. Drawings or concrete materials can be used to reveal how the procedures work. The advantages and disadvantages of different procedures can also be examined. For a particular procedure, problems can be created for which it might work well or for which it is inefficient.

Other techniques that encourage students to use more efficient procedures are using large numbers in problems so that inefficient counting procedures cannot easily be used and hiding one of the sets to stimulate a new way of thinking about the problem. Intervention studies indicate that teaching counting-on procedures in a conceptual way makes all single-digit sums accessible to U.S. first graders, including children who are learning disabled and those who do not speak English as their first language. 21 Providing support for children to improve their own procedures does not mean, however, that every child is taught to use all the procedures that other children develop. Nor does it mean that the teacher needs to provide every child in a class with

support and justification for different procedures. Rather, the research provides evidence that, at any one time, most children use a small number of procedures and that teachers can learn to identify them and help children learn procedures that are conceptually more efficient (such as counting on from the larger addend rather than counting all). 22

Mathematical proficiency with respect to single-digit addition encompasses not only the fluent performance of the operation but also conceptual understanding and the ability to identify and accurately represent situations in which addition is required. Providing word problems as contexts for adding and discussing the advantages and disadvantages of different addition procedures are ways of facilitating students&rsquo adaptive reasoning and improving their understanding of addition processes.

#### Single-Digit Subtraction

Subtraction follows a progression that generally parallels that for addition (see Box 6&ndash3). Some U.S. children also invent counting-down methods that model the taking away of numbers by counting back from the total. But counting down and counting backward are difficult for many children. 23

Box 6&ndash3 Learning Progression for Single-Digit Subtraction

A considerable number of children invent counting-up procedures for situations in which an unknown quantity is added to a known quantity. 24 Many of these children later count up in taking-away subtraction situations (13&ndash8=? becomes 8+?=13). When counting up is not introduced, many children may not invent it until the second or third grade, if at all. Intervention studies with U.S. first graders that helped them see subtraction situations as taking away the first x objects enabled them to learn and understand counting-up-to procedures for subtraction. Their subtraction accuracy became as high as that for addition. 25

Experiences that focus on part-part-whole relations have also been shown to help students develop more efficient thinking strategies, especially for subtraction. 26 Students examine a join or separate situation and identify which number represents the whole quantity and which numbers represent the parts. These experiences help students see how addition and subtraction are related and help them recognize when to add and when to subtract. For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplishments in arithmetic. 27

For students in grades K to 2, learning to see the part-whole relations in addition and subtraction situations is one of their most important accomplish-ments in arithmetic.

Examining the relationships between addition and subtraction and seeing subtraction as involving a known and an unknown addend are examples of adaptive reasoning. By providing experiences for young students to develop adaptive reasoning in addition and subtraction situations, teachers are also anticipating algebra as students begin to appreciate the inverse relationships between the two operations. 28

#### Single-Digit Multiplication

Much less research is available on single-digit multiplication and division than on single-digit addition and subtraction. U.S. children progress through a sequence of multiplication procedures that are somewhat similar to those for addition. 29 They make equal groups and count them all. They learn skip-count lists for different multipliers (e.g., they count 4, 8, 12, 16, 20,&hellipto multiply by four). They then count on and count down these lists using their fingers to keep track of different products. They invent thinking strategies in which they derive related products from products they know.

As with addition and subtraction, children invent many of the procedures they use for multiplication. They find patterns and use skip counting (e.g., multiplying 4×3 by counting &ldquo3, 6, 9, 12&rdquo). Finding and using patterns and other thinking strategies greatly simplifies the task of learning multiplication tables (see Box 6&ndash4 for some examples). 30 Moreover, finding and describing

Box 6&ndash4 Thinking Strategies for Single-Digit Multiplication

In single-digit arithmetic, there are 100 multiplication combinations that students must learn. Commutativity reduces that number by about half. Multiplication by 0 and by 1 may quickly be deduced from the meaning of multiplication. Multiplication by 2 consists of the &ldquodoubles&rdquo from addition. Single-digit multiplication by 9 is simplified by a pattern: in the product, the sum of the digits is 9. (For example, 9×7=63 and 6+3=9.) Multiplication by 5 may also be deduced through patterns or by first multiplying by 10 and then dividing by 2, since 5 is half of 10.

The remaining 15 multiplication combinations (and their commutative counterparts) may be computed by skip counting or by building on known combinations. For example, 3×6 must be 6 more than 2×6, which is 12. So 3×6 is 18. Similarly, 4×7 must be twice 2×7, which is 14. So 4×7 is 28. (Note that these strategies require proficiency with addition.) To compute multiples of 6, one can build on the multiples of 5. So, for example, 6×8 must be 8 more than 5×8, which is 40. So 6×8 is 48. If students are comfortable with such strategies for multiplication by 3, 4, and 6, only three multiplication combinations remain: 7×7, 7×8, and 8×8. These can be derived from known combinations in many creative ways.

patterns are a hallmark of mathematics. Thus, treating multiplication learning as pattern finding both simplifies the task and uses a core mathematical idea.

After children identify patterns, they still need much experience to produce skip-count lists and individual products rapidly. Little is known about how children acquire this fluency or what experiences might be of most help. A good deal of research remains to be done, in the United States and in other countries, to understand more about this process.

#### Single-Digit Division

Division arises from the two splitting situations described above. A collection is split into groups of a specified size or into a specified number of groups. Just as subtraction can be thought of using a part-part-whole relation, division can be thought of as splitting a number into two factors. Hence, divisions can also be approached as finding a missing factor in multiplication. For example, 72÷9=? can be thought of as 9÷?=72. But there is little

research concerning how best to introduce and use this relationship, or whether it is helpful to learn a division combination at the same time as the corresponding multiplication combination. Further, there is little research about how to help children learn and use easily all of the different symbols for division, such as 15÷3, and #### Practicing Single-Digit Calculations

Practicing single-digit calculations is essential for developing fluency with them. This practice can occur in many different contexts, including solving word problems. 31 Drill alone does not develop mastery of single-digit combinations. 32 Practice that follows substantial initial experiences that support understanding and emphasize &ldquothinking strategies&rdquo has been shown to improve student achievement with single-digit calculations. 33 This approach allows computation and understanding to develop together and facilitate each other. Explaining how procedures work and examining their benefits, as part of instruction, support retention and yield higher levels of performance. 34 In this way, computation practice remains integrated with the other strands of proficiency such as strategic competence and adaptive reasoning.

Practicing single-digit calculations is essential for developing fluency with them.

It is helpful for some practice to be targeted at recent learning. After students discuss a new procedure, they can benefit from practicing it. For example, if they have just discussed the make-a-10 procedure (see Box 6&ndash2), solving problems involving 8 or 9 in which the procedure can easily be used provides beneficial practice. It also is helpful for some practice to be cumulative, occurring well after initial learning and reviewing the more advanced procedures that have been learned.

Many U.S. students have had the experience of taking a timed test that might be a page of mixed addition, subtraction, multiplication, and division problems. This scattershot form of practice is, in our opinion, rarely the best use of practice time. Early in learning it can be discouraging for students who have learned only primitive, inefficient procedures. The experience can adversely affect students&rsquo disposition toward mathematics, especially if the tests are used to compare their performance. 35 If appropriately delayed, timed tests can benefit some students, but targeted forms of practice, with particular combinations that have yet to be mastered or on which efficient procedures can be used, are usually more effective. 36

#### Summary of Findings an Learning Single-Digit Arithmetic

For addition and subtraction, there is a well-documented progression of procedures used worldwide 37 by many children that stems from the sequential nature of the list of number words. This list is first used as a counting tool then it becomes a representational tool in which the number words themselves are the objects that are counted. 38 Counting becomes abbreviated and rapid, and students begin to develop procedures that take advantage of properties of arithmetic to simplify computation. During this progression, individual children use a range of different procedures on different problems and even on the same problem encountered at different times. 39 Even adults have been found to use a range of different procedures for simple addition problems. 40 Further, it takes an extended period of time before new and better strategies replace previously used strategies. 41 Learning-disabled children and others having difficulty with mathematics do not use procedures that differ from this progression. They are just slower than others in moving through it. 42

Instruction can help students progress. 43 Counting on is accessible to first graders it makes possible the rapid and accurate addition of all single-digit numbers. Single-digit subtraction is usually more difficult than addition for U.S. children. If children understand the relationship between addition and subtraction, perhaps by thinking of the problem in terms of part-part-whole, then they recognize that counting up can be used to solve subtraction problems. This recognition makes subtraction more accessible. 44

The procedures of counting on for addition and counting up for subtraction can be learned with relative ease. Multiplication and division are somewhat more difficult. Even adults might not have quick ways of reconstructing the answers to problems like 6×8=? or if they have forgotten the answers. Learning these combinations seems to require much specific pattern-based knowledge that needs to be orchestrated into accessible and rapid-enough products and quotients. As with addition and subtraction, children derive some multiplication and division combinations from others for example, they recall that 6×6=36 and use that combination to conclude that 6×7=42. Research into ways to support such pattern finding, along with the necessary follow-up thinking and practice, is needed if all U.S. children are to acquire higher levels of proficiency in single-digit arithmetic.

Acquiring proficiency with single-digit computations involves much more than rote memorization. This domain of number demonstrates how the different strands of proficiency contribute to each other. At this early point in

development, many of the linkages among strands result from children&rsquos natural inclination to make sense of things and to engage in actions that they understand. Children begin with conceptual understanding of number and the meanings of the operations. They develop increasingly sophisticated representations of the operations such as counting-on or counting-up procedures as they gain greater fluency. They also lean heavily on reasoning to use known answers such as doubles to generate unknown answers. Even in the early grades, students choose adaptively among different procedures and methods depending on the numbers involved or the context. 45 As long as the focus in the classroom is on sense making, they rarely make nonsensical errors, such as adding to find the answer when they should subtract. Proficiency comes from making progress within each strand and building connections among the strands. A productive disposition is generated by and supports this kind of learning because students recognize their competence at making sense of quantitative situations and solving arithmetic problems.

### Multidigit Whole Number Calculations

Step-by-step procedures for adding, subtracting, multiplying, or dividing numbers are called algorithms. For example, the first step in one algorithm for multiplying a three-digit number by a two-digit number is to write the three-digit number above the two-digit number and to begin by multiplying the one&rsquos digit in the top number by the one&rsquos digit in the bottom number (see Box 6&ndash5).

In the past, algorithms different from those taught today for addition, subtraction, multiplication, and division have been taught in U.S. schools. Also, algorithms different from those taught in the United States today are currently being taught in other countries. 46 Each algorithm has advantages

Box 6&ndash5 Beginning a multiplication algorithm and disadvantages. Therefore, it is important to think about which algorithms are taught and the reasons for teaching them.

Learning to use algorithms for computation with multidigit numbers is an important part of developing proficiency with numbers. Algorithms are procedures that can be executed in the same way to solve a variety of problems arising from different situations and involving different numbers. This feature has three important implications. First, it means that algorithms are useful tools&mdashdifferent procedures do not need to be invented for each problem. Second, algorithms illustrate a significant feature of mathematics: The structure of problems can be abstracted from their immediate context and compared to see whether different-looking problems can be solved in similar ways. Finally, the process of developing fluency with arithmetic algorithms in elementary school can contribute to progress in developing the other strands of proficiency if time is spent examining why algorithms work and comparing their advantages and disadvantages. Such analyses can boost conceptual understanding by revealing much about the structure of the number system itself and can facilitate understanding of place-value representations.

Research findings about learning algorithms for whole numbers can be summarized with seven important observations. First, the linkages among the strands of mathematical proficiency that are possible when children develop proficiency with single-digit arithmetic can be continued with multidigit arithmetic. For example, there can be a close connection between understanding and fluency. Conceptual knowledge that comes with understanding is important for the development of procedural fluency, while fluent procedural knowledge supports the development of further understanding and learning. When students fail to grasp the concepts that underlie procedures or cannot connect the concepts to the procedures, they frequently generate flawed procedures that result in systematic patterns of errors. 47 These so-called buggy algorithms are signs that the strands are not well connected. 48 When the initial computational procedures that students use to solve multidigit problems reflect their understanding of numbers, understanding and fluency develop together.

A second observation is that understanding and fluency are related. For multidigit addition and subtraction, given conventional instruction that emphasizes practicing procedures, a substantial percentage of children gain understanding of multidigit concepts before using a correct procedure, but another substantial minority do the opposite. 49 In contrast, instructional programs that emphasize understanding algorithms before using them have been shown to lead to increases in both conceptual and procedural knowledge. 50

So there is some evidence that understanding is the basis for developing procedural fluency. 51

A third observation is that proficiency with multidigit computation is more heavily influenced by instruction than single-digit computation is. Many features of multidigit procedures (e.g., the base-10 elements and how they are represented by place-value notation) are not part of children&rsquos everyday experience and need to be learned in the classroom. In fact, many students are likely to need help learning efficient forms of multidigit procedures. This means that students in different classrooms and receiving different instruction might follow different learning progressions use different procedures. 52 For single-digit addition and subtraction, the same learning progression occurs for many children in many countries regardless of the nature and extent of instruction. 53 But multidigit procedures, even those for addition and subtraction, depend much more on what is taught.

A fourth observation is that children can and do devise or invent algorithms for carrying out multidigit computations. 54 Opportunities to construct their own procedures provide students with opportunities to make connections between the strands of proficiency. Procedural fluency is built directly on their understanding. The invention itself is a kind of problem solving, and they must use reasoning to justify their invented procedure. Students who have invented their own correct procedures also approach mathematics with confidence rather than fear and hesitation. 55 Students invent many different computational procedures for solving problems with large numbers. For addition, they eventually develop a procedure that is consistent with the thinking that is used with standard algorithms. That thinking enables them to make sense of the algorithm as a record on paper of what they have already been thinking. For subtraction, many students can develop adding-up procedures and, if using concrete materials like base-10 blocks, can also develop ways of thinking that parallel algorithms usually taught today. 56 Some students need help to develop efficient algorithms, however, especially for multiplication and division. Consequently, for these students the process of learning algorithms involves listening to someone else explain an algorithm and trying it out, all the while trying to make sense of it. Research suggests that students are capable of listening to their peers and to the teacher and of making sense of an algorithm if it is explained and if the students have diagrams or concrete materials that support their understanding of the quantities involved. 57

Fifth, research has shown that students can learn well from a variety of different instructional approaches, including those that use physical materials to represent hundreds, tens, and ones, those that emphasize special counting

activities (e.g., count by tens beginning with any number), and those that focus on developing mental computation methods. 58 Although the data do not point to a single preferred instructional approach, they do suggest that effective approaches share some key features: The multidigit procedures that students use are easily understood students are encouraged to use algorithms that they understand instructional supports (classroom discussions, physical materials, etc.) are available to focus students&rsquo attention on the base-10 structure of the number system and on how that structure is used in the algorithm and students are helped to progress to using reasonably efficient but still comprehensible algorithms. 59

Sixth, research on symbolic learning argues that, to be helpful, manipulatives or other physical models used in teaching must be represented by a learner both as the objects that they are and as symbols that stand for something else. 60 The physical characteristics of these materials can be initially distracting to children, and it takes time for them to develop mathematical meaning for any kind of physical model and to use it effectively. These findings suggest that sustained experience with any physical models that students are expected to use may be more effective than limited experience with a variety of different models. 61

In view of the attention given to the use of concrete models in U.S. school mathematics classes, we offer a special note regarding their effective use in multidigit arithmetic. Research indicates that students&rsquo experiences using physical models to represent hundreds, tens, and ones can be effective if the materials help them think about how to combine quantities and, eventually, how these processes connect with written procedures. The models, however, are not automatically meaningful for students the meaning must be constructed as they work with the materials. Given time to develop meaning for a model and connect it with the written procedure, students have shown high levels of performance using the written procedure and the ability to give good explanations for how they got their answers. 62 In order to support understanding, however, the physical models need to show tens to be collections of ten ones and to show hundreds to be simultaneously 10 tens and 100 ones. For example, base-10 blocks have that quality, but chips all of the same size but with different colors for hundreds, tens, and ones do not.

A seventh and final observation is that the English number words and the Hindu-Arabic base-10 place-value system for writing numbers complicate the teaching and learning of multidigit algorithms in much the same way, as discussed in Chapter 5, that they complicate the learning of early number concepts. 63 Closely related to the difficulties posed by the irregu-

larities with number words are difficulties posed by the complexity of the system for writing numbers. As we said in chapter 3, the base-10 place-value system is very efficient. It allows one to write very large numbers using only 10 symbols, the digits 0 through 9. The same digit has a different meaning depending on its place in the numeral. Although this system is familiar and seems obvious to adults, its intricacies are not so obvious to children. These intricacies are important because research has shown that it is difficult to develop procedural fluency with multidigit arithmetic without an understanding of the base-10 system. 64 If such understanding is missing, students make many different errors in multidigit computations. 65

This conclusion does not imply that students must master place value before they can begin computing with multidigit numbers. In fact, the evidence shows that students can develop an understanding of both the base-10 system and computation procedures when they have opportunities to explore how and why the procedures work. 66 That should not be surprising it simply confirms the thesis of this report and the claim we made near the beginning of this chapter. Proficiency develops as the strands connect and interact.

The six observations can be illustrated and supported by examining briefly each of the arithmetic operations. As is the case for single-digit operations, research provides a more complete picture for addition and subtraction than for multiplication and division.

The progression followed by students who construct their own procedures is similar in some ways to the progression that can be used to help students learn a standard algorithm with understanding. To illustrate the nature of these progressions, it is useful to examine some specific procedures in detail.

The episode in Box 6&ndash6 from a third-grade class illustrates both how physical materials can support the development of thinking strategies about multidigit algorithms and one type of procedure commonly invented by children. 67 The episode comes from a discussion of students&rsquo solutions to a word problem involving the sum 54+48.

The episode suggests that students&rsquo invented procedures can be constructed through progressive abstraction of their modeling strategies with blocks. First, the objects in the problem were represented directly with the blocks. Then, the quantity representing the first set was abstracted, and only the blocks representing the second set were counted. Finally, the counting words were themselves counted by keeping track of the counts on fingers.

## Arithmetic properties of integers

Below is a table of some of the properties of integers undergoing arithmetic operations. The properties in the table are dependent on a and b being integers.

Closure a + b is an integer a × b is an integer
Commutativity a + b = b + a a × b = b × a
Associativity a + (b + c) = (a + b) + c a × (b × c) = (a × b) × c
Existence of an identity a + 0 = a a × 1 = a
Existence of an inverse a + (-a) = 0 only -1 and 1 are invertible
Distributivity a × (b + c) = a × b + a × c

## Properties Of Integers

In these lessons and examples, we will learn about digits, integers, even and odd integers, operations on even and odd numbers, prime numbers and composite numbers. We will also learn the following properties of Integers: Commutative Property for Addition, Associative Property for Addition, Distributive Property, Identity Property for Addition, Identity Property for Multiplication, Inverse Property for Addition and Zero Property for Multiplication.

### Introduction To Integer

#### Digits

Digits are the first concept of integers. There are ten digits namely: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

In our number system, the position of the digits are important. For example, consider the number 3,027. This can be represented in a place value table as follows: (For the SAT, the units digit and the ones digit refer to the same digit in a number).

#### Integers

Integers are like whole numbers but they also include negative numbers, for example, –4, –3, –2, –1, 0, 1, 2, 3, 4, &hellip

Positive integers are all the whole numbers greater than zero, ie: 1, 2, 3, 4, 5, &hellip We say that its sign is positive.

Negative integers are the numbers less than zero, ie: –1, –2, –3, –4, –5, &hellip We say that its sign is negative.

Integers extend infinitely in both positive and negative directions. This can be represented on the number line.

Zero is an integer that is neither positive nor negative.

#### Consecutive Integers

Consecutive integers are integers that follow in sequence, each number being 1 more than the previous number, for example 22, 23, 24, 25, &hellip

Consecutive integers can be more generally represented by n, n +1, n + 2, n + 3, &hellip, where n is any integer.

#### Even And Odd Integers

Even integers are integers that can be divided evenly by 2, for example, –4, –2, 0, 2, 4, &hellip An even integer always ends in 0, 2, 4, 6, or 8.

Zero is considered an even integer.

Odd integers are integers that cannot be divided evenly by 2, for example, –5, –3, –1, 1, 3, 5, &hellip An odd integer always ends in 1, 3, 5, 7, or 9.

To tell whether an integer is even or odd, look at the digit in the ones place. That single digit will tell you whether the entire integer is odd or even. For example, the integer 3,255 is an odd integer because it ends in 5, an odd integer. Likewise, 702 is an even integer because it ends in 2.

The following table shows the operations with even and odd integers. #### Prime Numbers

A prime number is a positive integer that has exactly two factors, 1 and itself, for example 29 has exactly two factors which are 1 and 29. So 29 is a prime number.

On the other hand, 28 has six factors which are 1, 2, 4, 7, 14, and 28. So 28 is not a prime number. It is called a composite number. Some examples of prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, &hellip Since the number 1 has only one factor (namely 1 itself), it is not a prime number.

The number 2 is the only prime that is even. Other even numbers will have 2 have as a factor and so will not be a prime.

A number that is not prime is called a composite number .

#### Properties Of Integers

The following are some of the properties of integers. Scroll down the page for more examples and explanations of the different properties of integers. #### Operations With Even And Odd Numbers

Add two even numbers and the result is even.
Add two odd numbers and the result is even.
Add one even and one odd and the result is odd.
Multiply two even numbers and the result is even.
Multiply two odd numbers and the result is odd.
Multiply one even and one odd and the result is even.

#### How To Distinguish Prime Numbers?

A prime number is a number greater than 1, which is only divisible by 1 and itself.

#### More Properties Of Integers

How to identify properties of Integers?
A property is a math rule that is always true.
Commutative Property for Addition, Associative Property for Addition, Distributive Property, Identity Property for Addition, Identity Property for Multiplication, Inverse Property for Addition and Zero Property for Multiplication.

#### Properties Of Integers

Three properties of integers are explained. Additive Identity, Additive Inverse, Opposite of a negative is positive. Examples are provided.

1. Additive Identity: Adding 0 to any integer does not change the value of the integer.
2. Additive Inverse: Each integer has an opposing number (opposite sign). When you add a number and its additive inverse, you get 0.
3. The opposite of a negative is a positive. Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.