# 1.2: Operations with Real Numbers - Mathematics

Learning Objectives

• Review the properties of real numbers.
• Simplify expressions involving grouping symbols and exponents.
• Simplify using the correct order of operations.

## Working with Real Numbers

In this section, we continue to review the properties of real numbers and their operations. The result of adding real numbers is called the sum53 and the result of subtracting is called the difference54. Given any real numbers a, b, and c, we have the following properties of addition:

Additive Identity Property:55 a+0=0+a=a a+(−a)=(−a)+a=0 (a+b)+c=a+(b+c) a+b=b+a

It is important to note that addition is commutative and subtraction is not. In other words, the order in which we add does not matter and will yield the same result. However, this is not true of subtraction.

(5+10=10+5) (5−10≠10−5)

(15=15) (−5≠5)

We use these properties, along with the double-negative property for real numbers, to perform more involved sequential operations. To simplify things, make it a general rule to first replace all sequential operations with either addition or subtraction and then perform each operation in order from left to right.

Example (PageIndex{1}):

Simplify: (−10−(−10)+(−5)).

Solution

Replace the sequential operations and then perform them from left to right.

(−10−(−10)+(−5)=−10+10−5) (color{Cerulean}{Replace −(−) with addition (+)}).

(color{Cerulean}{Replace +(−) with subtraction (-).})

(=0−5)

(=−5)

(−5)

Adding or subtracting fractions requires a common denominator59. Assume the common denominator c is a nonzero integer and we have

(frac{a}{c}+frac{b}{c}=frac{a+b}{c}) and (frac{a}{c}−frac{b}{c}=frac{a−b}{c})

Example (PageIndex{2}):

Simplify: (frac{2}{9}−frac{1}{15}+frac{8}{45}).

Solution

First determine the least common multiple (LCM) of (9, 15, and 45). The least common multiple of all the denominators is called the least common denominator60 (LCD). We begin by listing the multiples of each given denominator:

({9,18,27,36,45,54,63,72,81,90,dots}) (color{Cerulean}{Multiples of 9})

({15,30,45,60,75,90,dots}) (color{Cerulean}{Multiples of 15})

({45,90,135dots}) (color{Cerulean}{Multiples of : 45})

Here we see that the LCM((9, 15, 45) = 45). Multiply the numerator and the denominator of each fraction by values that result in equivalent fractions with the determined common denominator.

(frac{2}{9}−frac{1}{15}+frac{8}{45}=frac{2}{9}⋅color{Cerulean}{frac{5}{5}})(−frac{1}{15}⋅color{Cerulean}{frac{3}{3}})(+frac{8}{45})

(=frac{10}{45}−frac{3}{45}+frac{8}{45})

Once we have equivalent fractions, with a common denominator, we can perform the operations on the numerators and write the result over the common denominator.

(=frac{10−3+8}{45})

(=frac{15}{45})

And then reduce if necessary,

(=frac{15color{Cerulean}{÷15}}{45color{Cerulean}{÷15}})

(=frac{1}{3})

(frac{1}{3})

Finding the LCM using lists of multiples, as described in the previous example, is often very cumbersome. For example, try making a list of multiples for (12) and (81). We can streamline the process of finding the LCM by using prime factors.

(12=2^{2}⋅3)

(81=3^{4})

The least common multiple is the product of each prime factor raised to the highest power. In this case,

(LCM(12,81)=2^{2}⋅3^{4}=324)

Often we will find the need to translate English sentences involving addition and subtraction to mathematical statements. Below are some common translations.

(n+2 color{Cerulean}{The: sum: of: a: number: and: 2.})

(2−n color{Cerulean}{The: difference: of: 2: and: a: number.})

(n−2 color{Cerulean}{Here: 2: is: subtracted: from: a: number.})

Example (PageIndex{3}):

What is (8) subtracted from the sum of (3) and (frac{1}{2})?

Solution

We know that subtraction is not commutative; therefore, we must take care to subtract in the correct order. First, add (3) and (frac{1}{2}) and then subtract (8) as follows: Perform the indicated operations.

((3+frac{1}{2})−8=(frac{3}{1}⋅color{Cerulean}{frac{2}{2}})(+frac{1}{2})−8)

(=(frac{6+1}{2})−8)

(=frac{7}{2}−frac{8}{1}⋅color{Cerulean}{frac{2}{2}})

(=frac{7−16}{2})

(=−frac{9}{2})

(−frac{9}{2})

The result of multiplying real numbers is called the product61 and the result of dividing is called the quotient62. Given any real numbers a, b, and c, we have the following properties of multiplication:

Zero Factor Property:63 a⋅0=0⋅a=0 a⋅1=1⋅a=a (a⋅b)⋅c=a⋅(b⋅c) a⋅b=b⋅a

It is important to note that multiplication is commutative and division is not. In other words, the order in which we multiply does not matter and will yield the same result. However, this is not true of division.

(5⋅10=10⋅5) (5÷10≠10÷5)

(50=50) (0.5≠2)

We will use these properties to perform sequential operations involving multiplication and division. Recall that the product of a positive number and a negative number is negative. Also, the product of two negative numbers is positive.

Example (PageIndex{4}):

Multiply: 5(−3)(−2)(−4).

Solution

Multiply two numbers at a time as follows: (−120)

Because multiplication is commutative, the order in which we multiply does not affect the final answer. However, when sequential operations involve multiplication and division, order does matter; hence we must work the operations from left to right to obtain a correct result.

Example (PageIndex{5}):

Simplify: 10÷(−2)(−5).

Solution

Perform the division first; otherwise the result will be incorrect. Notice that the order in which we multiply and divide does affect the result. Therefore, it is important to perform the operations of multiplication and division as they appear from left to right.

(25)

The product of two fractions is the fraction formed by the product of the numerators and the product of the denominators. In other words, to multiply fractions, multiply the numerators and multiply the denominators:

(frac{a}{b}⋅frac{c}{d}=frac{ac}{bd})

Example (PageIndex{6}):

Multiply (−frac{4}{5}⋅frac{25}{12}).

Solution

Multiply the numerators and multiply the denominators. Reduce by dividing out any common factors. (−frac{5}{3})

Two real numbers whose product is (1) are called reciprocals67. Therefore, (frac{a}{b}) and (frac{b}{a}) are reciprocals because (frac{a}{b}⋅frac{b}{a}=frac{ab}{ab}=1). For example,

(frac{2}{3}⋅frac{3}{2}=frac{6}{6}=1)

Because their product is (1, frac{2}{3}) and (frac{3}{2}) are reciprocals. Some other reciprocals are listed below:

(frac{5}{8}) and (frac{8}{5}) (7) and (frac{1}{7}) (−frac{4}{5}) and (−frac{5}{4})

This definition is important because dividing fractions requires that you multiply the dividend by the reciprocal of the divisor.

(frac{a}{b}÷color{Cerulean}{frac{c}{d}}) (=frac { frac { a } { b } } { frac { c } { d } } cdot color{OliveGreen}{frac { frac { d } { c } } { frac { d } { c } }}) (=frac { frac { a } { b } cdot frac { d } { c } } { 1 } = frac { a } { b } cdot color{Cerulean}{frac { d } { c }})

In general,

(frac { a } { b } div color{Cerulean}{frac { c } { d }}) (= frac { a } { b } cdot color{Cerulean}{frac { d } { c }}) (= frac { a d } { b c })

Example (PageIndex{7}):

Simplify: (frac{5}{4}÷frac{3}{5}⋅frac{1}{2}).

Solution

Perform the multiplication and division from left to right.

(frac{5}{4}÷color{Cerulean}{frac{3}{5}}) (⋅frac{1}{2}=frac{5}{4}⋅color{Cerulean}{frac{5}{3}}) (⋅frac{1}{2})

(=frac{5⋅5⋅1}{4⋅3⋅2})

(=frac{25}{24})

In algebra, it is often preferable to work with improper fractions. In this case, we leave the answer expressed as an improper fraction.

(frac{25}{24})

Exercise (PageIndex{1})

Simplify: (frac{1}{2}⋅frac{3}{4}÷frac{1}{8}).

## Grouping Symbols and Exponents

In a computation where more than one operation is involved, grouping symbols help tell us which operations to perform first. The grouping symbols68 commonly used in algebra are:

(( )) (color{Cerulean}{Parentheses})

([ ]) (color{Cerulean}{Brackets})

({ }) (color{Cerulean}{Braces})

(-) (color{Cerulean}{Fraction: bar})

All of the above grouping symbols, as well as absolute value, have the same order of precedence. Perform operations inside the innermost grouping symbol or absolute value first.

Example (PageIndex{8}):

Simplify: (2−(frac{4}{5}−frac{2}{15})).

Solution

Perform the operations within the parentheses first.

(2−(frac{4}{5}−frac{2}{15})= 2−(frac{4}{5}⋅color{Cerulean}{frac{3}{3}})(−frac{2}{15}))

(=2−(frac{12}{15}−frac{2}{15}))

(=2−(frac{10}{15}))

(=frac{2}{1}⋅color{Cerulean}{frac{3}{3}})(−frac{2}{3})

(=frac{6-2}{3})

(=frac{4}{3})

(frac{4}{3})

Example (PageIndex{9}):

Simplify: (frac { 5 - | 4 - ( - 3 ) | } { | - 3 | - ( 5 - 7 ) }).

Solution

The fraction bar groups the numerator and denominator. Hence, they should be simplified separately.

(−frac{2}{5})

If a number is repeated as a factor numerous times, then we can write the product in a more compact form using exponential notation69. For example,

(5⋅5⋅5⋅5=54)

The base70 is the factor and the positive integer exponent71 indicates the number of times the base is repeated as a factor. In the above example, the base is (5) and the exponent is (4). Exponents are sometimes indicated with the caret (^) symbol found on the keyboard, (5^4 = 5*5*5*5). In general, if a is the base that is repeated as a factor n times, then When the exponent is (2) we call the result a square72, and when the exponent is (3) we call the result a cube73. For example,

(5^{2}=5⋅5=25) (color{Cerulean}{"5: squared”})

(5^{3}=5⋅5⋅5=125) (color{Cerulean}{“5: cubed”})

If the exponent is greater than (3), then the notation (a^{n}) is read, “a raised to the nth power.” The base can be any real number,

((2.5)^{2}=(2.5)(2.5)=6.25)

((−frac{2}{3})^{3}=(−frac{2}{3})(−frac{2}{3})(−frac{2}{3})=−frac{8}{27})

((−2)^4=(−2)(−2)(−2)(−2)=16)

(−2^{4}=−1⋅2⋅2⋅2⋅2=−16)

Notice that the result of a negative base with an even exponent is positive. The result of a negative base with an odd exponent is negative. These facts are often confused when negative numbers are involved. Study the following four examples carefully:

The base is ((−3)).

The base is (3).

((−3)^{4}=(−3)(−3)(−3)(−3)=+81)

((−3)^{3}=(−3)(−3)(−3)=−27)

(−3^{4}=−1⋅3⋅3⋅3⋅3=−81)

(−3^{3}=−1⋅3⋅3⋅3=−27)

Table (PageIndex{3})

The parentheses indicate that the negative number is to be used as the base.

Example (PageIndex{10}):

Calculate:

1. ((−frac{1}{3})^{3})
2. ((−frac{1}{3})^{4})

Solution

Here (−frac{1}{3}) is the base for both problems.

1.Use the base as a factor three times.

((−frac{1}{3})^{3}=(−frac{1}{3})(−frac{1}{3})(−frac{1}{3}))

(=−frac{1}{27})

2.Use the base as a factor four times.

((−frac{1}{3})^{4}=(−frac{1}{3})(−frac{1}{3})(−frac{1}{3})(−frac{1}{3})

(=+frac{1}{81})

1. −(frac{12}{7})
2. (frac{1}{81})

Exercise (PageIndex{2})

Simplify:

1. (−2^{4})
2. ((−2)^{4})

1. −16

2. 16

## Order of Operations

When several operations are to be applied within a calculation, we must follow a specific order to ensure a single correct result.

1. Perform all calculations within the innermost parentheses or grouping symbol first.
2. Evaluate all exponents.
3. Apply multiplication and division from left to right.
4. Perform all remaining addition and subtraction operations last from left to right.

Note that multiplication and division should be worked from left to right. Because of this, it is often reasonable to perform division before multiplication.

Example (PageIndex{11}):

Simplify: (5^{3} − 24 ÷ 6 ⋅ frac{1}{2} + 2.)

Solution

First, evaluate (5^{3}) and then perform multiplication and division as they appear from left to right.

egin{aligned} 5 ^ { 3 } - 24 div 6 cdot frac { 1 } { 2 } + 2 & = 5 ^ { 3 } - 24 div 6 cdot frac { 1 } { 2 } + 2 & = 125 - 24 div 6 cdot frac { 1 } { 2 } + 2 & = 125 - 4 cdot frac { 1 } { 2 } + 2 & = 125 - 2 + 2 & = 123 + 2 & = 125 end{aligned}

Multiplying first would have led to an incorrect result. (125)

Example (PageIndex{12}):

Simplify: (- 10 - 5 ^ { 2 } + ( - 3 ) ^ { 4 }).

Solution

Take care to correctly identify the base when squaring.

(46)

We are less likely to make a mistake if we work one operation at a time. Some problems may involve an absolute value, in which case we assign it the same order of precedence as parentheses.

Example (PageIndex{13}):

Simplify: (7 - 5 left| - 2 ^ { 2 } + ( - 3 ) ^ { 2 } ight.).

Solution

Begin by performing the operations within the absolute value first.

Subtracting (7−5) first will lead to incorrect results. (−18)

Exercise (PageIndex{3})

Simplify: (- 6 ^ { 2 } - left[ - 15 - ( - 2 ) ^ { 3 } ight] - ( - 2 ) ^ { 4 }).

(-45)

## Key Takeaways

• Addition is commutative and subtraction is not. Furthermore, multiplication is commutative and division is not.
• Adding or subtracting fractions requires a common denominator; multiplying or dividing fractions does not.
• Grouping symbols indicate which operations to perform first. We usually group mathematical operations with parentheses, brackets, braces, and the fraction bar. We also group operations within absolute values. All groupings have the same order of precedence: the operations within the innermost grouping are performed first.
• When using exponential notation (a^{n}), the base a is used as a factor n times. Parentheses indicate that a negative number is to be used as the base. For example, ((−5)^{2}) is positive and (−5^{2}) is negative.
• To ensure a single correct result when applying operations within a calculation, follow the order of operations. First, perform operations in the innermost parentheses or groupings. Next, simplify all exponents. Perform multiplication and division operations from left to right. Finally, perform addition and subtraction operations from left to right.

Exercise (PageIndex{4})

Perform the operations. Reduce all fractions to lowest terms.

1. (33−(−15)+(−8))
2. (−10−9+(−6))
3. (−23+(−7)−(−10))
4. (−1−(−1)−1)
5. (frac{1}{2}+frac{1}{3}−frac{1}{6})
6. (−frac{1}{5}+frac{1}{2}−frac{1}{10})
7. (frac{2}{3}−(−frac{1}{4})−frac{1}{6})
8. (−frac{3}{2}−(−frac{2}{9})−frac{5}{6})
9. (frac{3}{4}−(−frac{1}{2})−frac{5}{8})
10. (−frac{1}{5}−frac{3}{2}−(−frac{7}{10}))
11. Subtract (3) from (10).
12. Subtract (−2) from (16).
13. Subtract (−frac{5}{6}) from (4).
14. Subtract (−frac{1}{2}) from (frac{3}{2}).
15. Calculate the sum of (−10) and (25).
16. Calculate the sum of (−30) and (−20).
17. Find the difference of (10) and (5).
18. Find the difference of (−17) and (−3).

1. (40)

3. (−20)

5. (frac{2}{3})

7. (frac{3}{4})

9. (frac{5}{8})

11. (7)

13. (frac{29}{6})

15. (15)

17. (5)

Exercise (PageIndex{5})

The formula (d = | b − a |) gives the distance between any two points on a number line. Determine the distance between the given numbers on a number line.

1. (10) and (15)
2. (6) and (22)
3. (0) and (12)
4. (−8) and (0)
5. (−5) and (−25)
6. (−12) and (−3)

1. 5 units

3. 12 units

5. 20 units

Exercise (PageIndex{6})

Determine the reciprocal of the following.

1. (frac{1}{3})
2. (frac{2}{5})
3. (−frac{3}{4})
4. (−12)
5. (a) where (a ≠ 0)
6. (frac{1}{a})
7. (frac{a}{b}) where (a ≠ 0)
8. (frac{1}{ab})

1. (3)

3. (−frac{4}{3})

5. (frac{1}{a})

7. (frac{b}{a})

Exercise (PageIndex{7})

Perform the operations.

1. (−4 (−5) ÷ 2)
2. ((−15) (−3) ÷ (−9))
3. (−22 ÷ (−11) (−2))
4. (50 ÷ (−25) (−4))
5. (frac{2}{3} (−frac{9}{10}))
6. (−frac{5}{8} (−frac{16}{25}))
7. (frac{7}{6} (−frac{6}{7}))
8. (−frac{15}{9} (frac{9}{5}))
9. (frac{4}{5} (−frac{2}{5}) ÷ frac{16}{25})
10. ((−frac{9}{2}) (−frac{3}{2}) ÷ frac{27}{16})
11. (frac{8}{5} ÷ frac{5}{2} ⋅ frac{15}{40})
12. (frac{3}{16} ÷ frac{5}{8} ⋅ frac{1}{2})
13. Find the product of (12) and (7).
14. Find the product of (−frac{2}{3}) and (12).
15. Find the quotient of (−36) and (12).
16. Find the quotient of (−frac{3}{4}) and (9).
17. Subtract (10) from the sum of (8) and (−5).
18. Subtract (−2) from the sum of (−5) and (−3).
19. Joe earns ($18.00) per hour and “time and a half” for every hour he works over (40) hours. What is his pay for (45) hours of work this week? 20. Billy purchased (12) bottles of water at ($0.75) per bottle, (5) pounds of assorted candy at ($4.50) per pound, and (15) packages of microwave popcorn costing ($0.50) each for his party. What was his total bill?
21. James and Mary carpooled home from college for the Thanksgiving holiday. They shared the driving, but Mary drove twice as far as James. If Mary drove for (210) miles, then how many miles was the entire trip?
22. A (6 frac{3}{4}) foot plank is to be cut into (3) pieces of equal length. What will be the length of each piece?
23. A student earned (72, 78, 84,) and (90) points on her first four algebra exams. What was her average test score? (Recall that the average is calculated by adding all the values in a set and dividing that result by the number of elements in the set.)
24. The coldest temperature on Earth, (−129)°F, was recorded in (1983) at Vostok Station, Antarctica. The hottest temperature on Earth, (136)°F, was recorded in (1922) at Al’ Aziziyah, Libya. Calculate the temperature range on Earth.

1. (10)

3. (−4)

5. (−frac{3}{5})

7. (−1)

9. (−frac{1}{2})

11. (frac{6}{25})

13. (84)

15. (−3)

17. (−7)

## 1.2: Operations with Real Numbers - Mathematics

Introduce concepts in a simple context and then generalize them in such a way that rules and facts that are true in the simple context remain true in the more general context.

### The Natural Numbers.

We can add or multiply two natural numbers and obtain another natural number. However, the difference or the ratio of two natural numbers is not always a natural number. For example, 5-2 and 12/3 are natural numbers, but 3-5 and 3/12 are not.

1. a + b = b + a   The commutative law of addition.
2. a * b = b * a   The commutative law of multiplication.
3. (a + b) + c = a + (b + c)   The associative law of addition.
4. (a * b) * c = a * (b * c)   The associative law of multiplication.
5. (a + b) * c = a * c + b * c   The distributive law.

These laws are true for all and any natural numbers a, b, c. Actually they hold for all numbers we will encounter, but the whole point of building the number system is that we do this in such a way that the above rules remain true.

The distributive law connects multiplication and addition and is the most crucial, and the most misused and misunderstood law in the above list.

## The Integers

The set of integers can be thought of as having been obtained by expanding the set of natural numbers to make subtraction always possible. Of course we have to define what we mean by the sum, difference, product, and ratio of two integers. This is done in the familiar way with the guiding principle being that the laws listed above remain valid. (That principle for example leads to the requirement that the product of two negative numbers is positive.)

The sum, difference, and product of two integers is always an integer. The ratio, however, is not, which gives rise to the next level:

## The Rational Numbers

The result of adding, subtracting, multiplying, or dividing rational numbers (so long as we don't divide by zero) is another rational number. We say that the set of rational numbers is closed under addition, subtraction, multiplication, and division.

Of course, after extending the integers to the rational numbers, we again need to define what we mean by the sum, difference, product, and ratio of two rational numbers. This is discussed in detail on the page on fractions.

## The Real Numbers

It can be shown that there is no rational number whose square equals 2. Hence the number system needs to be extended once more. For our purposes a number is a decimal expression whose digits may or may not terminate or repeat. It can also be shown that a real number is rational if and only if its digit repeat or terminate. Real numbers that aren't rational are irrational

## Hierarchy of Numbers

Each number set contains the number sets it surrounds. For example the set of rational numbers contains all natural numbers (and all integers). The Figure also indicates which operations are possible in each set. For example, we can add, subtract, and multiply integers, and the result will be an integer. (The result of dividing two integers is not always an integer, for example 5/2 is not.) The examples given are in the set shown, but not in a smaller set. For example, 2/3 is a rational number. It's also a real number, but it's not an integer, and it's not a natural number.

## Real numbers in word problems: Write a real world problem involving the multiplication of a fraction and a whole number with a product that is between 8 and 10 then solve the problem Find two imaginary numbers whose sum is a real number. How are the two imaginary numbers related? What is its sum? Is equal following terms? -9 21 = (-9) 21 Are these numbers 2i, 4i, 2i + 1, 8i, 2i + 3, 4 + 7i, 8i, 8i + 4, 5i, 6i, 3i complex? The diameter of the Earth is 12,756 kilometers. If Mercury's diameter is about 7,876.6 kilometers shorter than that of the Earth, what is the diameter of Mercury? How many solutions has the equation ? in the real numbers? Solve equation: log13(7x + 12) = 0 Determine missing coordinate of the point M [x, 120] of the graph of the function f bv rule: y = 5 x Determine the numbers b, c that the numbers x1 = -1 and x2 = 3 were roots of quadratic equation: -3x 2 + b x + c = 0 Eequation f(x) = 0 has roots x1 = 64, x2 = 100, x3 = 25, x4 = 49. How many roots have equation f(x 2 ) = 0 ? Subtract twice the number -23.6 from the difference of the numbers -130 and -40.2. Marlon drew a scale drawing of a summer camp. In real life, the sand volleyball court is 8 meters wide. It is 4 centimeters wide in the drawing. What is the drawing's scale factor? Simplify your answer and write it as a ratio, using a colon. Solve exponential equation (in real numbers): 9 8x-2 =9 Open intervals A = (x-2 2x-1) and B = (3x-4 4) are given. Find the largest real number for which A ⊂ B applies. Calculate the geometric mean of numbers a=15.2 and b=25.6. Determine the mean by construction where a and b are the length of the lines. Find cube root of 18 Cube, which consists of 8 small cubes with edge 3 dm has volume:

## Properties of Real Numbers Worksheets

What Are the Properties of Real Numbers? In the number system, real numbers are the combination of irrational and rational numbers. You can easily perform all arithmetic operations on these numbers and can represent them on the number line. On the other hand, imaginary numbers are the un-real numbers and cannot be represented on the number line. These imaginary numbers are typically used to describe complex numbers. Here we have discussed the critical properties of real numbers which help solve algebraic problems. Commutative Properties - The commutative property of addition says that you can add numbers in any order. That means that you will get the same result even If you change the order of the numbers. The commutative property of multiplication follows the same rule that you can multiply numbers in any order. Addition: a + b = b + a Multiplication: a x b = b x a Associative Property - Addition and multiplication can be performed on more than two numbers. So, if we have two or more numbers in a number sentence, we have to figure out which two numbers to associate or group first. The associative property says that we can group numbers in any order and still get the same result. Addition: a + (b + c) = (a + b) + c Multiplication: a x (b x c) = (a x b) x c Distributive Property - The distributive property is used when both multiplication and addition are involved in the same number sentence. It says that when multiplying a term by the terms within the parenthesis, we multiply each term of the parenthesis with the outside term. a x (b + c) = a x b + a x c Identity Property - The identity property tells us that when we add zero with a term, we get the same term as a result. Zero is known as the additive identity. Identity property of multiplication tells us that when we multiply 1 with any numbers, we get the same number as a result. Addition: a + 0 = a Multiplication: a x 1 = a.

### Basic Lesson

Demonstrates how to find the equation of a line when given slope and an intercept. Two elements are interchanging a + b = b + a Therefore it is the commutative property of addition.

### Intermediate Lesson

Explores how to determine the nature of complex equations. Practice problems are provided. An expression plus its negation give the identity element(0) a + b + (-a + -b) = 0 Therefore it is the additive inverse property.

### Independent Practice 1

Contains 20 Properties of Real Numbers problems. The answers can be found below. Identify the property of real numbers that is demonstrated.

### Independent Practice 2

Features another 20 Properties of Real Numbers problems.

### Homework Worksheet

Properties of Real Numbers problems for students to work on at home. As here first expression times each component of the second expression a. (b + c) = a. b + a. c Therefore it is distributive property. Example problems are provided and explained.

### Topic Quiz

10 Properties of Real Numbers problems. A math scoring matrix is included.

### Homework and Quiz Answer Key

Answers for the homework and quiz.

### Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

### Basic Lesson

The following is an example of which property of numbers? x + 4y = 4y + x. Two elements are interchanging a + b = b + a Therefore, it is commutative property of addition.

### Intermediate Lesson

The following is an example of which property of numbers? (3x + z) + (-3x + -z) =

### Independent Practice 1

The following are examples of which property of numbers? The answers can be found below.

### Independent Practice 2

Features another 20 Properties of Real Numbers problems.

### Homework Worksheet

Properties of Real Numbers problems for students to work on at home. As here first expression times each component of the second expression a. (b + c) = a. b + a. c Therefore it is distributive property. Example problems are provided and explained.

### Topic Quiz

10 Properties of Real Numbers problems. A math scoring matrix is included.

### Homework and Quiz Answer Key

Answers for the homework and quiz.

### Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

#### The Basics

In algebraic expressions, letters stand for numbers. Substituting a number for each variable and performing the operations is called "evaluating the expression." Replace each variable with a number value and follow the order of operations.

#### Who is He?

This mathematician has kept the mathematical world up in arms saying that he proved the question that x n + y n = z n has no solution when n is greater than 2. Answer: Pierre De Fermat.

## Mathematical Number Sets

• Natural Numbers are nothing more than your counting numbers: 1, 2, 3, …
• Whole Numbers are your counting numbers but it also includes zero: 0, 1, 2, 3, …
• Integers are the Natural Numbers and their opposites, or negatives: …-3, -2, -1, 0, 1, 2, 3…
• Rational Numbers are Integers that can be expressed as terminating or repeating decimal (i.e, simple fraction).
• Irrational Numbers are numbers that cannot be written as a simple fraction because their decimals never terminate or repeat.
• Real Numbers are all the numbers on the Number Line and include all the Rational and Irrational Numbers
• Complex Numbers are the set of Real Numbers and Imaginary Numbers.

## Unpacking Documents for High School Course Domains:

### High School Geometry

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not. Develop definitions of rotations, reflections, and translations in terms of points, angles, circles, perpendicular lines, parallel lines, and line segments. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Similarity, Right Triangles, and Trigonometry

Justify and apply the formula A=1/2 ab sin (C) to find the area of any triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

### High School Algebra I

Number and Quantity- The Real Number System

Perform all four arithmetic operations and apply properties to generate equivalent forms of rational numbers and square roots.

Algebra - Seeing Structure in Expressions

Interpret expressions that represent a quantity in terms of context. Use the properties of exponents to rewrite exponential expressions.

### High School Algebra II

Number and Quantity - The Complex Number System