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 sum^{53} and the result of subtracting is called the difference^{54}. Given any real numbers a, b, and c, we have the following properties of addition:
Additive Identity Property:^{55} | a+0=0+a=a |
---|---|
Additive Inverse Property:^{56} | a+(−a)=(−a)+a=0 |
Associative Property:^{57} | (a+b)+c=a+(b+c) |
Commutative Property:^{58} | 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)
Answer
(−5)
Adding or subtracting fractions requires a common denominator^{59}. 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 denominator^{60} (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})
Answer
(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})
Answer
(−frac{9}{2})
The result of multiplying real numbers is called the product^{61} and the result of dividing is called the quotient^{62}. Given any real numbers a, b, and c, we have the following properties of multiplication:
Zero Factor Property:^{63} | a⋅0=0⋅a=0 |
---|---|
Multiplicative Identity Property:^{64} | a⋅1=1⋅a=a |
Associative Property:^{65} | (a⋅b)⋅c=a⋅(b⋅c) |
Commutative Property:^{66} | 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:
Answer
(−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.
Answer
(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.
Answer:
(−frac{5}{3})
Two real numbers whose product is (1) are called reciprocals^{67}. 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.
Answer
(frac{25}{24})
Exercise (PageIndex{1})
Simplify: (frac{1}{2}⋅frac{3}{4}÷frac{1}{8}).
- Answer
(3) www.youtube.com/v/4zV-fyepZkk
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 symbols^{68} 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})
Answer:
(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.
Answer:
(−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 notation^{69}. For example,
(5⋅5⋅5⋅5=54)
The base^{70} is the factor and the positive integer exponent^{71} 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 square^{72}, and when the exponent is (3) we call the result a cube^{73}. 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) |
The parentheses indicate that the negative number is to be used as the base.
Example (PageIndex{10}):
Calculate:
- ((−frac{1}{3})^{3})
- ((−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})
Answers:
- −(frac{12}{7})
- (frac{1}{81})
Exercise (PageIndex{2})
Simplify:
- (−2^{4})
- ((−2)^{4})
- Answer
1. −16
2. 16
www.youtube.com/v/o3X52psRJtg
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.
- Perform all calculations within the innermost parentheses or grouping symbol first.
- Evaluate all exponents.
- Apply multiplication and division from left to right.
- 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.
Answer:
(125)
Example (PageIndex{12}):
Simplify: (- 10 - 5 ^ { 2 } + ( - 3 ) ^ { 4 }).
Solution
Take care to correctly identify the base when squaring.
Answer:
(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.
Answer:
(−18)
Exercise (PageIndex{3})
Simplify: (- 6 ^ { 2 } - left[ - 15 - ( - 2 ) ^ { 3 } ight] - ( - 2 ) ^ { 4 }).
- Answer
(-45)
www.youtube.com/v/DnaviQzLPA0
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.
- (33−(−15)+(−8))
- (−10−9+(−6))
- (−23+(−7)−(−10))
- (−1−(−1)−1)
- (frac{1}{2}+frac{1}{3}−frac{1}{6})
- (−frac{1}{5}+frac{1}{2}−frac{1}{10})
- (frac{2}{3}−(−frac{1}{4})−frac{1}{6})
- (−frac{3}{2}−(−frac{2}{9})−frac{5}{6})
- (frac{3}{4}−(−frac{1}{2})−frac{5}{8})
- (−frac{1}{5}−frac{3}{2}−(−frac{7}{10}))
- Subtract (3) from (10).
- Subtract (−2) from (16).
- Subtract (−frac{5}{6}) from (4).
- Subtract (−frac{1}{2}) from (frac{3}{2}).
- Calculate the sum of (−10) and (25).
- Calculate the sum of (−30) and (−20).
- Find the difference of (10) and (5).
- Find the difference of (−17) and (−3).
- Answer
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.
- (10) and (15)
- (6) and (22)
- (0) and (12)
- (−8) and (0)
- (−5) and (−25)
- (−12) and (−3)
- Answer
1. 5 units
3. 12 units
5. 20 units
Exercise (PageIndex{6})
Determine the reciprocal of the following.
- (frac{1}{3})
- (frac{2}{5})
- (−frac{3}{4})
- (−12)
- (a) where (a ≠ 0)
- (frac{1}{a})
- (frac{a}{b}) where (a ≠ 0)
- (frac{1}{ab})
- Answer
1. (3)
3. (−frac{4}{3})
5. (frac{1}{a})
7. (frac{b}{a})
Exercise (PageIndex{7})
Perform the operations.
- (−4 (−5) ÷ 2)
- ((−15) (−3) ÷ (−9))
- (−22 ÷ (−11) (−2))
- (50 ÷ (−25) (−4))
- (frac{2}{3} (−frac{9}{10}))
- (−frac{5}{8} (−frac{16}{25}))
- (frac{7}{6} (−frac{6}{7}))
- (−frac{15}{9} (frac{9}{5}))
- (frac{4}{5} (−frac{2}{5}) ÷ frac{16}{25})
- ((−frac{9}{2}) (−frac{3}{2}) ÷ frac{27}{16})
- (frac{8}{5} ÷ frac{5}{2} ⋅ frac{15}{40})
- (frac{3}{16} ÷ frac{5}{8} ⋅ frac{1}{2})
- Find the product of (12) and (7).
- Find the product of (−frac{2}{3}) and (12).
- Find the quotient of (−36) and (12).
- Find the quotient of (−frac{3}{4}) and (9).
- Subtract (10) from the sum of (8) and (−5).
- Subtract (−2) from the sum of (−5) and (−3).
- 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?
- 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?
- 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?
- 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?
- 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.)
- 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.
- Answer
1. (10)
3. (−4)
5. (−frac{3}{5})
7. (−1)
9. (−frac{1}{2})
11. (frac{6}{25})
13. (84)
15. (−3)
17. (−7)
19. ($855)
21. (315) miles
23. (81) points
Exercise (PageIndex{8})
Perform the operations.
- (7 − {3 − [−6 − (10)]})
- (− (9 − 12) − [6 − (−8 − 3)])
- (frac{1}{2} {5 − (10 − 3)})
- (frac{2}{3} {−6 + (6 − 9)})
- (5 {2 [3 (4 − frac{3}{2} )]})
- (frac{1}{2} {−6 [− (frac{1}{2} − frac{5}{3})]})
- (frac { 5 - | 5 - ( - 6 ) | } { | - 5 | - | - 3 | })
- (frac { | 9 - 12 | - ( - 3 ) } { | - 16 | - 3 ( 4 ) })
- (frac { - | - 5 - ( - 7 ) | - ( - 2 ) } { | - 2 | + | - 3 | })
- (frac { 1 - | 9 - ( 3 - 4 ) | } { - | - 2 | + ( - 8 - ( - 10 ) ) })
- Answer
1. (−1)2
3. (−1)
5. (75)
7. (−3)
9. (0)
Exercise (PageIndex{9})
Perform the operations.
- (12^{2})
- ((−12)^{2})
- (−12^{2})
- (−(−12)^{2})
- (−5^{4})
- ((−5)^{4})
- ((−frac{1}{2})^{3})
- (−(−frac{1}{2})^{3})
- (−(−frac{3}{4})^{2})
- (−(−frac{5}{2})^{3})
- ((−1)^{22})
- ((−1)^{13})
- (−(−1)^{12})
- (−(−1)^{5})
- (−10^{2})
- (−10^{4})
- Answer
1. (144)
3. (−144)
5. (−625)
7. (−frac{1}{8})
9. (−frac{9}{16})
11. (1)
13. (−1)
15. (−100)
Exercise (PageIndex{10})
Simplify.
- (5 − 3 (4 − 3^{2}))
- (8 − 5 (3 − 3^{2}))
- ((−5)^{2} + 3 (2 − 4^{2}))
- (6 − 2 (−5^{2} + 4 ⋅ 7))
- (5 − 3 [3 (2 − 3^{2}) + (−3)^{2}])
- (10 − 5 [(2 − 5)^{2} − 3])
- ([5^{2} − 3^{2} ] − [2 − (5 + (−4)^{2} )])
- (−7^{2} − [ (2 − 7)^{2} − (−8)^{2} ])
- (frac{3}{16} ÷ (frac{5}{12} −frac{1}{2} +frac{2}{3}) ⋅ 4)
- (6 cdot left[ left( frac { 2 } { 3 } ight) ^ { 2 } - left( frac { 1 } { 2 } ight) ^ { 2 } ight] div ( - 2 ) ^ { 2 })
- (frac { 3 - 2 cdot 5 + 4 } { 2 ^ { 2 } - 3 ^ { 2 } })
- (frac { left( 3 + ( - 2 ) ^ { 2 } ight) cdot 4 - 3 } { - 4 ^ { 2 } + 1 })
- (frac { - 5 ^ { 2 } + ( - 3 ) ^ { 2 } cdot 2 - 3 } { 8 ^ { 2 } + 6 ( - 10 ) })
- (frac { ( - 4 ) ^ { 2 } + ( - 3 ) ^ { 3 } } { - 9 ^ { 2 } - left( - 12 + 2 ^ { 2 } ight) * 10 })
- (−5^{2} − 2 |−5| )
- (−2^{4} + 6 | 2^{4} − 5^{2} |)
- (− (4− | 7^{2} − 8^{2} |))
- (−3 (5 − 2 |−6|))
- ((−3)^{2}− | −2 + (−3)^{3} | − 4^{2})
- (−5^{2} − 2 | 3^{3} − 2^{4} | − (−2)^{5})
- (5 ⋅ |−5| − (2 − |−7|)^{3})
- (10^{2} + 2 ( |−5|^{3} − 6^{3}))
- (frac{2}{3} − | frac{1}{2} − (−frac{4}{3})^{2} |)
- (−24 | frac{10}{3} − frac{1}{2} ÷ frac{1}{5} |)
- Calculate the sum of the squares of the first three consecutive positive odd integers.
- Calculate the sum of the squares of the first three consecutive positive even integers.
- What is (6) subtracted from the sum of the squares of (5) and (8)?
- What is (5) subtracted from the sum of the cubes of (2) and (3)?
- Answer
1. (20)
3. (−17)
5. (41)
7. (35)
9. (frac{9}{7})
11. (frac{3}{5})
13. (−frac{5}{2})
15. (−35)
17. (11)
19. (−36)
21. (150)
23. (−frac{11}{18})
25. (35)
27. (83)
Exercise (PageIndex{11})
- What is PEMDAS and what is it missing?
- Does (0) have a reciprocal? Explain.
- Explain why we need a common denominator in order to add or subtract fractions.
- Explain why ((−10)^{4}) is positive and (−10^{4}) is negative.
- Answer
1. Answer may vary
3. Answer may vary
Footnotes
^{53}The result of adding.
^{54}The result of subtracting.
^{55}Given any real number (a, a + 0 = 0 + a = a).
^{56}Given any real number (a, a + (−a) = (−a) + a = 0).
^{57}Given real numbers (a, b) and (c, (a + b) + c = a + (b + c)).
^{58}Given real numbers (a) and (b), (a + b = b + a).
^{59}A denominator that is shared by more than one fraction.
^{60}The least common multiple of a set of denominators.
^{61}The result of multiplying.
^{62}The result of dividing.
^{63}Given any real number (a, a ⋅ 0 = 0 ⋅ a = 0 .)
^{64}Given any real number (a, a ⋅ 1 = 1 ⋅ a = a .)
^{65}Given any real numbers (a, b) and (c, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) . )
^{66}Given any real numbers (a) and (b, a ⋅ b = b ⋅ a.)
^{67}Two real numbers whose product is (1).
^{68}Parentheses, brackets, braces, and the fraction bar are the common symbols used to group expressions and mathematical operations within a computation.
^{69}The compact notation (a^{n}) used when a factor (a) is repeated (n) times.
^{70}The factor (a) in the exponential notation (a^{n}).
^{71}The positive integer (n) in the exponential notation (a^{n}) that indicates the number of times the base is used as a factor.
^{72}The result when the exponent of any real number is (2).
^{73}The result when the exponent of any real number is (3).
1.2: Operations with Real Numbers - Mathematics
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- "Proper" Subset (left) - 1st format
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- "Proper" Subset (left) - 2nd format
- "Proper" Subset (right) - 1st format
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- "Proper" Subset (right) - 2nd format
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Math & Numbers
- Overview of Real Numbers
- Comparing Two Integers on a Number Line
- Comparing Two Decimals on a Number Line
- Comparing Two Fractions on a Number Line
- Comparing Two Fractions Without Using a Number Line
- Comparing Two Numbers using Percents
- Comparing Two Different Units of Measurement
- Comparing Numbers which have a Margin of Error
- Comparing Numbers which have Rounding Errors
- Comparing Numbers from Different Time Periods
- Comparing Numbers computed with Different Methodologies
Properties of Numbers
- Associative Property
- Commutative Property
- Distributive Property
- Identity Property
- Inverse Property
- Closure & Density Property
- Equivalence Relationships
- Equivalence Properties
- Equivalence Examples
- Trichotomy Property of Inequality
- Transitive Property of Inequality
- Reversal Property of Inequality
- Additive Property of Inequality
- Multiplicative Property of Inequality
- Exponents and Roots Properties of Inequality
Exponents, Radicals, & Roots
- Raising numbers to a Power
- Multiplying Numbers With Exponents
- Dividing Numbers With Exponents
- Distributive Property of Exponents
- Negative Exponents
- Zero Exponent
- Exponent Videos & Free Resources
- Adding & Subtracting Radicals
- Multiplying Radicals
- Dividing Radicals
- Rationalize the Denominator
- Fractional Exponents & Radicals
- Simplifying Radicals
- Calculate Square Root Without Using a Calculator
- Calculate Roots Using Equations
- Radical Videos & Free Resources
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Example Problems - Geometric Sequence
Example Problems - Arithmetic Sequence
Example Problems - Rationalize the Denominator
Example Problems - Quadratic Equations
Example Problems - Work Rate Problems
Example Problems - statistics
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1.2: Operations with Real Numbers - Mathematics
Intermediate Algebra
Tutorial 4: Operations on Numbers
- Add and subtract real numbers.
- Multiply and divide real numbers.
- Evaluate a numeric exponential expression.
- Evaluate a numeric radical expression.
- Use the order of operations appropriately.
Even in this day and age of calculators, it is very important to know these basic rules of operations on real numbers. Even if you are using a calculator, you are the one that is putting the information into it, so you need to know things like when you are subtracting versus multiplying and the order that you need to put it in. Also, if you are using a calculator you should have a rough idea as to what the answer should be. You never know, you may hit a wrong key and get a wrong answer (it happens to the best of us). Also, your batteries in your calculator may run out and you may have to do a problem by hand (scary. ). You want to be prepared for those Murphy's Law moments.
We will start with the individual operations and then mix ‘em up using order of operations.
Adding Real Numbers |
Adding Real Numbers with the Same Sign |
Step 2: Attach their common sign to sum .
Example 1: Find the sum -6 + (-8).
The sum of the absolute values would be 14 and their common sign is -. That is how we get the answer of -14.
You can also think of this as money - I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 6 dollars and then having lost another 8 dollars for a total loss of 14 dollars.
The sum of the absolute values would be 14.2 and their common sign is -. That is how we get the answer of -14.2.
You can also think of this as money - I know we can all relate to that. Think of the negative as a loss. In this example, you can think of it as having lost 5.5 dollars and then having lost another 8.7 dollars for a total loss of 14.2 dollars.
Adding Real Numbers with Opposite Signs |
Step 2: Attach the sign of the number that has the higher absolute value.
The difference between 8 and 6 is 2 and the sign of 8 (the larger absolute value) is -. That is how we get the answer of -2.
Thinking in terms of money: we lost 8 dollars and got back 6 dollars, so we are still in the hole 2 dollars.
*Mult. top and bottom of first fraction
by 2 to get the LCD of 6
*Take the difference of the numerators
and write over common denominator 6
Thinking in terms of money: we had 2/3 of a dollar and lost 1/6 of a dollar, so we would come out ahead 1/2 of a dollar.
Note that if you need help on fractions, click on this link: Fractions
Subtracting Real Numbers a - b = a + (-b) or a - (-b) = a + b |
Now, you do not have to write it out like this if you are already comfortable with it. This just gives you the thought behind it.
Subtracting 5 is the same as adding a -5.
Once it is written as addition, I just follow the rules for addition, as shown above, to complete for an answer of -8.
Subtracting -5 is the same as adding 5.
Once it is written as addition, I just follow the rules for addition, as shown above, to complete for an answer of 2.
Multiplying or Dividing Real Numbers |
Step 1: Multiply or divide their absolute values.
Step 2: Put the correct sign.
If they have opposite signs, the product or quotient is negative.
The product of the absolute values 4 x 3 is 12 and they have opposite signs, so our answer is -12.
Note that if you need help on fractions, click on this link: Fractions
The quotient of the absolute values 10/2 is 5 and they have the same signs, so our answer is 5.
*Mult. num. together
*Mult. den. together
*(+)(-) = -
Note that if you need help on fractions, click on this link: Fractions
0/a = 0 (when a does not equal 0)
Multiplying any expression by 0 results in an answer of 0.
Dividing 0 by any expression other than 0 results in an answer of 0.
Example 13: Find the quotient 5/0.
Dividing by 0 results in an undefined answer.
(note there are n x's in the product)
The exponent tells you how many times a base appears in a PRODUCT.
- When there is no index number n , it is understood to be a 2 or square root.
- When looking for the nth radical or nth root, you want the expression that when you raise it to the nth power you would get the radicand (what is inside the radical sign).
Note that we are only interested in the principal root and since 9 is positive and there is not a sign in front of the radical, our answer is positive 3. If there had been a negative in front of the radical our answer would have been -3.
Now we are looking for the third or cube root of -1/8, which means we are looking for a number that when we cube it we get -1/8.
Since -1/2 cubed is -1/8, our answer is going to be -1/2.
Please P arenthesis or grouping symbols
Excuse E xponents (and radicals)
My Dear M ultiplication/ D ivision left to right
Aunt Sally A ddition/ S ubtraction left to right
When you do have more than one mathematical operation, you need to use the order of operations as listed above. You may have already heard of the saying "Please Excuse My Dear Aunt Sally". It is just a way to help you remember the order you need to go in when applying the order of operations.
Multiplication
The product of two real numbers is always a rational number. Hence R is closed under multiplication.
If a and b any two real numbers, then
5 x 9 = 45 is a real number
Commutative Property :
Multiplication of real numbers is commutative.
If a and b are any two real numbers, then
Therefore, Commutative Property is true for multiplication of real numbers.
Associative Property :
Multiplication of real numbers is associative.
If a, b and c are any three real numbers, then
Therefore,ਊssociative Property is true for multiplication.
Multiplicative Identity :
The product of any real number and 1 is the rational number itself. ‘One’ is the multiplicative identity for real numbers.
If a is any real number number, then
Every real number multiplied by 0 gives the result 0.
If a is any real number, then
Multiplicative Inverse or Reciprocal :
For every real number a, a ≠ 0, there exists a real number 1/a such that a ⋅ 1 /a = 1. Then 1/a is the multiplicative inverse of a.
If 'a' is a real number, then 1/a is the multiplicative inverse or reciprocal of it.
The reciprocal of 2 is 1/2.
The reciprocal of 1/3 is 3.
The reciprocal of 3 is 1/3.
The reciprocal of 0 is undefined.
Irrational numbers $mathbb$
We have seen that any rational number can be expressed as an integer, decimal or exact decimal number.
However, not all decimal numbers are exact or recurring decimals, and therefore not all decimal numbers can be expressed as a fraction of two integers.
These decimal numbers which are neither exact nor recurring decimals are characterized by infinite nonperiodic decimal digits, ie that never end nor have a repeating pattern.
Note that the set of irrational numbers is the complementary of the set of rational numbers.
Some examples of irrational numbers are $sqrt<2>,pi,sqrt[3]<5>,$ and for example $pi=3,1415926535ldots$ comes from the relationship between the length of a circle and its diameter.
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.
- a + b = b + a The commutative law of addition.
- a * b = b * a The commutative law of multiplication.
- (a + b) + c = a + (b + c) The associative law of addition.
- (a * b) * c = a * (b * c) The associative law of multiplication.
- (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: log_{13}(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 x_{1} = -1 and x_{2} = 3 were roots of quadratic equation: -3x 2 + b x + c = 0
Eequation f(x) = 0 has roots x_{1} = 64, x_{2} = 100, x_{3} = 25, x_{4} = 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