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Basic Properties of Real Numbers: Symbols and Notations

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. Topics covered in this module: understand the difference between variables and constants, be familiar with the symbols of operation, equality, and inequality, be familiar with grouping symbols, be able to correctly use the order of operations.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Overview

  • Variables and Constants
  • Symbols of Operation, Equality, and Inequality
  • Grouping Symbols
  • The Order of Operations

Variables and Constants

A basic characteristic of algebra is the use of symbols (usually letters) to represent numbers.

Variable

A letter or symbol that represents any member of a collection of two or more numbers is called a variable.

Constant

A letter or symbol that represents a specific number, known or unknown is called a constant.

In the following examples, the letter x x is a variable since it can be any member of the collection of numbers {35, 25, 10} {35, 25, 10} . The letter h h is a constant since it can assume only the value 5890.

Example 1

Suppose that the streets on your way from home to school have speed limits of 35 mph, 25 mph, and 10 mph. In algebra we can let the letter x x represent our speed as we travel from home to school. The maximum value of x x depends on what section of street we are on. The letter x x can assume any one of the various values 35,25,10.

Example 2

Suppose that in writing a term paper for a geography class we need to specify the height of Mount Kilimanjaro. If we do not happen to know the height of the mountain, we can represent it (at least temporarily) on our paper with the letter h h . Later, we look up the height in a reference book and find it to be 5890 meters. The letter h h can assume only the one value, 5890, and no others. The value of h h is constant.

Symbols of Operation, Equality, and Inequality

Binary Operation

A binary operation on a collection of numbers is a process that assigns a number to two given numbers in the collection. The binary operations used in algebra are addition, subtraction, multiplication, and division.

Symbols of Operation

If we let x x and y y each represent a number, we have the following notations:
 Addition x+ y Subtraction x-y Multiplication x⋅y (x)(y) x(y) xy Division xy x/y x÷y yx Addition x+ y Subtraction x-y Multiplication x⋅y (x)(y) x(y) xy Division xy x/y x÷y yx

Sample Set A

Example 3

a+b a+b represents the sum of a a and b b .

Example 4

4+y 4+y represents the sum of 4 and y y .

Example 5

8−x 8−x represents the difference of 8 and x x .

Example 6

6x 6x represents the product of 6 and x x .

Example 7

ab ab represents the product of a a and b b .

Example 8

h3 h3 represents the product of h h and 3.

Example 9

(14.2)a (14.2)a represents the product of 14.2 14.2 and a a .

Example 10

(8) (24) (8) (24) represents the product of 8 and 24.

Example 11

5⋅6(b) 5⋅6(b) represents the product of 5,6, and b b .

Example 12

6 x 6 x represents the quotient of 6 and x x .

Practice Set A

Exercise 1

Represent the product of 29 and x x five different ways.

Solution

29⋅x, 29x, (29)(x), 29(x), (29)x 29⋅x, 29x, (29)(x), 29(x), (29)x

If we let a a and b b represent two numbers, then a a and b b are related in exactly one of three ways:

Equality and Inequality Symbols

a=b a and b are equal a>b a is strictly greater than b a<b a is strictly less than b a=b a and b are equal a>b a is strictly greater than b a<b a is strictly less than b

Some variations of these symbols include

a≠b a is not equal to b  a≥b a is greater than or equal to b a≤b a is less than or equal to b a≠b a is not equal to b  a≥b a is greater than or equal to b a≤b a is less than or equal to b

The last five of the above symbols are inequality symbols. We can negate (change to the opposite) any of the above statements by drawing a line through the relation symbol (as in a≠b a≠b ), as shown below:

a a is not greater than b b can be expressed as either

a > b or a≤b. a > b or a≤b.

a a is not less b b than can be expressed as either

a < b or a≥b. a < b or a≥b.

a < b a<b and a ≥ b a≥b both indicate that aa is less than bb.

Grouping Symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in algebra are

Parentheses: ( ) Brackets: [ ] Braces: { } Bar:    ¯ Parentheses: ( ) Brackets: [ ] Braces: { } Bar:    ¯

In a computation in which more than one operation is involved, grouping symbols help tell us which operations to perform first. If possible, we perform operations inside grouping symbols first.

Sample Set B

Example 13

(4+17)−6=21−6=15 (4+17)−6=21−6=15

Example 14

8(3+6)=8(9)=72 8(3+6)=8(9)=72

Example 15

5[8+(10−4)]=5[8+6]=5[14]=70 5[8+(10−4)]=5[8+6]=5[14]=70

Example 16

2{3[4(17−11)]}=2{3[4(6)]}=2{3[24]}=2{72}=144 2{3[4(17−11)]}=2{3[4(6)]}=2{3[24]}=2{72}=144

Example 17

9(5+1) 24+3 9(5+1) 24+3 .

The fraction bar separates the two groups of numbers 9(5+1) 9(5+1) and 24+3 24+3 . Perform the operations in the numerator and denominator separately.

9(5+1) 24+3 = 9(6) 24+3 = 54 24+3 = 54 27 =2 9(5+1) 24+3 = 9(6) 24+3 = 54 24+3 = 54 27 =2

Practice Set B

Use the grouping symbols to help perform the following operations.

Exercise 2

3(1+8) 3(1+8)

Solution

27

Exercise 3

4[2(11−5)] 4[2(11−5)]

Solution

48

Exercise 4

6{2[2(10−9)]} 6{2[2(10−9)]}

Solution

24

Exercise 5

1+19 2+3 1+19 2+3

Solution

4

The following examples show how to use algebraic notation to write each expression.

Example 18

9 minus y y becomes 9−y 9−y

Example 19

46 times x x becomes 46x 46x

Example 20

7 times (x+y) (x+y) becomes 7(x+y) 7(x+y)

Example 21

4 divided by 3, times z z becomes ( 4 3 ) z ( 4 3 ) z

Example 22

(a−b) (a−b) times (b−a) (b−a) divided by (2 times a a ) becomes (a−b) (b−a) 2a (a−b) (b−a) 2a

Example 23

Introduce a variable (any letter will do but here we’ll let x x represent the number) and use appropriate algebraic symbols to write the statement: A number plus 4 is strictly greater than 6. The answer is x+4>6 x+4>6 .

The Order of Operations

Suppose we wish to find the value of 16+4⋅9 16+4⋅9 . We could

  1. add 16 and 4, then multiply this sum by 9.
     16+4⋅9=20⋅9=180 16+4⋅9=20⋅9=180
  2. multiply 4 and 9, then add 16 to this product.
     16+4⋅9=16+36=52 16+4⋅9=16+36=52

We now have two values for one number. To determine the correct value we must use the standard order of operations.

Order of Operations

  1. Perform all operations inside grouping symbols, beginning with the innermost set.
  2. Perform all multiplications and divisions, as you come to them, moving left-to-right.
  3. Perform all additions and subtractions, as you come to them, moving left-to-right.

As we proceed in our study of algebra, we will come upon another operation, exponentiation, that will need to be inserted before multiplication and division. (See Section (Reference).)

Sample Set C

USe the order of operations to find the value of each number.

Example 24

16+4⋅9 Multiply first. =16+36 Now add. =52 16+4⋅9 Multiply first. =16+36 Now add. =52

Example 25

(27−8)+7(6+12) Combine within parentheses. =19+7(18) Multiply. =19+126 Now add. =145 (27−8)+7(6+12) Combine within parentheses. =19+7(18) Multiply. =19+126 Now add. =145

Example 26

8+2[4+3(6−1)] Begin with the innermost set of grouping symbols, ( ). =8+2[4+3(5)] Now work within the next set of grouping symbols, [ ]. =8+2[4+15] =8+2[19] =8+38 =46 8+2[4+3(6−1)] Begin with the innermost set of grouping symbols, ( ). =8+2[4+3(5)] Now work within the next set of grouping symbols, [ ]. =8+2[4+15] =8+2[19] =8+38 =46

Example 27

6+4[2+3(19−17)] 18−2[2(3)+2] = 6+4[2+3(2)] 18−2[6+2] = 6+4[2+6] 18−2[8] = 6+4[8] 18−16 = 6+32 2 = 38 2 =19 6+4[2+3(19−17)] 18−2[2(3)+2] = 6+4[2+3(2)] 18−2[6+2] = 6+4[2+6] 18−2[8] = 6+4[8] 18−16 = 6+32 2 = 38 2 =19

Practice Set C

Use the order of operations to find each value.

Exercise 6

25+8(3) 25+8(3)

Solution

49

Exercise 7

2+3(18−5⋅2) 2+3(18−5⋅2)

Solution

26

Exercise 8

4+3[2+3(1+8÷4)] 4+3[2+3(1+8÷4)]

Solution

37

Exercise 9

19+2{5+2[18+6(4+1)]} 5⋅6−3(5)−2 19+2{5+2[18+6(4+1)]} 5⋅6−3(5)−2

Solution

17

Exercises

For the following problems, use the order of operations to find each value.

Exercise 10

2+3(6) 2+3(6)

Solution

20

Exercise 11

18−7(8−3) 18−7(8−3)

Exercise 12

8⋅4÷16+5 8⋅4÷16+5

Solution

7

Exercise 13

(21+4)÷5⋅2 (21+4)÷5⋅2

Exercise 14

3(8+2)÷6+3 3(8+2)÷6+3

Solution

8

Exercise 15

6(4+1)÷(16÷8)−15 6(4+1)÷(16÷8)−15

Exercise 16

6(4−1)+8(3+7)−20 6(4−1)+8(3+7)−20

Solution

78

Exercise 17

(8) (5)+2(14)+(1) (10) (8) (5)+2(14)+(1) (10)

Exercise 18

61−22+4[3(10)+11] 61−22+4[3(10)+11]

Solution

203

Exercise 19

(1+16−3) 7 +5(12) (1+16−3) 7 +5(12)

Exercise 20

8(6+20) 8 + 3(6+16) 22 8(6+20) 8 + 3(6+16) 22

Solution

29

Exercise 21

18÷2+55 18÷2+55

Exercise 22

21÷7÷3 21÷7÷3

Solution

1

Exercise 23

85÷5⋅5−85 85÷5⋅5−85

Exercise 24

(300−25)÷(6−3) (300−25)÷(6−3)

Solution

91 2 3 91 2 3

Exercise 25

4⋅3+8⋅28−(3+17)+11(6) 4⋅3+8⋅28−(3+17)+11(6)

Exercise 26

2{(7+7)+6[4(8+2)]} 2{(7+7)+6[4(8+2)]}

Solution

508

Exercise 27

0+10(0)+15[4(3)+1] 0+10(0)+15[4(3)+1]

Exercise 28

6.1(2.2+1.8) 6.1(2.2+1.8)

Solution

24.4 24.4

Exercise 29

5.9 2 +0.6 5.9 2 +0.6

Exercise 30

(4+7) (8−3) (4+7) (8−3)

Solution

55

Exercise 31

(10+5) (10+5)−4(60−4) (10+5) (10+5)−4(60−4)

Exercise 32

( 5 12 − 1 4 )+( 1 6 + 2 3 ) ( 5 12 − 1 4 )+( 1 6 + 2 3 )

Solution

1

Exercise 33

4( 3 5 − 8 15 )+9( 1 3 + 1 4 ) 4( 3 5 − 8 15 )+9( 1 3 + 1 4 )

Exercise 34

0 5 + 0 1 +0[2+4(0)] 0 5 + 0 1 +0[2+4(0)]

Solution

0

Exercise 35

0⋅9+4⋅0÷7+0[2(2−2)] 0⋅9+4⋅0÷7+0[2(2−2)]

For the following problems, state whether the given statements are the same or different.

Exercise 36

x≥y and x>y x≥y and x>y

Solution

different

Exercise 37

x is strictly less than y and x is not greater than or equal to y

Exercise 38

x=y and y=x x=y and y=x

Solution

same

Exercise 39

Represent the product of 3 and x x five different ways.

Exercise 40

Represent the sum of a a and b b two different ways.

Solution

a+b, b+a a+b, b+a

For the following problems, rewrite each phrase using algebraic notation.

Exercise 41

Ten minus three

Exercise 42

x x plus sixteen

Solution

x+16 x+16

Exercise 43

51 divided by a a

Exercise 44

81 times x x

Solution

81x 81x

Exercise 45

3 times (x+y) (x+y)

Exercise 46

(x+b) (x+b) times (x+7) (x+7)

Solution

( x+b )( x+7 ) ( x+b )( x+7 )

Exercise 47

3 times x x times y y

Exercise 48

x x divided by (7 times b b )

Solution

x 7b x 7b

Exercise 49

(a+b) (a+b) divided by (a+4) (a+4)

For the following problems, introduce a variable (any letter will do) and use appropriate algebraic symbols to write the given statement.

Exercise 50

A number minus eight equals seventeen.

Solution

x−8=17 x−8=17

Exercise 51

Five times a number, minus one, equals zero.

Exercise 52

A number divided by six is greater than or equal to forty-four.

Solution

x 6 ≥44 x 6 ≥44

Exercise 53

Sixteen minus twice a number equals five.

Determine whether the statements for the following problems are true or false.

Exercise 54

6−4(4) (1)≤10 6−4(4) (1)≤10

Solution

true

Exercise 55

5(4+2⋅10)≥110 5(4+2⋅10)≥110

Exercise 56

8⋅6−48≤0 8⋅6−48≤0

Solution

true

Exercise 57

20+4.3 16 <5 20+4.3 16 <5

Exercise 58

2[6(1+4)−8]>3(11+6) 2[6(1+4)−8]>3(11+6)

Solution

false

Exercise 59

6 [ 4 + 8 + 3 ( 26 - 15 ) ] 6[4+8+3(26-15)] Is not less than or equal to 3 [ 7 ( 10 - 4 ) ] 3[7(10-4)]

Exercise 60

The number of different ways 5 people can be arranged in a row is 5⋅4⋅3⋅2⋅1 5⋅4⋅3⋅2⋅1 . How many ways is this?

Solution

120

Exercise 61

A box contains 10 computer chips. Three chips are to be chosen at random. The number of ways this can be done is

 10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 3⋅2⋅1⋅7⋅6⋅5⋅4⋅3⋅2⋅1 10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1 3⋅2⋅1⋅7⋅6⋅5⋅4⋅3⋅2⋅1

How many ways is this?

Exercise 62

The probability of obtaining four of a kind in a five-card poker hand is

 13⋅48 (52⋅51⋅50⋅49⋅48)÷(5⋅4⋅3⋅2⋅1) 13⋅48 (52⋅51⋅50⋅49⋅48)÷(5⋅4⋅3⋅2⋅1)

What is this probability?

Solution

0.00024, or  1 4165 0.00024, or  1 4165

Exercise 63

Three people are on an elevator in a five story building. If each person randomly selects a floor on which to get off, the probability that at least two people get off on the same floor is

 1− 5⋅4⋅3 5⋅5⋅5 1− 5⋅4⋅3 5⋅5⋅5

What is this probability?

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