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Signed Numbers: Variables, Constants, and Real Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses variables, constants, and real numbers. By the end of the module students should be able to distinguish between variables and constants, be able to recognize a real number and particular subsets of the real numbers and understand the ordering of the real numbers.

Section Overview

  • Variables and Constants
  • Real Numbers
  • Subsets of Real Numbers
  • Ordering Real Numbers

Variables and Constants

A basic distinction between algebra and arithmetic is the use of symbols (usually letters) in algebra to represent numbers. So, algebra is a generalization of arithme­tic. Let us look at two examples of situations in which letters are substituted for numbers:

  1. Suppose that a student is taking four college classes, and each class can have at most 1 exam per week. In any 1-week period, the student may have 0, 1, 2, 3, or 4 exams. In algebra, we can let the letter xx size 12{x} {} represent the number of exams this student may have in a 1-week period. The letter xx size 12{x} {} may assume any of the various values 0, 1, 2, 3, 4.
  2. Suppose that in writing a term paper for a biology class a student needs to specify the average lifetime, in days, of a male housefly. If she does not know this number off the top of her head, she might represent it (at least temporarily) on her paper with the letter tt size 12{t} {} (which reminds her of time). Later, she could look up the average time in a reference book and find it to be 17 days. The letter tt size 12{t} {} can assume only the one value, 17, and no other values. The value tt size 12{t} {} is constant.

Variable, Constant

  1. A letter or symbol that represents any member of a collection of two or more numbers is called a variable.
  2. A letter or symbol that represents one specific number, known or unknown, is called a constant.

In example 1, the letter xx size 12{x} {} is a variable since it can represent any of the numbers 0, 1, 2, 3, 4. The letter tt size 12{t} {} example 2 is a constant since it can only have the value 17.

Real Numbers

Real Number Line

The study of mathematics requires the use of several collections of numbers. The real number line allows us to visually display (graph) the numbers in which we are interested.

A line is composed of infinitely many points. To each point we can associate a unique number, and with each number, we can associate a particular point.

Coordinate

The number associated with a point on the number line is called the coordinate of the point.

Graph

The point on a number line that is associated with a particular number is called the graph of that number.

Constructing a Real Number Line

We construct a real number line as follows:

  1. Draw a horizontal line.

    A horizontal line with arrows on the end.

  2. Origin

    Choose any point on the line and label it 0. This point is called the origin.

    A horizontal line with arrows on the end. The center has a hash mark labeled 0.

  3. Choose a convenient length. Starting at 0, mark this length off in both direc­tions, being careful to have the lengths look like they are about the same.

    A horizontal line with arrows on the end. The center has a hash mark labeled 0. There are numerous evenly-spaced hash marks on either side of the zero.

We now define a real number.

Real Number

A real number is any number that is the coordinate of a point on the real number line.

Positive Numbers, Negative Numbers

Real numbers whose graphs are to the right of 0 are called positive real numbers, or more simply, positive numbers. Real numbers whose graphs appear to the left of 0 are called negative real numbers, or more simply, negative numbers.

A horizontal line with arrows on the end. The center has a hash mark labeled 0. On the right side is a bracket, labeled Positive numbers. On the left side is a bracket, labeled Negative numbers.

The number 0 is neither positive nor negative.

Subsets of Real Numbers

The set of real numbers has many subsets. Some of the subsets that are of interest in the study of algebra are listed below along with their notations and graphs.

Natural Numbers, Counting Numbers

The natural or counting numbers ( N N): 1, 2, 3, 4, . . . Read “and so on.”

A number line containing dots on the hash marks for numbers one through seven.

Whole Numbers

The whole numbers ( W W): 0, 1, 2, 3, 4, . . .

A number line containing dots on the hash marks for numbers zero through seven.

Notice that every natural number is a whole number.

Integers

The integers ( Z Z): . . . -3, -2, -1, 0, 1, 2, 3, . . .

A number line containing dots on the hash marks for numbers -4 through 4.

Notice that every whole number is an integer.

Rational Numbers (Fractions)

The rational numbers ( Q Q): Rational numbers are sometimes called fractions. They are numbers that can be written as the quotient of two integers. They have decimal representations that either terminate or do not terminate but contain a repeating block of digits. Some examples are

- 3 4 = - 0.75 ︸ Terminating - 3 4 = - 0.75 ︸ Terminating       8 11 27 = 8.407407407 . . . ︸ Nonterminating, but repeating 8 11 27 = 8.407407407 . . . ︸ Nonterminating, but repeating

Some rational numbers are graphed below.

A number line containing hash marks for numbers -3 through 4. There are dots for negative three and one-eighths, negative one-half, two-fifths, two divided by one, and three and one-half.

Notice that every integer is a rational number.

Notice that there are still a great many points on the number line that have not yet been assigned a type of number. We will not examine these other types of numbers in this text. They are examined in detail in algebra. An example of these numbers is the number ππ, whose decimal representation does not terminate nor contain a repeating block of digits. An approximation for ππ is 3.14.

Sample Set A

Example 1

Is every whole number a natural number?

No. The number 0 is a whole number but it is not a natural number.

Example 2

Is there an integer that is not a natural number?

Yes. Some examples are 0, -1, -2, -3, and -4.

Example 3

Is there an integer that is a whole number?

Yes. In fact, every whole number is an integer.

Practice Set A

Exercise 1

Is every natural number a whole number?

Solution

yes

Exercise 2

Is every whole number an integer?

Solution

yes

Exercise 3

Is every integer a real number?

Solution

yes

Exercise 4

Is there an integer that is a whole number?

Solution

yes

Exercise 5

Is there an integer that is not a natural number?

Solution

yes

Ordering Real Numbers

Ordering Real Numbers

A real number bb size 12{b} {} is said to be greater than a real number aa size 12{a} {}, denoted b>ab>a size 12{b>a} {}, if bb size 12{b} {} is to the right of aa size 12{a} {} on the number line. Thus, as we would expect, 5>25>2 size 12{5>2} {} since 5 is to the right of 2 on the number line. Also, - 2 > -5- 2 > -5 size 12{"- 2 ">"-5"} {} since -2 is to the right of -5 on the number line.

A number line containing hash marks for numbers -5 through 5. There are dots on the hash marks for -5, -2, 2, and 5. Above the left side of the number line is the expression -2 > -5, and on the left side is 5 > 2.

If we let aa size 12{a} {} and bb size 12{b} {} represent two numbers, then aa size 12{a} {} and bb size 12{b} {} are related in exactly one of three ways: Either

Equality Symbol

a=b a  and  b  are equal (8=8) a=b a  and  b  are equal (8=8)

Inequality Symbols

a>b a   is greater than  b (8>5) a<b a  is less than  b (5<8) Some variations of these symbols are a≠b a   is not equal to  b (8≠5) a≥b a   is greater than or equal to  b (a≥8) a≤b a   is less than or equal to  b (a≤8) a>b a   is greater than  b (8>5) a<b a  is less than  b (5<8) Some variations of these symbols are a≠b a   is not equal to  b (8≠5) a≥b a   is greater than or equal to  b (a≥8) a≤b a   is less than or equal to  b (a≤8)

Sample Set B

Example 4

What integers can replace x x so that the following statement is true?

- 3 ≤ x < 2 - 3 ≤ x < 2 size 12{"-3" <= " x"<" 2"} {}

A number line containing hash marks for numbers -5 through 5. There are dots on the hash marks for -3, -2, -1, 0, 1.

The integers are -3, -2, -1, 0, 1.

Example 5

Draw a number line that extends from -3 to 5. Place points at all whole numbers between and including -1 and 3.

A number line containing hash marks for numbers -3 through 5. There are dots on the hash marks for 0, 1, 2, and 3.

-1 is not a whole number

Practice Set B

Exercise 6

What integers can replace x x so that the following statement is true? - 5≤ x< 2- 5≤ x< 2 size 12{"-5" <= " x"<" 2"} {}

Solution

-5, -4, -3, -2, -1, 0

Exercise 7

Draw a number line that extends from -4 to 3. Place points at all natural numbers between, but not including, -2 to 2.

A horizontal line with arrows on the end.

Solution

A number line with hash marks for the numbers -4 to 3, and a dot on the hash mark for 1.

Exercises

For the following 8problems, next to each real number, note all collections to which it belongs by writing NN size 12{N} {} for natu­ral number, WW size 12{W} {} for whole number, or ZZ size 12{Z} {} for integer. Some numbers may belong to more than one collec­tion.

Exercise 8

6

Solution

N, W, Z

Exercise 9

12

Exercise 10

0

Solution

W, Z

Exercise 11

1

Exercise 13

-7

Exercise 14

-805

Solution

Z

Exercise 15

-900

Exercise 16

Is the number 0 a positive number, a negative number, neither, or both?

Solution

Neither

Exercise 17

An integer is an even integer if it is evenly divisi­ble by 2. Draw a number line that extends from -5 to 5 and place points at all negative even integers and all positive odd integers.

Exercise 18

Draw a number line that extends from -5 to 5. Place points at all integers that satisfy - 3≤ x < 4- 3≤ x < 4 size 12{"–3" <= " x "<" 4"} {}.

Solution

A number line with hash marks for the numbers -5 to 5. There is a solid dot on the hash mark for -3, an open dot on the hash mark for 4. There is a thick line drawn in between the dots on the line.

Exercise 19

Is there a largest two digit number? If so, what is it?

Exercise 20

Is there a smallest two digit number? If so, what is it?

Solution

Yes, 10

For the pairs of real numbers in the following 5 problems, write the appropriate symbol (<, >, =) in place of the □.

Exercise 21

-7 □ -2

Exercise 22

-5 □ 0

Solution

<

Exercise 23

-1 □ 4

Exercise 24

6 □ -1

Solution

>

Exercise 25

10 □ 10

For the following 5 problems, what numbers can replace m so that the following statements are true?

Exercise 26

- 1 ≤ m ≤- 5- 1 ≤ m ≤- 5 size 12{"–1 " <= " m " <= "–5"} {}, mm size 12{m} {} an integer.

Solution

{-1, 0, 1, 2, 3, 4, 5}

Exercise 27

- 7 < m <- 1- 7 < m <- 1 size 12{"–7 "<" m "<"–1"} {}, mm size 12{m} {} an integer.

Exercise 28

−3≤m<2−3≤m<2 size 12{ - 3 <= m<2} {}, mm size 12{m} {} a natural number.

Solution

{1}

Exercise 29

- 15 < m≤- 1- 15 < m≤- 1 size 12{"–15 "<" m" <= " –1"} {}, mm size 12{m} {} a natural number.

Exercise 30

−5≤m<5−5≤m<5 size 12{ - 5 <= m<5} {}, mm size 12{m} {} a whole number.

Solution

{0, 1, 2, 3, 4}

For the following 10 problems, on the number line, how many units are there between the given pair of numbers?

Exercise 31

0 and 3

Exercise 32

-4 and 0

Solution

4

Exercise 33

-1 and 6

Exercise 34

-6 and 2

Solution

8

Exercise 35

-3 and 3

Exercise 36

Are all positive numbers greater than zero?

Solution

yes

Exercise 37

Are all positive numbers greater than all nega­tive numbers?

Exercise 38

Is 0 greater than all negative number?

Solution

yes

Exercise 39

Is there a largest natural number?

Exercise 40

Is there a largest negative integer?

Solution

yes, -1

Exercises for Review

Exercise 41

((Reference)) Convert 658658 size 12{6 { {5} over {8} } } {} to an improper fraction.

Exercise 42

((Reference)) Find the value: 311311 size 12{ { {3} over {"11"} } } {} of 335335 size 12{ { {"33"} over {5} } } {}.

Solution

95 or 145 or 1.895 or 145 or 1.8 size 12{ { {9} over {5} } " or 1" { {4} over {5} } " or 1" "." 8} {}

Exercise 43

((Reference)) Find the sum of 45+3845+38 size 12{ { {4} over {5} } + { {3} over {8} } } {}.

Exercise 44

((Reference)) Convert 30.06 cm to m.

Solution

0.3006 m

Exercise 45

((Reference)) Find the area of the triangle.

A triangle with base 16mm and height 3mm

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