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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 ): 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 ): 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 ): . . . -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 ): 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       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 π , whose decimal representation does not terminate nor contain a repeating block of digits. An approximation for π is 3.14.

Sample set a

Is every whole number a natural number?

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

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Is there an integer that is not a natural number?

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

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Is there an integer that is a whole number?

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

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Practice set a

Is every natural number a whole number?

yes

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Is every whole number an integer?

yes

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Is every integer a real number?

yes

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Is there an integer that is a whole number?

yes

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Is there an integer that is not a natural number?

yes

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Ordering real numbers

Ordering real numbers

A real number b size 12{b} {} is said to be greater than a real number a size 12{a} {} , denoted b > a size 12{b>a} {} , if b size 12{b} {} is to the right of a size 12{a} {} on the number line. Thus, as we would expect, 5 > 2 size 12{5>2} {} since 5 is to the right of 2 on the number line. Also, - 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 a size 12{a} {} and b size 12{b} {} represent two numbers, then a size 12{a} {} and b size 12{b} {} are related in exactly one of three ways: Either

Equality symbol

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 )

Sample set b

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

- 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.

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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

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Practice set b

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

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

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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.

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

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Exercises

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

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

Neither

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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.

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

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.

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

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

Yes, 10

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

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

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

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

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

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

{1}

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

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

{0, 1, 2, 3, 4}

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

Are all positive numbers greater than zero?

yes

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Are all positive numbers greater than all nega­tive numbers?

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Is 0 greater than all negative number?

yes

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Is there a largest natural number?

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Is there a largest negative integer?

yes, -1

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Exercises for review

( [link] ) Convert 6 5 8 size 12{6 { {5} over {8} } } {} to an improper fraction.

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( [link] ) Find the value: 3 11 size 12{ { {3} over {"11"} } } {} of 33 5 size 12{ { {"33"} over {5} } } {} .

9 5 or 1 4 5 or 1 . 8 size 12{ { {9} over {5} } " or 1" { {4} over {5} } " or 1" "." 8} {}

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( [link] ) Find the sum of 4 5 + 3 8 size 12{ { {4} over {5} } + { {3} over {8} } } {} .

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( [link] ) Convert 30.06 cm to m.

0.3006 m

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( [link] ) Find the area of the triangle.

A triangle with base 16mm and height 3mm

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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