PROPERTIES

OF

REAL NUMBERS

1 ¾ .215 -7 PI

Subsets of real numbers – REVIEW

Natural numbers

numbers used for counting

1, 2, 3, 4, 5, ….

Whole numbers

the natural numbers plus zero

0, 1, 2, 3, 4, 5, …

Integers

the natural numbers ( positive integers ), zero, plus the negative integers

…,-4, -3, -2, -1, 0, 1, 2, 3, 4, …

Rational numbers

numbers that can be written as fractions

decimal representations can either terminate

or repeat

Examples:

fractions: 7/5 -3/2 -4/5

Any whole number can be written as a fraction by placing it over the number 1

8 = 8/1 100 = 100/1

terminating decimals

¼ = .25 2/5 = .4

Repeating decimals

1/3 = .3 2/3 = .6

These will always have a bar over the repeating section.

Irrational numbers

Cannot be written as fractions

Decimal representations do not terminate or repeat

if the positive rational number is not a perfect square, then its square root is irrational

Examples:

Pi - non-repeating decimal

2 - not a perfect square

Rational numbers Irrational numbers

Integers

Whole numbers

Natural numbers

THE REAL NUMBERS

Graphing on a number line

- 2 .3 -2 ¼

Tip: Best to put them as all decimals

Put the square root in the calculator and find its equivalent

-1.414… .333……… -2.25

-3 -2 -1 0 1 2 3

Ordering numbers

Use the < , >, and = symbols

Compare - .08 and - .1

Here again for square roots put them in the calculator and get their equivalents

-.08 = -.282842712475 - .1 = -.316227766017

So: - .1 < - .08 or - .08 > - .1

Properties of Real Numbers

Opposite or additive inverse

sum of opposites or additive inverses is 0

Examples:

400 4 1/5 - .002 - 4/9

-400

Additive inverse of any number a is -a

- 4 1/5 . 002 4/9

Reciprocal or multiplicative inverse

product of reciprocals equal 1

Examples:

400 4 1/5 - .002 - 4/9

1/400

Multiplicative inverse of any number a is 1/a

5/21 - 500 - 9/4

Other Properties:

Addition:

Closure a + b is a real number

Commutative a + b = b + a

4 + 3 = 7` 3 + 4 = 7

numbers can be moved in addition

Associative (a + b) + c = a + (b + c)

(1 + 2) + 3 = 6 1+ (2 + 3) = 6

3 + 3 = 6 1 + 5 = 6

the order in which we add the numbers

does not matter in addition

Identity a + 0 = a

7 + 0 = 7

when you add nothing to a number you

still only have that number

Inverse a + -a = 0

7 + -7 = 0

Multiplication

Closure ab is a real number

Commutative ab = ba

6(4) = 24 4 (6) = 24

When multiplying the numbers may be

switched around, will not affect product

Associative (ab)c = a(bc)

The order in which they are multiplied

does not affect the outcome of the product

(3*4)5 = 60 3(4*5) = 60

12(5) = 60 3(20) = 60

Identity a * 1 = a

One times any number is the number itself

7 * 1 = 7

Inverse a * 1/a = 1

Product of reciprocals is one

7 * 1/7 = 7/7 = 1

DISTRIBUTIVE Property

Combines addition and multiplication

a(b + c) = ab + ac

2(3 + 4) = 2(3) + 2(4)

6 + 8

14

ABSOLUTE VALUE

Absolute value is its distance from zero on the number line.

Absolute value is always positive because distance is always positive

Examples:

-4 =

0 =

-1 * -2 =

4

0

2

Assignment

Page 8 – 9

Problems

34 – 60 even