Exploring Real Numbers

Objectives:

(1)To classify numbers

(2)To compare numbers

Number Groups

• Natural Numbers– 1, 2, 3, …

• Whole Numbers– 0, 1, 2, …

• Integers– … -3, -2, -1, 0, 1, 2, 3, …

• Rational Numbers:– Integers, fractions, finite decimals, repeating decimals

• Irrational Numbers:– Infinite, non-repeating decimals

• Real Numbers:– All rational and irrational numbers

How are repeating decimals rational?

We use the 10x – x rule.

Ex: x = 0.99999….

10x = 9.999999999…- x = - 0.999999999…

9x = 9

9 9

x = 1

How are repeating decimals rational?

Wait a minute…

If x = 0.999… and x = 1, then ????

0.999… = 1

Definitions

• Counterexample: An example that proves a statement false.

• Inequality: a mathematical sentence that compares the value of two expressions using an inequality symbol, such as <, >, or ≠

• Opposites: two numbers that have the same distance from zero

• Absolute Value: a number’s distance from zero

Example #1: Classifying Numbers

a. - (17/31)* Rational

b. 23* Natural, Whole, Integers, Rational

c. 0* Whole, integers, rational

d. 4.581

* rational

Example #2: Using Counterexamples

Is each statement true or false? If it is false, give a counterexample.

a. All whole number are rational numbers.

- True, the easiest way to turn a whole number into a fraction is to put it over 1.

b. The square of a number is always greater than the number.

- False: 0.52 = 0.25. 0.5 is our counterexample

Example #3: Ordering Fractions

Write -3/8, -1/2, and -5/12 in order from least to greatest.

1st Step: write each fraction as a decimal

-3/8 = -0.375

-1/2 = -0.5

-5/12 = -0.41666…

2nd Step: order the decimals from least to greatest

-0.5, -0.41666…, -0.375

3rd Step: replace decimals with their fraction equivalents

-1/2, -5/12, -3/8

Example #4: Finding Absolute Value

Find each absolute value:

.a 12 12

.b -5.6 5.6

.c 5 – 8 -3

3