1. Rational and Irrational Numbers
4-8
The Real Numbers
Evaluating Algebraic Expressions
Rational and Irrational Numbers

2. Rational and Irrational Numbers Essential Question
4-8
The Real Numbers
Rational and Irrational Numbers Essential Question
Evaluating Algebraic Expressions
How do I distinguish between rational and irrational numbers?

3. Evaluating Algebraic Expressions
4-8
The Real Numbers
Vocabulary
Evaluating Algebraic Expressions
real number
irrational number

4. Evaluating Algebraic Expressions
4-8
The Real Numbers
The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.
Evaluating Algebraic Expressions
Irrational numbers
Rational numbers
Real Numbers
Integers
Whole
numbers

5. The Real Number System
Natural Numbers - The set of natural numbers is often referred to as the set of counting numbers, because they are those numbers that we use to count.
  Also notice that 0 is not included in the natural numbers.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …∞} 5

6. The Real Number System
Whole Numbers - the natural numbers together with 0.
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …∞} 6

7. The Real Number System
Integers - If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers.
{…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …} 7

8. Evaluating Algebraic Expressions
4-8
The Real Numbers
Evaluating Algebraic Expressions
Our next group of numbers is called the rational numbers. Rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.
4 5 23
3 = 3.8
= 0.6
1.44 = 1.2
So...rational numbers include fractions, terminating decimals, and repeating decimals.

9. Evaluating Algebraic Expressions
4-8
The Real Numbers
So, the rational numbers include the natural numbers, whole numbers, and integers.
Evaluating Algebraic Expressions
Reals
Rationals
1/2
-2.65
Integers -3
-19
Wholes
1, 2, 3...
Naturals

10. Rational Numbers {p/q : p and q are integers, q is not zero} Number
As a fraction
Rational? 5
5/1
.75
.001
1/1000 …
2/9 … ? 10

11. Evaluating Algebraic Expressions
4-8
The Real Numbers
Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers (a simple fraction). If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.
Evaluating Algebraic Expressions
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

12. What other examples of irrational numbers are there?
4-8
The Real Numbers
What other examples of irrational numbers are there?
Evaluating Algebraic Expressions 12

13. Evaluating Algebraic Expressions
4-8
The Real Numbers
Let’s Review all of our sets of numbers in the real number system - Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.
Evaluating Algebraic Expressions
Reals
Rationals
-2.65
Integers -3
-19
Wholes
Irrationals
Naturals
1, 2, 3...

14. Evaluating Algebraic Expressions
4-8
The Real Numbers
Additional Example 1: Classifying Real Numbers
Write all classifications that apply to each number.
Evaluating Algebraic Expressions A. 5
5 is a whole number that is not a perfect square.
irrational, real B.
–12.75
–12.75 is a terminating decimal.
rational, real
16 2
= = 2
4 2
16 2 C.
whole, integer, rational, real

15. Evaluating Algebraic Expressions
4-8
The Real Numbers
Check It Out! Example 1
Write all classifications that apply to each number.
Evaluating Algebraic Expressions A. 9
9 = 3
whole, integer, rational, real B.
–35.9
–35.9 is a terminating decimal.
rational, real
81 3
= = 3
9 3
81 3 C.
whole, integer, rational, real

16. Evaluating Algebraic Expressions
4-8
The Real Numbers
Evaluating Algebraic Expressions
A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

17. Evaluating Algebraic Expressions
4-8
The Real Numbers
Additional Example 2: Determining the Classification of All Numbers
Evaluating Algebraic Expressions
State if each number is rational, irrational, or not a real number. A. 21
irrational
0 3
0 3
= 0 B.
rational

18. Evaluating Algebraic Expressions
4-8
The Real Numbers
Additional Example 2: Determining the Classification of All Numbers
Evaluating Algebraic Expressions
State if each number is rational, irrational, or not a real number.
4 0 C.
not a real number

19. Evaluating Algebraic Expressions
4-8
The Real Numbers
Check It Out! Example 2
Evaluating Algebraic Expressions
State if each number is rational, irrational, or not a real number. A. 23
23 is a whole number that is not a perfect square.
irrational
9 0 B.
undefined, so not a real number

20. Evaluating Algebraic Expressions
4-8
The Real Numbers
Check It Out! Example 2
Evaluating Algebraic Expressions
State if each number is rational, irrational, or not a real number.
8 9 = C.
rational