1. §5.4, Irrational Numbers

2. Learning Targets I will define the irrational numbers
Learning Targets I will define the irrational numbers. I will simplify square roots. I will perform operations with square roots. I will rationalize the denominator.

3. The Irrational Numbers
The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating.
For example, a well-known irrational number is π because there is no last digit in its decimal representation, and it is not a repeating decimal:
π ≈ …

4. Square Roots
The principal square root of a nonnegative number n, written , is the positive number that when multiplied by itself gives n.
For example,
because 6 · 6 = 36.
Notice that is a rational number because 6 is a terminating decimal.
Not all square roots are irrational.

5. Square Roots
A perfect square is a number that is the square of a whole number.
For example, here are a few perfect squares:
0 = 02
1 = 12
4 = 22
9 = 32
The square root of a perfect square is a whole number:

6. The Product Rule For Square Roots
If a and b represent nonnegative numbers, then The square root of a product is the product of the square roots.

7. Example 1: Simplifying Square Roots
Simplify, if possible: a. b. c.
Because 17 has no perfect square factors (other than 1), it cannot be simplified.

8. Multiplying Square Roots
If a and b are nonnegative, then we can use the product rule to multiply square roots. The product of the square roots is the square root of the product.

9. Example 2: Multiplying Square Roots
Multiply: a. b. c.

10. Dividing Square Roots The Quotient Rule
If a and b represent nonnegative real numbers and b ≠ 0, then The quotient of two square roots is the square root of the quotient.

11. Example 3: Dividing Square Roots
Find the quotient: a. b.

12. Adding and Subtracting Square Roots
The number that multiplies a square root is called the square root’s coefficient.
Square roots with the same radicand can be added or subtracted by adding or subtracting their coefficients:

13. Example 4: Adding and Subtracting Square Roots
Add or subtract as indicated:
a. b.
Solution:

14. Rationalizing the Denominator
We rationalize the denominator to rewrite the expression so that the denominator no longer contains any radicals.

15. Example 6: Rationalizing Denominators
Rationalize the denominator: a. b.

16. Homework
Page 272, #18 – 40 (e), 50 – 70 (e)