1. Irrational Numbers

2. Objectives 1.Define irrational numbers 2.Simplify square roots. 3.Perform operations with square roots. 4.Rationalize the denominator.

3. What are Irrational Number? Irrational numbers: – Numbers which cannot be expressed as ration of whole numbers – numbers whose decimal representations are neither terminating nor repeating. Example: π has no last digit in its decimal representation, and it is not a repeating decimal: π ≈ 3.1415926535897932384626433832795…

4. How to Memorize PI “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.” 314159265358979 Guinness World Records for memorizing digits of π: 67,890 digits Latest calculation: 12 ∙ 10 12 digits--December 28, 2013:

5. Square Roots Square root of n, written : a number that when multiplied by itself gives n. i.e., · = n E.g.,, because 6 · 6 = 36. Notice that is a rational number, since 6 is a rational number. Not all square roots are irrational. But √2, is an irrational number.

6. Perfect Square Perfect Square: number which is the square of a whole number E.g., 0 = 0 2 1 = 1 2 4 = 2 2 9 = 3 2 16 = 4 2 The square root of perfect square is a whole number

7. 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.

8. Example Simplify a) b) c)

9. Multiplying Square Roots The product of square roots is the square root of the product.

10. Example a) b) c)

11. Quotient Rule for Square Roots 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

12. Examples a) b)

13. Adding & Subtracting Square Roots Recall the distributive rule for real numbers: a ∙ c + b ∙ c = (a + b) ∙ c Terms with the same square roots can be combined.

14. Examples Simplify: a)b) Solution: a) b)

15. Rationalizing the Denominator a) b)

16. Your Turn Rationalize the following: a)(6 √10) / √2 b)7 √6 ------ √3