1. Thinking Mathematically Number Theory and the Real Number System 5.4 The Irrational Numbers

2. The Irrational Numbers The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating. An irrational number cannot be written as the ratio of two integers. One of the most well known irrational numbers is the ratio between the circumference and diameter of a circle known as “pi” and written π.

3. The Golden Ratio The golden ratio has the value Fibonnaci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34 which is approximately 1.618 033 988

4. Square Roots The principal “square root” of a positive number, n, is the positive number that when multiplied times itself produces n. This number is written  n. The square root of zero is zero. Some square roots are rational numbers. For example since 6 x 6 = 36,  36 = 6. But, the square root operation provides us with many examples of irrational numbers. E.g.  2

5. Example: Square Roots Exercise Set 5.4 #3, #11  25 = ?  173 = ?

6. Simplify by looking for perfect square factors Exercise Set 5.4 #19 Simplify  80 Simplifying Square Roots

7. The Product Rule for Square Roots If a and b represent nonnegative numbers, then √(ab) =  a  b and  a  b =  (ab). Exercise Set 5.4 #29  3 x  6 = ?

8. The 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.

9. Example: Quotients of Square Roots Exercise Set 5.4 #35

10. Adding and Subtracting Square Roots a  c + b  c = (a + b)  c a  c - b  c = (a - b)  c Important: Exercise Set 5.4 #49  50 -  18 = ?

11. Rationalizing the Denominator The process of rewriting a radical expression to remove the square root from the denominator without changing the value of the expression is called rationalizing the denominator. Exercise Set 5.4 #59 Rationalize the denominator of

12. Thinking Mathematically Number Theory and the Real Number System 5.4 The Irrational Numbers