Thinking Mathematically

1. Thinking Mathematically The Irrational Numbers

2. The Irrational Numbers The set of irrational numbers is the set of number whose decimal representations are neither terminating nor repeating. 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.   3.14159

3. 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. Sometimes square roots are whole numbers. For example since 6 x 6 = 36, 36 = 6. If a square root is not a whole number, then it is an irrational number and cannot be written as a ratio of integers.

4. The Product Rule for Square Roots If a and b represent nonnegative numbers, then √(ab) = a• b and a• b = (ab). Example: 2• 5 = (2•5) = 10 7• 7 = (7•7) = 49 = 7 6• 12 = (6•12) = 72 = 36• 2 = 62

5. 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.

6. Adding and Subtracting Square Roots ac + bc = (a + b)c ac - bc = (a - b)c Example: 7 2 + 5 2 = (7 + 5) 2 = 12 2 2 5 - 6 5 = (2 - 6) 5 = -4 5 3 7 + 9 7 - 7 = (3 + 9 - 1) 7 = 11 7

7. 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. Example:

8. Discuss the square root of 2 and the Pythagoreans (500-300 BC) and show it is irrational • Other roots (e.g. cube roots) • Exponents and roots