The Irrational Numbers and the Real Number System 5.4 The Irrational Numbers and the Real Number System

Pythagorean Theorem Pythagoras, a Greek mathematician, is credited with proving that in any right triangle, the square of the length of one side (a2) added to the square of the length of the other side (b2) equals the square of the length of the hypotenuse (c2) . a2 + b2 = c2

Irrational Numbers An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number. Examples of irrational numbers:

Radicals are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.

Principal Square Root The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n. For example,

Perfect Square Any number that is the square of a natural number is said to be a perfect square. The numbers 1, 4, 9, 16, 25, 36, and 49 are the first few perfect squares.

Product Rule for Radicals Simplify: a) b)

Addition and Subtraction of Irrational Numbers To add or subtract two or more square roots with the same radicand, add or subtract their coefficients. The answer is the sum or difference of the coefficients multiplied by the common radical.

Example: Adding or Subtracting Irrational Numbers Simplify: Simplify:

Multiplication of Irrational Numbers Simplify:

Quotient Rule for Radicals

Example: Division Divide: Solution: Divide: Solution:

Rationalizing the Denominator A denominator is rationalized when it contains no radical expressions. To rationalize the denominator, multiply BOTH the numerator and the denominator by a number that will result in the radicand in the denominator becoming a perfect square. Then simplify the result.

Example: Rationalize Rationalize the denominator of Solution:

Homework P. 249 # 9 – 66 (x3)