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

2. 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

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

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

5. 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,

6. 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.

7. Product Rule for Radicals
Simplify: a) b)

9. 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.

10. Example: Adding or Subtracting Irrational Numbers
Simplify:
Simplify:

12. Multiplication of Irrational Numbers
Simplify:

14. Quotient Rule for Radicals

15. Example: Division
Divide:
Solution:
Divide:
Solution:

17. 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.

18. Example: Rationalize
Rationalize the denominator of
Solution:

20. Homework
P. 249 # 9 – 66 (x3)