Section 5.4 The Irrational Numbers and the Real Number System

What You Will Learn Irrational Numbers Radicals Rationalizing Denominators

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: 5.12639537… 6.1011011101111… 0.525225222…

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.

Coefficient The number that multiplies a radical is called the radical’s coefficient. In , the 3 is the coefficient.

Product Rule for Radicals

Simplifying Radicals Write the radical as a product of two radicals. One of the radicals should contain the greatest perfect square that is a factor of the radicand in the original expression. Then simplify the radical containing the perfect square factor.

Example 1: Simplifying Radicals

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 3: Subtracting Radicals with Different Radicands Simplify Solution Simplify so they have the same radicand

Multiplication of Irrational Numbers Use of the product rule for radicals. After the radicands are multiplied, simplify the remaining radical when possible.

Example 4: Multiplying Radicals Simplify.

Quotient Rule for Radicals

Example 5: Dividing Radicals Divide.

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 6: Rationalizing the Denominator Rationalize the denominator.

Example 7: Estimating Square Roots Without a Calculator The following diagram is a sketch of a 16-in. ruler marked using ½ inches. Indicate between which two adjacent ruler marks each of the following irrational numbers will fall.

Example 7: Estimating Square Roots Without a Calculator Solution 7 is close to 9 than to 4, is closer to 3 than to 2. is between 2.5 and 3.

Example 7: Estimating Square Roots Without a Calculator Solution 89 is close to 81, is closer to 9 than to 10. is between 9 and 9.5.

Example 8: Approximating Square Roots Please see your calculator manual for instructions on using the key. Use a scientific calculator to approximate the following square roots. Round each answer to two decimal places.