1 / 16

CHAPTER 5

CHAPTER 5. Number Theory and the Real Number System. 5.4. The Irrational Numbers. Objectives Define the irrational numbers. Simplify square roots. Perform operations with square roots. Rationalize the denominator. The Irrational Numbers.

fallont
Download Presentation

CHAPTER 5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CHAPTER 5 Number Theory and the Real Number System

  2. 5.4 • The Irrational Numbers

  3. Objectives • Define the irrational numbers. • Simplify square roots. • Perform operations with square roots. • Rationalize the denominator.

  4. The Irrational Numbers • The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating. • For example, a well-known irrational number is π because there is no last digit in its decimal representation, and it is not a repeating decimal: π≈3.1415926535897932384626433832795…

  5. Square Roots • The principal square root of a nonnegative number n, written , is the positive number that when multiplied by itself gives n. • For example, • because 6 · 6 = 36. • Notice that is a rational number because 6 is a terminating decimal. • Not all square roots are irrational.

  6. Square Roots • A perfect square is a number that is the square of a whole number. • For example, here are a few perfect squares: • 0 = 02 • 1 = 12 • 4 = 22 • 9 = 32 • The square root of a perfect square is a whole number:

  7. The Product Rule For Square Roots • If a and b represent nonnegative numbers, then • The square root of a product is the product of the square roots.

  8. Example: Simplifying Square Roots • Simplify, if possible: • a. b. • c. Because 17 has no perfect square factors (other than 1), it cannot be simplified.

  9. Multiplying Square Roots • If a and b are nonnegative, then we can use the product rule • to multiply square roots. • The product of the square roots is the square root of the product.

  10. Example: Multiplying Square Roots • Multiply: • a. • b. • c.

  11. Dividing Square RootsThe Quotient Rule • 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.

  12. Example: Dividing Square Roots • Find the quotient: • a. • b.

  13. Adding and Subtracting Square Roots • The number that multiplies a square root is called the square root’s coefficient. • Square roots with the same radicand can be added or subtracted by adding or subtracting their coefficients:

  14. Example: Adding and Subtracting Square Roots • Add or subtract as indicated: • a. b. • Solution:

  15. Rationalizing the Denominator • We rationalize the denominator to rewrite the expression so that the denominator no longer contains any radicals.

  16. Example: Rationalizing Denominators • Rationalize the denominator: • a. • b.

More Related