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Mastering Irrational Numbers: Theory and Operations

Learn about irrational numbers, simplify square roots, perform operations, rationalize denominators, and enhance math skills.

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Mastering Irrational Numbers: Theory and Operations

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  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.

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