Presentation on the real number system
- 1. THE REAL NUMBER SYSTEM Natural Numbers: N = { 1, 2, 3, …} Whole Numbers: W = { 0, 1, 2 , 3, ...} Integers: I = {….. -3, -2, -1, 0, 1, 2, 3, ...} Rational Numbers: Irrational Numbers: Q = {non-terminating, non-repeating decimals} π, e ,√2 , √ 3 ... Real Numbers: R = {all rational and irrational} Imaginary Numbers: i = {square roots of negative numbers} Complex Numbers: C = { real and imaginary numbers} Q = a b | a,b ∈I,b ≠0
- 2. Natural Numbers Whole Numbers Integers Rational Numbers IrrationalNumbers Real Numbers ImaginaryNumbers Complex Numbers
- 3. 1.1.4
- 5. Index Radicand When the index of the radical is not shown then it is understood to be an index of 2 Radical =
- 6. EXAMPLE 1: a) Give 4 examples of radicals b) Use a different radicand and index for each radical c) Explain the meaning of the index of each radical
- 7. Evaluate each radical: = 0.5 = 6 = 2 = = 5 EXAMPLE 2:
- 8. Choose values of n and x so that is: a) A whole number b) A negative integer c) A rational number d) An approximate decimal • = 4 = 5/4 = 1.4141… = -3 EXAMPLE 3:
- 10. WORK WITH YOUR PARTNER 1. How are radicals that are rational numbers different from radicals that are not rational numbers? Rational Numbers: These are rational numbers: These are NOT rational numbers: Q = a b | a,b ∈I,b ≠0
- 11. 2. Which of these radicals are rational numbers? Which ones are not rational numbers? How do you know? WORK WITH YOUR PARTNER
- 12. RATIONAL NUMBERS a. Can be written in the form b. Radicals that are square roots of perfect squares, cube roots of perfect cubes etc.. c. They have decimal representation which terminate or repeats Q = a b | a,b ∈I,b ≠0
- 13. IRATIONAL NUMBERS a. Can not be written in the form b. They are non-repeating and non-terminating decimals Q = a b | a,b ∈I,b ≠0
- 14. EXAMPLE 1: Tell whether each number is rational or irrational. Explain how do you know. Rational, because 8/27 is a perfect cube. Also, 2/3 or 0.666… is a repeating decimal. Irrational, because 14 is not a perfect square. Also, √14 is NOT a repeating decimal and DOES NOT terminate Rational, because 0.5 terminates. Irrational, because π is not a repeating decimal and does not terminates
- 15. POWER POINT PRACTICE PROBLEM Tell whether each number is rational or irrational. Explain how do you know.
- 16. EXAMPLE 2: Use a number line to order these numbers from least to greatest Use Calculators!
- 17. -2 -1 0 1 2 3 4 5 EXAMPLE 2: Use a number line to order these numbers from least to greatest
- 18. POWERPOINT PRACTICE PROBLEM Use a number line to order these numbers from least to greatest
- 19. Index Radicand Review of Radicals When the index of the radical is not shown then it is understood to be an index of 2. Radical =
- 20. MULTIPLICATION PROPERTY of RADICALS Use Your Calculator to calculate: What do you notice?
- 21. • WE USE THIS PROPERTY TO: Simplify square roots and cube roots that are not perfect squares or perfect cubes, but have factors that are perfect MULTIPLICATION PROPERTY of RADICALS where n is a natural number, and a and b are real numbers
- 22. Example 1: •
- 23. Example 2: •
- 24. Simplify each radical. Write each radical as a product of prime factors, then simplify. Since √80 is a square root. Look for factors that appear twice
- 25. Simplify each radical. Write each radical as a product of prime factors, then simplify. Since 144∛ is a cube root. Look for factors that appear three times
- 26. Simplify each radical. Write each radical as a product of prime factors, then simplify. Since 162∜ is a fourth root. Look for factors that appear four times
- 27. POWERPOINT PRACTICE PROBLEM Simplify each radical.
- 28. Some numbers such as 200 have more than one perfect square factor: For example, the factors of 200 are: 1, 2 ,4, 5, 8, 10, 20, 25, 40, 50, 100, 200 Since 1, 4, 16, 25, 100, and 400 are perfect squares, we can simplify √400 in several ways: Writing Radicals in Simplest Form
- 29. Writing Radicals in Simplest Form 10√2 is in simplest form because the radical contains no perfect square factors other than 1
- 30. Mixed Radical: the product of a number and a radical Entire Radical: the product of one and a radical 4 6 72
- 31. Writing Mixed Radicals as Entire Radicals Any number can be written as the square root of its square! 2 = 45 = 100 = Any number can be also written as the cube root of its cube, or the fourth root of its perfect fourth! 2 = 45 =
- 32. • Writing Mixed Radicals as Entire Radicals
- 33. Write each mixed radical as an entire radical
- 34. POWERPOINT PRACTICE PROBLEM Write each mixed radical as an entire radical