1. 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}

2. Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Imaginary Numbers Complex Numbers

3. 1.1.4

4. Review

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. EXAMPLE 3:

9. 4.2 Irrational Numbers

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:

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

13. IRATIONAL NUMBERS a.Can not be written in the form b.They are non-repeating and non-terminating decimals

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. HOMEWORK O PAGES: 211 - 212 O PROBLEMS: 3 – 6, 9, 15, 20, 18, 19 4.2

20. 4.3 Mixed and Entire Radicals

22. 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

23. MULTIPLICATION PROPERTY of RADICALS What do you notice?

24. MULTIPLICATION PROPERTY of RADICALS where n is a natural number, and a and b are real numbers

25. Example 1:

26. Example 2:

27. 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

28. 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

29. 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

30. POWERPOINT PRACTICE PROBLEM Simplify each radical.

31. 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

32. 10√2 is in simplest form because the radical contains no perfect square factors other than 1

33. Mixed Radical: the product of a number and a radical Entire Radical: the product of one and a radical

34. Writing Mixed Radicals as Entire Radicals Any number can be written as the square root of its square! Any number can be also written as the cube root of its cube, or the fourth root of its perfect fourth!

35. Writing Mixed Radicals as Entire Radicals

36. Write each mixed radical as an entire radical

37. POWERPOINT PRACTICE PROBLEM Write each mixed radical as an entire radical