Numbers and Operations

1. Numbers and Operations

2. Families of numbers

3. Natural Numbers • Counting Numbers • 1, 2, 3, 4, 5, …

4. Whole Numbers • Counting Numbers & Zero • 0, 1, 2, 3, 4, 5, …

5. Integers • Positive and Negative Numbers and Zero • …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

6. Rational Numbers • Can be expressed as the ratio of 2 integers

7. Irrational Numbers • Cannot be expressed as the ratio of 2 integers • Non-terminating, non-repeating integers • Π

8. The Numbrella Complex numbers Real Numbers Imaginary Numbers | Rational Numbers Irrational Numbers | Integers | Whole Numbers | Natural Numbers a+bi Has a real and an imaginary component i—or bi Can be expressed as a fraction Can’t be expressed as a fraction All “non-decimal” values All positive integers and zero All positive integers

9. Examples: 1. Which value is closest to the square root of 273? A. 16.5 B. 136.5 C. 546 D. 74,529 2. Determine the approximate value of the point: 1 2 3 4 5 6 7 8

10. 3. Given the numbers below, what is the order of these numbers from least to greatest? 3.3 A. 3.3 B. 3.3 C. 3.3 D. 3.3

11. Scientific Notation

12. Examples Expand: 2.15 x 10-3 .00215 Write in scientific notation: 1. 3,145,062 2. 2,230,000 3. .000345

13. Examples Simplify: • (1.5 x 102)(4 x 103) 2. 30 x 106 . 10 x 108

14. The diameter of a red blood cell, in inches, is 3 x 10-4. This expression is the same as which of the following numbers? A. 0.00003 B. 0.0003 C. 0.003 D. 3,000 2. A. 0.5 x 102 B. 2 x 102 C. 2 x 100.5 D. 0.5 x 10-2 E. 2 x 10-2

15. Percent

16. Percentages • Convert 20% to a decimal • Convert .45 to a percentage • Convert ¾ to a percentage

17. Examples: • What is 7 percent of 50? • A CD that normally costs $15 is on sale for 20% off. What will you pay

18. Order of Operations

19. PEMDAS A R A N T H E S I S X P O N E N T S M UL T & D I V A D D & S U B From left to right

20. Examples: 30 ÷ 10 • (20 – 15)2

21. GCF and LCM

22. Examples • GCF—greatest common factor • What is the largest number that divides all the given numbers evenly 20 35 60 24 WHAT DO THEY SHARE?

23. Examples • LCM—least common multiple • What is the smallest number that the given numbers go into evenly 20 35 60 24 WHAT IS THE LARGEST VALUE SHOWN IN EACH?

24. Using Proportions

25. What is a proportion and how can you solve a problem with it? • In the town of Centerville, 2.1 centimeters of snow fell in 3 hours. Snow continued to fall at the same rate. How many centimeters of snow had fallen after 7 hours? • 0.3 cm • 2.8 cm • 4.9 cm • 14.7 cm

26. Distance Problems

27. Distance problems

28. Example • It took the Smith’s 5 hours to go 275 miles. What was their average rate of speed? D=rt

29. Estimation What are the critical terms for estimation?

30. Examples: • Mrs. Ditters and her son went to lunch. Their bill came to $27.29. If a fair tip is between 15 and 20 percent, what would be a fair tip to leave their waiter? • $2.00 • $2.72 • $5.00 • $20.00

31. Rule of Exponents

32. Exponent • xn • Base: the number to be multiplied by itself • Exponent: how many times the base is to multiplied by itself => EXPONENT => BASE

33. Examples x2 – 2x2 • (23)4 • (23)(24) • x2 + 2x2 X0 =1

34. Negative Exponent x-n = 1 . xn And A negative exponent means the items belongs on the other side of the fraction bar

35. Example: simplify

36. What is the value of the expression 2( ¼ )2. A. 1/16 B. 1/8 C. 2 2/8 D. 8

37. Simplifying Square Roots

38. Perfect Squares and cubes • Be familiar with the following values

39. Square Roots • Simplifying Square roots use a factor tree and remove groups of 2 (if you are working with cube roots, remove in groups of 3, etc.)

40. Addition and Subtraction • The same number must be under the radical in order to add or subtract

41. Multiplication or Division • Radicals must have the same index in order to multiply or divide