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Numbers and Operations

Numbers and Operations. Families of numbers. 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.

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Numbers and Operations

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  1. Numbers and Operations

  2. Families of numbers

  3. 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

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

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

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

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

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

  9. Examples: The approximate value of √7: √4 = 2 √9 = 3 so √7 is approx. 2.6 Determine the approximate value of the point: 1 2 3 4 5 6 7 8 The point is about 3.4

  10. Scientific Notation

  11. Significant Digits Rules • 1-9 are significant • 0’s between digits are significant • 0’s at the end suggest rounding and are not significant • Leading 0’s are not significant • 0’s at the end of a decimal indicate the level of precision • Every digit in scientific notation is significant

  12. Examples • 1024 4 Significant Digits • 1000 1 Significant Digit • .0005 1 Significant Digit • ALWAYS HAVE ONE SIGNIFICANT DIGIT IN FRONT OF THE DECIMAL FOR SCIENTIFIC NOTATION

  13. Examples Expand: 2.15 x 10-3 2.15 x 103 a negative exponent tells you to move the decimal to the left .00215 2150 Write in scientific notation: 3,145,062 2,230,000 .000345 move the decimal so that there is only one digit in front and count the number of spaces you have moved—moving left is positive here and right is negative. 3.145062 x 106 2.23 x 106 3.45 x 10-4

  14. Examples Simplify: do the math on the numeric portion as you normally would, use the rules of exponents on the powers of ten, place in standard scientific notation to finish (one digit before the decimal) (2.75 x 102)(4 x 103) 11 x 105 1.1 x 106 5 x 106 . 10 x 108 .5 x 10-2 5 x 10-1

  15. Percent

  16. Percentages • Convert 20% to a decimal • 20/100= .2 • Convert .45 to a percentage • .45 * 100= 45% • Convert ¾ to a percentage • ¾= .75 .75 * 100=75%

  17. Examples: • What is 7 percent of 50? • .07 * 50 = 3.5 • A CD that normally costs $15 is on sale for 20% off. What will you pay • Option 1 • .2 * 15 = 3 15-3= 12 • Option 2 • If it is 20% off you will pay 80% • .8 * 15 = 12

  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 30 ÷ 10 • 52 30 ÷ 10 • 25 30 Parenthesis Exponents then mult and div From left to right

  21. Absolute Value

  22. Formal definition Absolute value is the distance from the origin and distance is always positive.

  23. Examples • |6| |-7| |-9-3| 6 7 |-12| 12

  24. GCF and LCM

  25. Examples • GCF—greatest common factor • What is the largest number that divides all the given numbers evenly 20 35 60 24 5 4 5 7 6 10 3 8 2 2 2 3 2 5 2 4 2 2 22* 5 5*7 22*3*5 23*3 WHAT DO THEY SHARE? 5 22* 3=12

  26. Examples • LCM—least common multiple • What is the smallest number that the given number go into evenly 20 35 60 24 5 4 5 7 6 10 3 8 2 2 2 3 2 5 2 4 2 2 22* 5 5*7 22*3*5 23*3 WHAT IS THE LAGEST VALUE SHOWN IN EACH? 22*5*7=140 23*3*5=120

  27. Using Proportions

  28. What is a proportion and how can you solve a problem with it? • If Sue charges a flat rate each hour to babysit. If she ears $44 for 8 hours. What will she earn for 5 hours? • PRIMARY RULE: • If you put the $ amount in the numerator on one side put the same value in the numerator on the other side. Etc. cross mult. 220 = 8x 27.5= x Sue will earn $27.50 for 5 hours.

  29. Distance and Work Problems

  30. Distance problems

  31. Example • It took the Smith’s 5 hours to go 275 miles. What was their average rate of speed? D=rt 275 = r(5) 55 = r They went about 55 mph

  32. Work problems • Use the reciprocal of the time for the rate of work W for 1st person=hours worked * rate of work W for 2nd person=hours worked * rate of work Total job always =1 1 = W for 1st person + W for 2ndperson

  33. Example: • John and Sam decide to build a bird house. John and build the bird house in 5 hours working alone. Sam can do it in 8 hours alone. How long will it take if they work together? • It will take them 3.08 hours to make the bird house.

  34. Estimation What are the critical terms for estimation?

  35. Precision • The “detail” associated with a measurement

  36. Calculations with two different levels of precision can only be accurate to the least precise measure.

  37. Accuracy • How correct a measurement is • The smaller the unit of measure the more accurate your measurement

  38. Error • The amount of difference between your measurement and the true value

  39. Examples: Jim bought 3 pounds of nails for $16.25. Which amount is closest to the price per pound? Round off and check above and below 15/3 = 5 and 18/3 = 6 A reasonable values would be between $5 and $6 but closer to $5

  40. Conversions

  41. Length Conversions • 1 inch = 2.54 cm • 12 inches = 1 foot • 3 feet = 1 yard • 5280 feet = 1 mile • How many inches are in 1 yard? • 1 yard = 3 feet 1 foot = 12 inches • 3x12 =36 inches

  42. Fluid Conversions • 3 Teaspoons = 1 Tablespoon • 2 Tablespoons = 1 ounce • 8 ounces = 1 cup • 2 cups = 1 pint • 2 pints = 1 quart • 4 quarts = 1 gallon

  43. Weight Conversions • 16 ounces = 1 pound • 2.2 pounds = 1 kilogram • 2000 pounds = 1 ton

  44. milli- • centi- • -meter = distance • -gram = weight • -liter = fluid • kilo-

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