CET MATHEMATICS REVIEWER

• Real Number System


  A positive, negative, or zero and can be classified as either rational or irrational number
• Types of Real Numbers

  • Rational
  • Irrational
• Subsets of Real Numbers

  • Integers
  • Whole
  • Natural
• Properties of Real Numbers

  • Closure
  • Commutative
  • Associative
  • Distributive
• Positive and Negative Numbers

  A number is denoted as positive if it is directly preceded by a "+" sign or no sign at all. A number is denoted as negative if it is directly preceded by a "-" sign.
• Opposites


  Numbers that are the same distance from zero on the number line but have opposite signs
• Absolute Value (Geometric)


  The distance from a to 0 on the number line
• Absolute Value (Algebraic)


  x, if x > 0

  • x, if x < 0

  0, if x = 0
• Absolute Value Examples

  • |-3|=3

  |x-4|=6 (x=10 or x=-2)

  |x+5|=-3 (no solution)
• Addition with 0

  0+any number=that particular number, that is, 0+0=a for any real number a
• Additive Identity


  0 is called the additive identity
• Reciprocals


  Two numbers are reciprocals of each other if their product is 1
• Negative Exponents


  x^(-n) = 1/x^n
• Laws on Signed Numbers

  • Add same signs: add numbers, copy sign

  Add different signs: subtract lower from larger, copy sign of larger

  Subtract: change sign of subtrahend, proceed to algebraic addition

  Multiply/Divide same signs: multiply/divide, result is positive

  Multiply/Divide different signs: multiply/divide, result is negative
• PEMDAS Examples


  • (42 √4-2)+(6+8÷2) = 30
  • 9² ÷ (-3)√9-58÷2 = -20

  [12(-3)-(-6)] ÷ 2 = -15

  • (-6)² -64-2]+[-42-12] = -51
• Decimal Operations


  Add/Subtract: line up decimal points, add/subtract

  Multiply: multiply numerators, multiply denominators

  Divide: multiply dividend by reciprocal of divisor
• Fraction Operations


  Add/Subtract same denominator: add/subtract numerators, copy denominator

  Add/Subtract different denominators: find LCD, convert fractions, then add/subtract

  Multiply: multiply numerators, multiply denominators

  Divide: multiply dividend by reciprocal of divisor
• Percent to Decimal


  Divide percent by 100
• Decimal to Percent


  Multiply by 100 and add % sign
• Fraction to Decimal


  Divide numerator by denominator
• Scientific Notation


  x = a * 10^c, where 1 <= a < 10 and c is an integer
• Scientific Notation Examples


  • 4,625,235 = 4.625235 * 10^6

  0.0645 = 6.45 * 10^-2
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• In changing fractions to decimals, divide the denominator to the numerator using long division.
• In changing percentages to decimals, divide the percent by 100%
• In changing decimals to percent, multiply the number by 100%
• Scientific notation


  Involves powers of 10. For any positive number x, it can be written as x = ax10° where 1 < a < 10 and c is an integer
• Changing Number to Scientific Notation


  When a number is expressed in this form, it is said to be written in scientific notation
• Samples of changing number to scientific notation

  • 4,625,235 = 4.625235×10
  • 0.0645 = 6.45×10-2
• Changing Scientific Notation to number (Standard Form)

  Move the decimal point in the first factor corresponding to places indicated by the exponent for the power of 10. The decimal point is moved to the right if the exponent is positive and to the left if the exponent is negative.
• Samples of changing scientific notation to standard form

  • 4.621×10 = 46,210
  • 3.695×10² = 0.03695
• Ratio


  Any fraction can be described as a ratio
• Proportion


  An equality of two ratios. It is denoted by a b = c d read as "a is to b as c is to d", where a and d are called extremes and b and c are called means.
• Samples of proportions


  • 4:58:10
  • 1:33:1
• Expressing a ratio in lowest terms

  27:81 → 27 27 1 ÷ 81 27 3
• Finding the value of x in a proportion

  1. 8 = x: 96
• Finding the shortest side in a similar triangle

  1. 15 = x: 21
• Finding the number of years for two brothers' ages to be in a 4:5 ratio

  1. 5 = 5+x:8+x