GMAT - basic math definitions

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GMAT - basic math definitions
2011-10-31 23:40:49
GMAT math definitions

Basic math definitions and properties
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  1. Greatest common factor
    Greatest common factor (GCF) of two numbers is a greatest number that divides both the given numbers
  2. Least common multiple
    The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.
  3. Composite number
    A positive integer that has factors other than just 1 and the number itself. For example, 4, 6, 8, 9, 10, 12, etc. are all composite numbers. The number 1 is not composite.
  4. Factor
    • Any integer which divides evenly into a given integer. For example, 8 is a factor of 24.
    • Factors of a number are all the terms that divide into the number cleanly.
    • Factors can be created by multiplying any combination of prime factors together.
    • Always count "1" as a factor!
  5. Factor tree
    A structure used to find the prime factorization of a positive integer
  6. Integer
    All positive and negative whole numbers (including zero). That is, the set {... , –3, –2, –1, 0, 1, 2, 3, ...}.
  7. Prime number
    A positive integer which has only 1 and the number itself as factors
  8. Real numbers
    • Includes all rational and irrational numbers
    • All numbers on the number line. This includes (but is not limited to) positives and negatives, integers and rational numbers, square roots, cube roots , π (pi), etc.
  9. Natural numbers
    • (counting numbers)
    • The numbers used for counting. That is, the numbers 1, 2, 3, 4, etc.
  10. Rational numbers
    All positive and negative fractions, including integers and so-called improper fractions. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator
  11. Irrational numbers
    Real numbers that are not rational. Irrational numbers include numbers such as , , , π, e, etc.
  12. Whole numbers
    • nonnegative integers
    • The numbers 0, 1, 2, 3, 4, 5, etc.
  13. Denominator
    • The bottom part of a fraction. For 2/5, the denominator is 5.
    • Cannot have zero in the denominator (e.g., 6/0 and 0/0 have no answer)
    • The fraction 10/0 can't equal anything. There is no number you can multiply by 0 and get 10 as your answer. The fraction 10/0 is undefined.
    • FYI: As a result we say is 0/0indeterminate, which is a special kind of undefined expression.
  14. Numerator
    • The top part of a fraction. For 12/31, the numerator is 12.
    • OK to have 0 on top (in the numerator)
    • The fraction 0/2 because 2·0 = 0.
    • FYI: As a result we say 0/0 is indeterminate, which is a special kind of undefined expression.
  15. Even number
    An integer that is a multiple of 2. The even numbers are { . . . , –4, –2, 0, 2, 4, 6, . . . }.
  16. Odd number
    An integer that is not a multiple of 2. The odd numbers are { . . . , –3, –1, 1, 3, 5, . . . }.
  17. Fraction
    A ratio of numbers or variables. Fractions may not have denominator 0.
  18. Proper fraction
    A fraction with a smaller numerator than denominator. For example, is a proper fraction.
  19. Improper fraction
    A fraction which has a larger numerator than denominator. For example, is an improper fraction.
  20. Mixed numbers
    • A number written as the sum of an integer and a proper fraction. For example, 5¾ is a mixed number. 5¾ is the sum 5 + ¾.
    • Note: In math courses beyond Algebra I, so-called improper fractions are usually preferred to mixed numbers.
  21. FOIL Method
    • A technique for distributing two binomials. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product. Inner means multiply the innermost two terms. Last means multiply the terms which occur last in each binomial. Then simplify the products and combine any like terms which may occur.
    • (x + 2)(x + 5) = x·x + x·5 + 2·x + 2·5 First Outer Inner Last
    • = x2 + 7x + 10
  22. Product
    The result of multiplying a set of numbers or expressions.
  23. Fundamental Theorem of Arithmetic
    The assertion that prime factorizations are unique. That is, if you have found a prime factorization for a positive integer then you have found the only such factorization. There is no different factorization lurking out there somewhere.
  24. Least common denominator
    The smallest whole number that can be used as a denominator for two or more fractions. The least common denominator is the least common multiple of the original denominators.
  25. Median
    The median of a set of numbers is the value for which half the numbers are larger and half are smaller. If there are two middle numbers, the median is the arithmetic mean of the two middle numbers.
  26. Mean
    Another word for average. Mean almost always refers to arithmetic mean. In certain contexts, however, it could refer to the geometric mean, harmonic mean, or root mean square.
  27. Mode
    • The number that occurs the most often in a list.
    • A data set can have no mode, one mode, or more than one mode.
  28. Reciprocal
    • Multiplicative Inverse of a Number
    • The reciprocal of x is 1/x. In other words, a reciprocal is a fraction flipped upside down. Multiplicative inverse means the same thing as reciprocal.
    • Note: The product of a number and its multiplicative inverse is 1.
  29. Product
    The result of multiplying a set of numbers or expressions.
  30. Inverse
    The quantity which cancels out the a given quantity. There are different kinds of inverses for different operations.
  31. Additive inverse of a number
    The opposite of a number. For example, the additive inverse of 12 is –12. The additive inverse of –3 is 3. Formally, the additive inverse of x is –x. Note: The sum of a number and its additive inverse is 0.
  32. Negative number
    A real number less than zero. Zero itself is neither negative nor positive.
  33. Positive number
    A real number greater than zero. Zero itself is not positive.
  34. Nonnegative number
    Not negative. That is, greater than or equal to zero.
  35. Number line
    A line representing the set of all real numbers. The number line is typically marked showing integer values.
  36. Perfect number
    • A number n for which the sum of all the positive integer factors of n which are less than n add up to n.
    • For example, 6 and 28 are perfect numbers. The number 6 has factors 1, 2, and 3, and 1 + 2 + 3 = 6. The number 28 has factors 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28.
  37. Perfect square
    Any number that is the square of a rational number. For example, 0, 1, 4, 9, 16, 25, etc. are all perfect squares
  38. Power
    The result of raising a base to an exponent. For example, 8 is a power of 2 since 8 is 2^3.
  39. Quotient
    • The result of dividing two numbers or expressions. For example, the 40 divided by 5 has a quotient of 8.
    • Note: 43 divided by 5 has a quotient of 8 and a remainder of 3.
  40. Ratio
    The result of dividing one number or expression by another. Sometimes a ratio is written as a proportion, such as 3:2 (three to two). More often, though, ratios are simplified according to the standard rules for simplifying fractions or rational expressions.
  41. Relatively prime
    Describes two numbers for which the only common factor is 1. In other words, relatively prime numbers have a greatest common factor (gcf) of 1. For example, 6 and 35 are relatively prime (gcf = 1). The numers 6 and 8 are not relatively prime (gcf = 2).
  42. Remainder
    The part left over after long division.
  43. Root of a number
    A term that can refer to the square root or nth root of a number.
  44. Square root
    • A nonnegative number that must be multiplied times itself to equal a given number.
    • Note: square root of x never refers to a negative number.
  45. Sum
    The result of adding a set of numbers or algebraic expressions.
  46. Twin primes
    Prime numbers that are two apart from each other, such as 3 and 5. Other examples are 11 and 13, 17 and 19, 101 and 103.
  47. Zero
    The number which indicates no quantity, size, or magnitude. Zero is neither negative nor positive. Note: Zero is the additive identity.
  48. Dividend
    • The amount that you want to divide up.
    • dividend ÷ divisor = quotient
  49. Divisor
    • The number you divide by.
    • dividend ÷ divisor = quotient
  50. PEMDAS
    • Parentheses, Exponents, Multiplication/Division and Addition/Subraction
    • Order of operations
  51. is, was, were will be, same
  52. difference, less
    • -
    • subtract
  53. of, times, product
    • x
    • multiply
  54. average of x and y
  55. total, sum, add
  56. y less than x
  57. quotient, proportion
  58. ratio of x to y
  59. "divisible by", "factor", "multiple" suggest ...
    usually divisibility problems
  60. Prime factor
    • the most basic multiplicative building blocks of a number
    • Example: x = 45
    • prime factors are 3x3x5
  61. Exponent
    • shorthand for multiplying or dividing the same number by itself multiple times.
    • Exponential terms consist of a base (number being multiplied) and the exponent (the number of times the number is being multipled)
  62. Natural numbers
    • The numbers we use to count
    • counting numbers
    • zero is NOT a natural number
  63. Rules for combining exponential terms
    • exponential terms can only be combined if they have a common base or a common exponent
    • * it is often necessary to change the base of a term in order to have the common bases necessary to combine terms
  64. Whole numbers
    • include all the natural numbers and ZERO
    • 0,1,2,3, etc.
  65. Least common denominator
    Similar concept to LCM, used with fractions in order to add or subtract fractions
  66. Product
    • The result of multiplying numbers.
    • E.g., 3x5 = 15 (product)
  67. Reciprocal fractions
    • Two fractions, in which the numerator of each fraction equals the denominator of the other
    • E.g., 2/3 and 3/2 are reciprocals, if multiplied together, they equal 1
  68. Complex fractions
    fractions that have a fraction in the numerator and/or the denominator
  69. Rules of multiplying fractions
    • All terms must be proper or improper fractions - mixed or whole numbers must be converted to improper fractions.
    • Cancel common factors from the numerators and denominators
    • Multiply numerators
    • Multiply denominators
  70. Rules of dividing fractions
    • All terms must be proper or improper fractions - convert any mixed or whole numbers to improper fractions
    • Change the division to multiplication and replace the divisor with its reciprocal
    • Cancel common factors from the numerators and denominators
    • Multiply numerators
    • Multiply denominators
  71. Multiplying decimal numbers
    • Multiply as if there were no decimal point
    • When you get the result, count the number of digits to the right of the deciaml points in the numbers that were multiplied
    • In the answer, place the decimal point the same number of digits to the right of the point as you counted in the previous step
  72. Converting a repeating decimal number to a fraction
    • Place the repeating digits over just as many 9s
    • E.g., .48 = 48/99 = 16/33
  73. Powers of a number
    • When several identical numbers are multiplied together the result is called a power of that number
    • The same number could be a power of more than one number.
    • E.g., 2x2 = 8
    • 2x2x2x2= 16
    • both 8 and 16 are powers of 2
    • *a power of a number can be less than its base, e.g., 0.5^2 = 0.25
  74. Fractional exponents
    • an exponent of 1/2 indidcates that the exponential represents a square root.
    • e.g., 36^1/2 = 6
  75. Commutative property
    • when the position of numbers can be changed in an expression without changing the result of an operation
    • addition and multiplication both have commutative properties
    • e.g., 5+2 = 2+5
  76. Associative property
    • When the order in which operations are performed does not affect the final result
    • addition and multiplication both have associative properties
    • e.g., 2+(3+4) = (2+3)+4
  77. Distributive property
    • When an operation can be distributed over all the terms within a parenthesis
    • Multiplication is distributive over addition and subtraction, e.g., 2(3+4) = (2x3) + (2x4)
  78. Inequality symbols < >
    • An inequality symbol always points to the smaller number
    • The "mouth" of the symbol wants to "Eat" the larger number
    • Inequality symbols can be used to describe a range of number, e.g., 0<n --> numbers greater than 0
    • 0<n<5 --> numbers between 0 and 5
  79. Ratio
    • A ratio is a comparison of two quantities
    • e.g., 8:10 or 8 to 10
    • A ratio is simplified the same way that a fraction is simplified
    • A ratio can be inverted and still be true
    • Two equal ratios form a proportion
  80. Proportion
    • Two equal ratios form a proportion
    • 50 mi / 4 gal = 25 mi / 2 gal
    • A proportion has equal cross-products
    • A different, but true proportion results when both fractions are replaced by their reciprocals
    • A different, but true proportion results when the numbers on opposite corners are swapped
  81. Percent
    • Per 100
    • 100% of something represents all of it
    • One percent (1%) of something represents 1/100 of it; 1% literally means 1 per hundred)
    • Percents let us compare different parts of a whole as if it were made up of 100 parts, no matter how many actual items make up the whole thing under consideration
  82. Percent --> decimal
    • A percent can be written as a decimal number by moving its decimal point two places to the left
    • 50% = 0.5
  83. Decimal --> percent
    • A decimal number can be written as a percent by moving its decimal point two places to the right
    • 0.25 = 25%
  84. Convert percent to a fraction
    • Place the percent over 100 and simplify
    • 5% = 5/100 = 1/20
  85. Convert a fraction to a percent
    • First convert the fraction to a decimal number
    • then convert the decimal to a percent
    • e.g., 1/10 = 0.1 = 10%
  86. Every percent problem has this format ...
    • percent of whole = part (decimal number X whole = part), where one of these three terms will be missing:
    • - Find 25% of 80 (find the PART)
    • - 25% of what number = 20 (find the WHOLE)
    • - What percent of 80 is 20? (Find the PERCENT)
    • "of" = "times" (multiply)
  87. Problems involving the percent of increase or decrease can be solved by ...
    • Percent x Initial Value = Increase/Decrease
    • e.g., if the value of something went from 80 to 100, by what percent did it increase?
    • ? x 80 = 20
    • Increased by 25%
  88. Probability
    The probability of an event occurring is the ratio of favorable outcomes to possible outcomes
  89. Range (of a data set)
    The difference between the highest score and the lowest score
  90. Line segment
    Is a line connecting two points. The name of a line segment is written with a line over the names of its two ends.
  91. Angles
    An angle is formed when two straight lines intersect
  92. Degree
    A degree is a unit of measurement in an angle
  93. 360 degree angle
    Formed by seapping a line segment to form a circle
  94. Right angle
    • 90 degree angle
    • a right angle is drawn with a small square at its center
  95. Acute angle
    Is between 0 degrees and 90 degrees
  96. Obtuse angle
    An angle greater than 90 degrees
  97. Straight angle
    180 degree angle, forming a straight line
  98. Perpendicular
    Lines that intersect at a 90 degree angle
  99. Parallel
    Lines that run in the same direction and never intersect one another
  100. Plane
    In geometry, a flat surface
  101. Polygon
    When lines enclose a flat space in a plane they create a polygon (a many-sided plane figure)
  102. Regular polygon
    When the sides of a polygon are all equal to one another
  103. Irregular polygon
    When all the sides of a polygon are not equal to one another
  104. Quadrilateral
    A polygon that has four sides
  105. Rectangle
    • Irregular polygon
    • Opposite sides are equal and parallel
    • Four 90 degree angles
  106. Square
    • Regular polygon, all four sides are equal
    • Opposite sides are paralle
    • Four 90 degree angles
  107. Parallelogram
    • Irregular polygon
    • Opposite sides are equal and parallel
    • Opposite angles are equal
  108. Rhombus
    • Regular parallelogram; all four sides are equal
    • Opposite sides are parallel
    • Opposite angles are equal
  109. Trapezoid
    • Irregular polygon
    • Two opposite sides are parallel
  110. Triangle
    Polygon that has three sides and three angles that add up to 180 degrees
  111. Right triangle
    • A triangle with one right angle
    • The side opposite the right angle is the hypotenuse of the triangle
  112. Hypotenuse
    The side opposite the right angle of a right triangle
  113. Isosceles triangle
    • A triangle with two equal sides
    • The two angles that are opposite the equal sides are equal to one another
  114. Equilateral triangle
    • Regular polygon
    • A triangle that has three equal sides
    • Each angle is 60 degrees
  115. Scalene triangle
    A triangle that has three unequal sides
  116. Pythagorean Theorem
    The square of a hypotenuse of a right triangle equals the sum of the squares of the other two sides
  117. Circumference
    • Of a circle, is the length of the outside edge of the circle
    • Circumference = pi x diameter
    • Circumference = 2 x pi x radius
  118. Radius
    Of a circle, is the distance from its center to its outer edge
  119. Diameter
    • Of a circle, is the length of a line that cuts the circle in half.
    • The diameter of a circl passes through the center of the circle and is twice as long as the radius
    • Diameter/radius = 2
    • Diameter = 2(radius)
  120. diameter/radius =
  121. circumference / diamter =
  122. Pi in fraction
    3 and 1/7 or 22/7
  123. Length
    measurement of distance
  124. Perimeter of a polygon
    • Is the sum of the lengths of all of its sides
    • The perimeter of a rectangle = 2 x(length + width)
    • The perimeter of a square = 4 x side
    • The perimeter of a rhombus = 4x side
    • The perimeter of an equilateral triangle = 3x side
  125. Area
    Is the measurement of a surface and is measured in square units
  126. Area of a rectangle
    length x width
  127. Area of a parallelogram
    base x height
  128. Area of a square
  129. Area of a triangle
    (b x h)/2
  130. Area of a circle
  131. Surface area
    Measure of the total area of every surface of a solid object
  132. Rectangular prism
    • brick-shaped object
    • the surface area is the sum of the areas of all six surfaces
    • to quickly find the area, find the area of the three visible surfaces and then double the result
    • The volume = length x width x height
  133. Volume
    • Measure of occupied space.
    • Measured in cubic units
  134. Cube
    • Rectangular prism having edges of equal length
    • The volume of a cube is length x width x height, or side^cubed
  135. Origin
    Zero (0) on a number line; the reference point
  136. Signed number
    • a positive or negative number
    • The sign tells us if the number is greater than zero (positive) or less than zero (negative)
  137. Absolute value
    • The numeric term in a signed number
    • The absolute value of any number is always positive because it is the distance of that number from zero, regardless of whether it is to the right or left
  138. Opposite numbers
    • The opposite of a signed number is the number with the same absolute value but a different sign
    • e.g., 5 and -5