
Greatest common factor
Greatest common factor (GCF) of two numbers is a greatest number that divides both the given numbers

Least common multiple
The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.

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.

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!

Factor tree
A structure used to find the prime factorization of a positive integer

Integer
All positive and negative whole numbers (including zero). That is, the set {... , –3, –2, –1, 0, 1, 2, 3, ...}.

Prime number
A positive integer which has only 1 and the number itself as factors

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.

Natural numbers
 (counting numbers)
 The numbers used for counting. That is, the numbers 1, 2, 3, 4, etc.

Rational numbers
All positive and negative fractions, including integers and socalled improper fractions. Formally, rational numbers are the set of all real numbers that can be written as a ratio of integers with nonzero denominator

Irrational numbers
Real numbers that are not rational. Irrational numbers include numbers such as , , , π, e, etc.

Whole numbers
 nonnegative integers
 The numbers 0, 1, 2, 3, 4, 5, etc.

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.

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.

Even number
An integer that is a multiple of 2. The even numbers are { . . . , –4, –2, 0, 2, 4, 6, . . . }.

Odd number
An integer that is not a multiple of 2. The odd numbers are { . . . , –3, –1, 1, 3, 5, . . . }.

Fraction
A ratio of numbers or variables. Fractions may not have denominator 0.

Proper fraction
A fraction with a smaller numerator than denominator. For example, is a proper fraction.

Improper fraction
A fraction which has a larger numerator than denominator. For example, is an improper fraction.

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, socalled improper fractions are usually preferred to mixed numbers.

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

Product
The result of multiplying a set of numbers or expressions.

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.

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.

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.

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.

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.

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.

Product
The result of multiplying a set of numbers or expressions.

Inverse
The quantity which cancels out the a given quantity. There are different kinds of inverses for different operations.

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.

Negative number
A real number less than zero. Zero itself is neither negative nor positive.

Positive number
A real number greater than zero. Zero itself is not positive.

Nonnegative number
Not negative. That is, greater than or equal to zero.

Number line
A line representing the set of all real numbers. The number line is typically marked showing integer values.

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.

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

Power
The result of raising a base to an exponent. For example, 8 is a power of 2 since 8 is 2^3.

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.

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.

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).

Remainder
The part left over after long division.

Root of a number
A term that can refer to the square root or nth root of a number.

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.

Sum
The result of adding a set of numbers or algebraic expressions.

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.

Zero
The number which indicates no quantity, size, or magnitude. Zero is neither negative nor positive. Note: Zero is the additive identity.

Dividend
 The amount that you want to divide up.
 dividend ÷ divisor = quotient

Divisor
 The number you divide by.
 dividend ÷ divisor = quotient

PEMDAS
 Parentheses, Exponents, Multiplication/Division and Addition/Subraction
 Order of operations

is, was, were will be, same
=



average of x and y
(x+y)/2





"divisible by", "factor", "multiple" suggest ...
usually divisibility problems

Prime factor
 the most basic multiplicative building blocks of a number
 Example: x = 45
 prime factors are 3x3x5

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)

Natural numbers
 The numbers we use to count
 counting numbers
 zero is NOT a natural number

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

Whole numbers
 include all the natural numbers and ZERO
 0,1,2,3, etc.

Least common denominator
Similar concept to LCM, used with fractions in order to add or subtract fractions

Product
 The result of multiplying numbers.
 E.g., 3x5 = 15 (product)

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

Complex fractions
fractions that have a fraction in the numerator and/or the denominator

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

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

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

Converting a repeating decimal number to a fraction
 Place the repeating digits over just as many 9s
 E.g., .48 = 48/99 = 16/33

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

Fractional exponents
 an exponent of 1/2 indidcates that the exponential represents a square root.
 e.g., 36^1/2 = 6

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

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

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)

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

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

Proportion
 Two equal ratios form a proportion
 50 mi / 4 gal = 25 mi / 2 gal
 A proportion has equal crossproducts
 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

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

Percent > decimal
 A percent can be written as a decimal number by moving its decimal point two places to the left
 50% = 0.5

Decimal > percent
 A decimal number can be written as a percent by moving its decimal point two places to the right
 0.25 = 25%

Convert percent to a fraction
 Place the percent over 100 and simplify
 5% = 5/100 = 1/20

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%

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)

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%

Probability
The probability of an event occurring is the ratio of favorable outcomes to possible outcomes

Range (of a data set)
The difference between the highest score and the lowest score

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.

Angles
An angle is formed when two straight lines intersect

Degree
A degree is a unit of measurement in an angle

360 degree angle
Formed by seapping a line segment to form a circle

Right angle
 90 degree angle
 a right angle is drawn with a small square at its center

Acute angle
Is between 0 degrees and 90 degrees

Obtuse angle
An angle greater than 90 degrees

Straight angle
180 degree angle, forming a straight line

Perpendicular
Lines that intersect at a 90 degree angle

Parallel
Lines that run in the same direction and never intersect one another

Plane
In geometry, a flat surface

Polygon
When lines enclose a flat space in a plane they create a polygon (a manysided plane figure)

Regular polygon
When the sides of a polygon are all equal to one another

Irregular polygon
When all the sides of a polygon are not equal to one another

Quadrilateral
A polygon that has four sides

Rectangle
 Irregular polygon
 Opposite sides are equal and parallel
 Four 90 degree angles

Square
 Regular polygon, all four sides are equal
 Opposite sides are paralle
 Four 90 degree angles

Parallelogram
 Irregular polygon
 Opposite sides are equal and parallel
 Opposite angles are equal

Rhombus
 Regular parallelogram; all four sides are equal
 Opposite sides are parallel
 Opposite angles are equal

Trapezoid
 Irregular polygon
 Two opposite sides are parallel

Triangle
Polygon that has three sides and three angles that add up to 180 degrees

Right triangle
 A triangle with one right angle
 The side opposite the right angle is the hypotenuse of the triangle

Hypotenuse
The side opposite the right angle of a right triangle

Isosceles triangle
 A triangle with two equal sides
 The two angles that are opposite the equal sides are equal to one another

Equilateral triangle
 Regular polygon
 A triangle that has three equal sides
 Each angle is 60 degrees

Scalene triangle
A triangle that has three unequal sides

Pythagorean Theorem
The square of a hypotenuse of a right triangle equals the sum of the squares of the other two sides

Circumference
 Of a circle, is the length of the outside edge of the circle
 Circumference = pi x diameter
 Circumference = 2 x pi x radius

Radius
Of a circle, is the distance from its center to its outer edge

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)


circumference / diamter =
pi

Pi in fraction
3 and 1/7 or 22/7

Length
measurement of distance

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

Area
Is the measurement of a surface and is measured in square units

Area of a rectangle
length x width

Area of a parallelogram
base x height

Area of a square
side^squared

Area of a triangle
(b x h)/2

Area of a circle
pi(r^squared)

Surface area
Measure of the total area of every surface of a solid object

Rectangular prism
 brickshaped 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

Volume
 Measure of occupied space.
 Measured in cubic units

Cube
 Rectangular prism having edges of equal length
 The volume of a cube is length x width x height, or side^cubed

Origin
Zero (0) on a number line; the reference point

Signed number
 a positive or negative number
 The sign tells us if the number is greater than zero (positive) or less than zero (negative)

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

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

