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Numbers, Sequences, Factors

Integers: . . . , -3, -2, -1, 0, 1, 2, 3, . . .
Rationals: fractions, that is, anything expressable as a ratio of integers
Reals: integers plus rationals plus special numbers such as √ 2, √3 and π

Order Of Operations: PEMDAS
(Parentheses / Exponents / Multiply / Divide / Add / Subtract) Arithmetic Sequences: each term is equal to the previous termplus d
Sequence: t1 , t1 + d, t1 + 2d, . . .

Example: d = 4 and t1 = 3 gives the sequence 3, 7, 11, 15, . . .

Geometric Sequences: each term is equal to the previous term times r

Sequence: t1 , t1 · r, t1 · r2 , . . .

Example: r = 2 and t1 = 3 gives the sequence 3, 6, 12, 24, . . .

Factors: the factors of a number divide into that number without a remainder

Example:the factors of 52 are 1, 2, 4, 13, 26, and 52

Multiples: the multiples of a number are divisible by that number without a remainder

Example: the positive multiples of 20 are 20, 40, 60, 80, . . .

Percents: use the following formula to find part, whole, or percent

part =

percent
100

× whole

Example: 75% of 300 is what?
Solve x = (75/100) × 300 to get 225

Example: 45is what percent of 60?
Solve 45 = (x/100) × 60 to get 75%

Example: 30 is 20% of what?
Solve 30 = (20/100) × x to get 150

average =

sum of terms
number of terms

average speed =

total distance
total time

sum = average × (number of terms)
mode = value in the list that appears most often median = middle value in the list (which must be sorted)
Example: median of {3, 10,9, 27, 50} = 10
Example: median of {3, 9, 10, 27} = (9 + 10)/2 = 9.5

Fundamental Counting Principle:

If an event can happen in N ways, and another, independent event can happen in M ways, then both events together can happen in
N × M ways.

Probability:

probability =

number of desired outcomes number of total outcomes

Example: each SAT math multiple choicequestion has five possible answers, one of which is the correct answer. If you guess the answer to a question completely at ran- dom, your probability of getting it right is 1/5 = 20%.

The probability of two different events A and B both happening is
P (A and B) = P (A) · P (B), as long as the events are independent
(not mutually exclusive).

Powers, Exponents, Roots

xa · xb = xa+b
(xa)b = xa·b

xa /xb = xa−b
· y
(xy)a = xa · ya

1/xb = x−b

n +1, if n is even;

x0 = 1

√xy = √x √

(−1) =

−1, if n is odd.

Factoring, Solving

(x + a)(x + b) = x2 + (b + a)x + ab “FOIL”

a2 − b2 = (a + b)(a − b) “Difference Of Squares”

a2 + 2ab + b2 = (a + b)(a + b)
a2 − 2ab + b2 = (a − b)(a − b)

To solve a quadratic such as x2 +bx+c = 0,first factor the left side to get (x+a1 )(x+a2 ) =
0, then set each part in parentheses equal to zero. E.g., x2 + 4x + 3 = (x + 3)(x + 1) = 0 so that x = −3 or x = −1.

To solve two linear equations in x and y: use the first equation to substitute for a variable in the second. E.g., suppose x + y = 3 and 4x − y = 2. The first equation gives y = 3 − x, so the second equation becomes 4x − (3 −x) = 2 ⇒ 5x − 3 = 2 ⇒ x = 1, y = 2.

Functions

A function is a rule to go from one number (x) to another number (y), usually written

y = f (x).

For any given value of x, there can only be one corresponding value y. If y = kx for some number k (example: f (x) = 0.5 · x), then y is said to be directly proportional to x. If
y = k/x (example: f (x) = 5/x), then y issaid to be inversely proportional to x.

Absolute value:

|x| =

+x, if x ≥ 0;
−x, if x < 0.

Lines (Linear Functions)

Consider the line that goes through points A(x1 , y1 ) and B(x2 , y2 ).
Distance from A to B: /(x2 − x1 )2 + (y2 − y1 )2

Mid-point of the segment AB:

x1 + x2
,
2

y1 + y2
2

Slope of the line:

y2 − y1
x2 − x1

rise
=
run...