Math

Integer

A number without fractional or decimal parts, including positive and negative while numbers and zero. All integers are multiples of 1

Commutative Laws of Addition and Multiplication

Addition and subtraction are both commutative, which means that switching the order of any two numbers being added or multiplied together does not affect the result.

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a + b = b + a

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xyz = zyx

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Division and subtraction are not commutative; switching the order of the numbers changes the results.

Associative Law of Addition and Multiplication

Addition and multiplication are also associative; regrouping the numbers does not affect the result.

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(a + b) + c = a + (b+c)

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(ab)c = a(bc)

The Distributive Law of Multiplication (and the numerator in division)

This allows you to distribute a factor over numbers that are added or subtracted. You do this by multiplying that factor by each number in the group.

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a(b + c) = ab + ac

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This law works for the numerator in division as well:

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(a + b) / c = a/c + b/c

To add two numbers that have the same sign…

add the number parts and keep the sign

To add two numbers that have different signs…

find the difference between the number parts and keep the sign of the number whose bumber part is larger

Multiplication and Division of Positive and Negative Numbers

same sign → positive result

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different sign → negative result

Absolute Value

the value of a number without its sign. The absolute value of a number can be thought of as the b=number’s distance from zero on the number line.

Properties of Zero

1\.) Adding zero to or subtracting it from a number does not change the number.

2\.) Subtracting a number from zero obviously changes the numbers sign.

3\.) The product of zero and any number is zero.

4\.) Division by zero is undefined or “It can’t be done”. When you are given a fraction that has an algebraic expression in the denominator, be sure that the expression cannot equal zero.

Properties of 1 and -1

1\.) Multiplying or dividing a number by 1 does not change the number.

2\.) Multiplying or dividing a nonzero number by -1 change the sign of the number.

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(a)(-1) = -a

(-1)(a) = -a

a / -1 = -a

Multiple

A multiple is the product of a specified number and an integer.

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Multiples do not have to be integers. All multiples must be the product of a specific number and an integer.

Simple Rules to Determine if one number is evenly divisible by another:

* An integer is divisible by 2 if its last digit is divisible by 2
* An integer is divisible by 3 if its digits add up to a multiple of 3
* An integer is divisible by 4 if its last two digits are a multiple of 4
* An integer is divisible by 5 if its last digit is 0 or 5
* An integer is divisible by 6 if its divisible by both 2 and 3
* An integer is divisible by 9 if its digits add up to a multiple of 9

Rules for Odds and Evens

* Odd + Odd = Even
* Even + Even = Even
* Odd + Even = Odd
* Odd x Odd = Odd
* Even x Even = Even
* Odd x Even = Even

Factors

The factors of an integer are the positive and negative integers by which it is evenly divisible (leaving no remainder)

Greatest Common Factor

The largest factors that a group of integers shares.

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To find the GCF, break down both integers into their prime factorizations and multiply all of the prime factors that they have in common.

Prime Number

An integer greater than 1 that has only two factors: itself and 1.

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The number 2 is the smallest prime number and the only even prime number.

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The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Prime Factorization

The expression of the number as the product of its prime factors (the factors that are prime numbers).

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Figure out one pair of factors and then determine their factors, continuing the process until you’re left with only prime numbers. Those primes will be the prime factorization.

Least Common Multiple

The smallest number that is a multiple of each of the integers.

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1\.) Determine the prime factorization of each integer

2\.) Write out each prime number the maximum number of times that it appears in any one of the prime factorizations

3\.) Multiply those prime numbers together to get the LCM of the original integers

Remainders

The remainder is what is “left over” in a division problem. A remainder is always smaller than the number you are dividing by.

Rules of Operations with Exponents

1\.) To multiply two powers with the same base, keep the base and add the exponents together.

2\.) To divide two powers with the same base, keep the base and subtract the exponent if the denominator from the exponent of the numerator.

3\.) To raise a power to another power, multiply the exponents.

4\.) To multiply two powers with different bases by the same power, multiply the bases together and raise to the power.

5\.) A base with a negative exponent indicates the reciprocal of that base to the positive value of the exponent.

6\.) Raising any nonzero number to an exponent of zero equals 1.

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Honorable mention:

* Raising a fraction between zero and one to a power produces a smaller result.
* Raising a negative number to an even power produces a positive result.
* Raising a negative number to an odd power gives a negative result.
* Raising an even number to a positive integer exponent gives an even number. Raising an odd number to any integer greater than or equal to 0 gives an odd number.

Addition and Subtraction of Radicals

Only like radicals can be added or subtracted from one another.

Multiplication and Division of Radicals

To multiply or divide one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs

Simplifying Radicals

If the number inside the radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square.

FOIL

First, Outside, Inside, Last

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Used for multiplying binomials.

Percent Decrease

Percent Decrease = (Actual Decrease / Original Amount) x 100%

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When two different positive values are increased or decreased by the same amount, the percent change will be greater for the smaller starting value.

Helpful Takeaways

* When a figure is described but does not appear, take a moment to draw it out. A concrete visualization can make it easier to grasp abstract concepts.
* The 7:24:25 Pythagorean triple may appear on the GRE. Knowing and recognizing the four Pythagorean triples of 3:4:5, 5:12:13, 8:15:17, and 7:24:25 can save time on Test Day.
* When presented with a line expressed in an unconventional form, the best course of action is often to convert it to slope-intercept form (y = mx + b).
* You may encounter points with negative values when working with figures in the coordinate plane. Remember that lengths, like any distance, are absolute values.
* In 30-60-90 triangle I, the hypotenuse, which is the longest side of a right triangle, is twice a long as the side opposite the 30° angle. So, Quantity A is 2 × 1 = 2. In 45-45-90 triangle II, The hypotenuse is 2‾√ times the length of either leg. So, Quantity B is 2‾√×2‾√=2. Since both quantities are 2, (C) is correct.

TAKEAWAY: Know the ratios of the sides for 30-60-90 and 45-45-90 right triangles.
* The formula for the area of a trapezoid is A = ((b1+b2) / 2) ×h, which is the average of the two bases times the height.
* When you can make the two quantities look alike by stating them in terms of the same variable or variables, the comparison is greatly simplified.
* The formula for the circumference of a circle is *C* = π*d*
* To find the number of terms in a sequence of consecutive integers, take the difference between the largest and the smallest and add 1. Here, 18 – 8 + 1 = 11.

Sets

groups of values that have some common property. The items within a set are called elements or members.

Finite vs. Infinite Sets

Finite: all the elements in a set can be counted

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Infinite: The elements in a set are limitless

Lists

a finite set where the order of the elements matters and duplicate members can be included

Intersection of two sets

a set that consists of all the elements that are contained in both sets (overlap) A^B

Union of two sets

the set of all the elements that are elements of either or both sets.

Mutually Exclusive

When sets have no common elements

Inclusion- Exclusion Principle

|A U B| = |A| + |B| - |A ^ B|

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|A U B| = |A| + |B|

The Combination Formula

n! / k! (n-k)!

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A combination question asks you how many unordered subgroups can be formed from a larger group.

Probability

the likelihood that an event will occur

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number of desired outcomes/ number of possible outcomes

Probability of Multiple Events

P(A or B) = P(A) + P(B) - P(A and B)

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P(A and B) = P(A) + P(B)

Vertex

the point at which two lines or line segments intersect to form an angle

Acute angle

measures less than 90 degrees

Obtuse angle

measures between 90 - 180 degrees

Complementary angles

their measures sum to 90 degrees

Supplementary angles

their measures sum to 180 degrees

Adjacent angles vs. Opposite angles

angles that are next to one another

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opposite angles are equal to one another because each of them is supplementary to the same adjacent angle

Transversal

a line that intersects two parallel lines

Polygons

A closed figure whose sides are straight line segments. Named according to their numbe of sides.

Perimeter

The distance around a shape; the sum of lengths of its sides.

Vertex of a polygon

A point where two sides intersect. Polygons are named by assigning each vertex a letter and listing them in order.

Diagonal of a polygon

A line segment connecting any two nonadjacent vertices

Regular polygon

a polygon with sides of equal length and interior angles of equal measures

Triangle

A polygon with three straight sides and three interior angles

Right triangle

A triangle with one interior angle of 90 degrees

Hypotenuse

The longest side of a right triangle. The hypotenuse is always opposite the right angle.

Isosceles Triangle

A triangle with two equal sides, which are opposite two equal angles.

Legs

The two equal sides of an isosceles triangle or the two shorter sides of a right triangle (the ones forming the right angle). Note: the third, unequal sides of an isosceles triangle is called the base.

Equilateral triangle

A triangle whose three sides are all equal in length and whose three interior angles each measure 60 degrees.

Triangle Inequality Theorem

the sum of the length of two sides of a triangle is greater than the length of the third side

Area of a triangle

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Perimeter of triangle

1/2(base)(height)

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No special equations; just sum up all three side lengths

Area of right triangle

1/2(Leg 1)(Leg 2)

Pythagorean Triples

3, 4, 5

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6, 8, 10

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5, 12, 13

Isosceles Right Triangle

45 - 45 - 90

x : x: xroot2

30 - 60 - 90 right triangle

30 - 60 - 90

x: xroot2: 2x

Quadrilateral

a four-sided polygon; the four interior angles sum to 360 degrees

Trapezoid

a quadrilateral with at least one pair of parallel sides

Parallelogram

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Area of a parallelogram

a quadrilateral with two pairs of parallel sides

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opposite sides are equal in length; opposite angles are equal in measure; angles that are not opposite are supplementary to each other

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Base x Height

Rectangle

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Perimeter of a rectangle

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Area of a rectangle

A parallelogram with four right angles

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opposite sides are equal; diagonals are equal

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2(Length + Width)

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Length x Width

Rhombus

a parallelogram with four equal sides

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opposite angles are equal to one another but they do not have to be right angles

Square

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Perimeter of a square

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Area of a square

a rectangle with equal sides

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4(Side)

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Side^2

Circle

The set of all pints in a plane at the same distance from a certain point

Diameter

A line segment that connects two points on the circle and passes through the center of the circle.

Radius

A line segment that connect the center of the circle to any point on the circle. The radius of a circle is one-half the length of the diameter.

Central Angle

An angle formed by two radii. The total degree measure of a circle is 360 degrees.

Chord

A line segment that joins two points on the circle. The longest chord of a circle is its diameter.

Tangent

A line that touches only one point on the circumference of a circle. A line drawn tangent to a circle is perpendicular to the radius at the point of tangency.

Circumference

The difference around of circle (similar to the perimeter of a polygon)

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C = (pi)(Diameter) = 2pi(r)

Arc

a section of the circumference of a circle

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Arc Length = n/360 (circumference)

Area of a circle

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Ara of a sector

A = (pi)r^2

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Area of a sector = n/360 (area of circle)

Plane

A flat surface that extends indefinitely in any direction

x-axis and y-axis

The horizontal (x) and vertical (y) lines that intersect perpendicularly to indicate location on a coordinate plane. Each axis is a number line.

Ordered Pair

Two numbers that designate distance from an axis in coordinate geometry. The first number is the x- coordinate; the second number is the y-coordinate.

Origin

the point where the x- and y- axes intersect; its coordinates are (0,0)

Solid

A three dimensional figure. The dimensions are usually called length, width, and height.

Substitution

Step 1: Solve one equation for one variable in terms of the other variable

Step 2: Substitute the result back into the other equation and solve.

Combination

Combine the equations in such a way that one of the variables cancels out. To eliminate a variable, you can add the equations or subtract one equation from another.

Median

The middle term in a group of terms that are arranged in numerical order

Mode

The term that appears most frequently in a set

Range

The distance between the greatest and least value in a group of data points.

Interquartile Range

The difference between the values of the third and first quartile values

Standard Deviation

Measures the dispersion in a given data set.

Face

The surface of a solid that lies in a particular plane

Rectangular solid

a solid with six rectangular faces. All edges meet at right angles.

Cube

a special rectangular solid in which all edges are of equal length and all faces are squares

Cylinder

A uniform solid whose horizontal cross sectional is a circle.

Volume of a rectangular solid

= (area of base)(height) = lwh

Surface area of a rectangular solid

= sum of area of faces = 2lw x 2lh x 2hw

Volume of a cube

= lwh = e^3

Surface area of a cube

= sum of area of faces = 6e^2

volume of a cylinder

= (area of base)(height) = pi r ^2 h

lateral surface area of a cylinder

= (circumference of base)(height) = 2 pi r h

Total surface area of a cylinder

= areas of circular ends + lateral surface area = 2 pi r ^2 + 2 pi r h

Percent Increase

To determine combined percent increase/decrease when no original; value is specified, use 100 as a starting value.

Percent Increase/Decrease: (Amount of Increase/Decrease) / (Original) x 100

Remainders

To solve a remainders problem, pick a number that fits the given conditions and see what happens.