Math Flashcards
Digit
The numbers that make up other numbers (0-9)
Names of digits
Designated by a place value. Ones, tens, hundreds, thousands, and ten thousands place
Number
Made up of either a digit or a collection of digits. Includes basically any collection of digits, like 0, negative numbers, fractions and decimals, and radicals
Integers
Numbers that have no fractional or decimal part. Can be negative and includes zero. Fractions are not integers. Integers are the only numbers that can be classified as even or odd
Distinct element
A unique digit, number, or integer
Is zero positive or negative?
Neither, but all other numbers can be classified as positive or negative
Zero
An integer that is not positive or negative. You can’t divide by zero. Zero is considered even and is not a prime number
Consecutive integers
Integers listed in order of value without any integers missing in between them (-6, -5, -4, -3, -2,..)
Absolute value
Equal to the number’s distance from 0 on a number line- it’s always positive
An integer is divisible by
Its factors
Remainder
An integer left over after division if one integer is not divisible by another
Multiples of a number
The multiples of an integer are all the integers that are the product of that integer and another integer. That number times an integer (times 1, times 2, times 3, etc). The multiples of 20 are 20, 40, 60, etc.
Factor
The factor of an integer is a number that divides evenly into that integer. There is no remainder after division. Includes 1 and the number- the factors of 40 include 1 and 40
Prime number
An integer that only has 2 factors- itself and 1. Prime numbers are positive integers. 2 is the only even prime number. 1 and 0 are not prime numbers
Sum
The result of addition
Difference
The result of subtraction
Product
The result of multiplication
Quotient
The result of division
Divisor
The number you divide by
Terms
The numbers and expressions used in an equation
Inclusive
A range of numbers that includes the numbers at the ends of the range
Between
A range of numbers that does not include the numbers at the ends of the range
Order of operations
Parentheses, exponents, multiplication and division, addition and subtraction
Mixed number
A number that is represented as an integer and a fraction
Associative law
When you are adding or multiplying a series of numbers, you can regroup the numbers in any way you’d like
Distributive law
Multiplying a number by a group of numbers added together is the same as doing each multiplication separately
Terminating decimal
A decimal that ends- not repeating
Is 1 a prime number?
No
When does the direction of the inequality sign change?
When you multiply or divide both sides of the inequality by a negative number
Roots
The values of the variable that make the equation equal to zero
When you see exponents that have equal bases and are being multiplied, what do you do?
Add the exponents
When exponents have equal bases and are being divided, what do you do?
Subtract the exponents
When an exponent is raised to a power, what do you do?
Multiply the exponents
Negative powers
Any term raised to a negative power is equal to the reciprocal of that term raised to the positive power.
Zero exponents
Any nonzero number raised to a power of zero is equal to 1
You can add or subtract square roots if
The values under the radical sign are equal
How to multiply and divide square roots
There are no restrictions on this, you just have multiply the terms outside the radical together and multiply the terms inside the radical together
Fractions with radicals can be rewritten as
An entire fraction in a radical can be rewritten as the numerator under a radical and the denominator under another radical
Ratio
Expresses a part to part relationship- does not mention the whole
Proportion
An equivalent relationship between two fractions or ratios- two fractions could be proportionate if they form equal ratios
Average
Arithmetic mean- the sum of all the numbers in the list divided by the number of numbers in the list
Median
The middle value in a list of numbers. There is an equal number of values above and below the median
Mode
The number in a list of numbers that occurs most frequently
Range
The difference between the greatest and least numbers in a list of numbers
Standard deviation
A measure of the variation of a set of data values. A low standard deviation indicates that the data values tend to be close to the mean. With a high standard deviation, the data is spread out over a wide range. Standard deviation focuses on the distance of a point from the mean, and it can never be negative. The GRE will never ask you to calculate the standard deviation, but you would need to know the mean to make an estimation. 68% of individuals are within one standard deviation of the mean, and 94% are within 2 standard deviations
FROZEN number
Fractions
Repeats
One
Zero
Extremes
Negatives
Good to use for plugging in
How many degrees in a line?
180
When two lines intersect, how many angles do they make?
4, they sum up to 360
Vertical angles
Angles that are across from each other when two lines intersect. The angles are equal if they are directly across from each other. It is possible for all 4 angles to be equal, as long as they add to 360.
Parallel lines
Lines that never intersect and have the same slope. When a pair of parallel lines is intersected by a third line, two types of angles are formed- big and small. Any big angle is equal to any other big angle and any small angle is equal to any other small angle. The sum of any big angle and any small angle is 180.
Equilateral triangle
A triangle in which all 3 sides are equal and all 3 angles are 60 degrees
Isosceles triangle
A triangle in which 2 of the 3 sides are equal and 2 of the 3 angles are also equal
How to tell which side of a triangle is longest and which one is shortest
The longest side is opposite the largest interior angle, and the shortest side is opposite the smallest interior angle. Equal sides are opposite equal angles
Third side rule
The length of any one side of a triangle must be less than the sum of the other two sides and greater than the difference between the two sides. If you know two sides of a right triangle, you can use the Pythagorean theorem to find the third. However, if the triangle is not a right triangle, it’s not possible to find the third side on the GRE. You can just estimate what it is
Right triangles
Triangles in which one interior angle is 90 degrees. They can be easily identified because there will be a box in the right angle. The other two angles have to sum to 90 degrees
Pythagorean triples (3)
Right triangles may have specific combinations of side measurements. The side lengths could also be the ratios of these numbers
3:4:5
6:8:10
5:12:13
30:60:90 right triangles
You can create a 30:60:90 triangle by drawing in the height of an equilateral triangle, which cuts the triangle in half. The shortest side of the new right triangle is x, the hypotenuse is 2x, and the height of the equilateral triangle is x radical 3. The side that is half the length of the hypotenuse is opposite the 30 degree angle
45:45:90 right triangles
If you take a square and cut it in half along its diagonal, you create an isosceles right triangle. Two sides of the square are the same and are the equal sides of the triangle. The two equal sides are x and the hypotenuse is x radical 2.
Chord
A line that connects two points on the circumference of a circle. The diameter of a circle is the longest chord in the circle. The radius is not a chord because it only touches one end
Arc
A section of the outside (circumference) of a circle
Central angle
Any angle formed by two radii
Regular figure
The figure has equal sides and angles
Polygons
Figures with at least 3 straight sides and angles, typically 5 or more. Includes pentagons, hexagons, and octagons
Trapezoid
A quadrilateral with only one set of parallel sides
Quadrants
Quadrant 1 is the top right, quadrant 2 top left, 3 is bottom left, 4 is bottom right
Goes counterclockwise
X intercept
The point where a line crosses the x axis. This is when y=0
Y intercept
The point where a line crosses the y axis. This is when x=0
Slope
The vertical change divided by the horizontal change
Perpendicular lines
Lines with slopes that are negative reciprocals of each other. The lines intersect at a right angle
Parallel lines
Lines with the same slope
For a parallelogram, angles on the same side
Add up to 180
Probability
Express the chance of a certain outcome occurring. The denominator is the total number of possible outcomes and the numerator is the number of outcomes that would satisfy the criteria
Probability of two events occurring
Multiply probability A and probability B
Probability of event A or event B occurring
Add probability A and probability B
Given event probability formula
Probability A+ Probability of not A= 1
Factorial
The factorial of a number is equal to that number times every positive whole number smaller than that number, down to 1. The factorial of 6 (6!) is 6x5x4x3x2x1= 720. On these problems, find a shortcut like canceling or factoring instead of doing a lot of multiplication
Remainder patterns when dividing by integers
When dividing by an integer s, there are s possible remainders, 0 through s-1. The remainders cycle in an infinite loop.
17/5 has an R of 2
18/5 has an R of 3
19/5 has an R of 4
20/5 has an R of 0
21/5 has an R of 1
Then the pattern repeats. There are s-1 (4) remainders when dividing by 5
Unit digit cycles
You might need to know the patterns of the ones digit when a number is raised to a power
9^1= 9
9^2= 81
9^3= 729
The pattern for 9 is 9, 1, 9. When 9 is raised to an even power, the digit is 1, when raised to an odd power, the digit is 9.
Permutation
An arrangement of things in a particular order. If asked to determine how many ways you could arrange 5 statues on the shelf, the answer is 5x4x3x2x1= 120. Figure out how many slots you have, write down the number of options for each slot, and multiply them
Combination
A group- different from a permutation because the order is irrelevant. Like if you have to bring home 3 different flavors of ice cream and there are 5 possible flavors to choose from. The order doesn’t matter in this case. Figure out how many slots you have and fill in the slots like you would with a permutation. Then, divide this number by the factorial of the number of slots
Combination problem with extra conditions
Find the number of ways to create each of the possibilities and add the two of them together
Divisibility rule for 3
Sum all of the digits for the number and then see whether the sum is divisible by 3
Divisibility rule for 4
Determine whether the last 2 digits (tens and ones place) is divisible by 4. Ignore all other digits
Divisibility rule for 6
The number has to be divisible by both 2 and 3- if it is, it is divisible by 6
Divisibility rule for 9
Sum all of the digits for the number and then see whether the sum is divisible by 9
Rational number
A terminating decimal- an integer is in the numerator and denominator. If the only prime factors of the denominator are 2 or 5, the decimal is terminating
Irrational number
A decimal that continues- found in non perfect squares like radical 2
Bowtie method
Cross multiply- whichever fraction has the greater number above it is the greatest fraction
Absolute value inequalities
Set the expression inside the absolute value bar greater than the positive value on the right of the inequality sign. Then, set the expression as less than the negative value on the right of the inequality sign
Absolute value inequality problems
Set up 2 equations. For example, if the absolute value of x is less than 3, the first equation is just x is less than 3- take away the absolute value symbol. For the second equation, the 3 becomes negative and you flip the inequality sign- x is greater than negative 3.
Steps of solving geometry problems
- Draw the figure
- Label any information you have
- Write down any formulas you need, like area or volume
Line segment
A line between 2 points without arrows on either end- it’s only considered a line if it has the arrows (lines go on forever)
Quadrilateral
Has 4 sides, all angles add up to 360. A parallelogram is a type of quadrilateral, but both sets of sides are parallel to each other. A rectangle (and square) is a type of parallelogram, but it has 4 equal angles
MADSPM exponents rules
- When you’re multiplying 2 exponents with the same base, you add the numbers
- When you’re dividing 2 exponents with the same base, you subtract the numbers
- When you have a power, you multiply the 2 exponent numbers together
“At least” probability questions
To find the probability of there being at least one woman on a committee of 3 people, you would have to find the probability of one woman, two women, and three women to find the answer.
Zero factorial
Equals 1
“Different groups” phrasing
Order doesn’t matter- multiply the terms, but divide by the factorial of the number of terms
Arithmetic sequence
Have a common difference (adding/subtracting a specific number)
Geometric sequence
Multiplying/dividing by a certain number- have a common ratio
Vertices
The corners of a solid
