Math Flashcards
integers
counting numbers including negative numbers (but not fractions/irrational numbers)
factor
a number that will divide evenly into the number in question
multiple
a number that is itself a factor of the number in question
prime number
integer whose only factors are itself and 1 (0 and 1 are NOT prime, no such thing as a negative prime)
prime numbers under 30
2,3,5,7,11,13,17,19,23,29
divisibility rules
1) divisible by 2 if its units digit is divisible by 2
2) divisible by 3 if the SUM of all its digits is divisible by 3
3) divisible by 4 if its last two digits form a number that is divisible by 4
4) divisible by 6 if the its divisible by both 2 and 3
5) divisible by 8 if the last 3 digits form a number divisible by 8
6) divisible by 9 if the SUM of its digits is divisible by 9
order of operations
P/E/MD/AS- parenthesis exponents multiplication/division addition/subtraction
associative law
when adding/multiplying a series of numbers you can regroup them however you want
distributive law
a(b+c)- = ab + ac and a(b-c) = ab- ac
Rules for exponents
1) a^2 = a * a
2) a^2 * a^3 = (aa)(aaa) = a^(2+3) = a^5
3) (a^2)^3 = (aa)(aa)(aa) = a^(23) = a^6
4) a^2/a^3 = aa/aaa = 1/a = a^(2-3) = a^(-1)
5) 15^12 - 15^11 = 15^11(15-1) = 15^11(14) CANT JUST SUBTRACT, FACTOR
more rules for exponents
1) raising a fraction between 0 and 1 to a power greater than 1 results in a SMALLER fraction
2) a negative number raised to an EVEN power because POSITIVE, a negative number raised to an ODD power remains NEGATIVE
3) a number raised to a negative power = 1 over the number to the positive power (ie. 2^-2 = 1/2^2)
4) a number raised to the 0 power = 1 0^0 = undefined
5) a number to the 1 power = the number itself
square root rule
1) when ETS asks for square root of 16 they want 4, not -4
2) multiply/divide like normal; multiply/divide the bases, then deal with the square root sign
3) can’t add/subtract unless the bases are the same, if the bases are different estimate their values and then add
square roots to know
square root 1 = 1
square root 2 = 1.4
square root 3 = 1.7
square root 4 = 2
Quadratic equations- 3 types
1) (x+y)(x-y) = x^2 - y^2
2) x^2 + 2xy +y^2 = (x+y)^2
3) x^2 - 2xy + y^2 = (x-y)^2
how to solve quadratic equations
F- first
I- inside
O- outside
L- last
steps to solve quadratic equations
1) separate the x^2 into (x )(x )
2) find the factors of the 3rd term that when added/subtracted give the 2nd term
3) figure out the +/- for the terms, they have to yield the middle term when added/subtracted and the last term when multiplied
4) if a number answer is required, set whole equation = 0; solve; set each ( ) = 0 and solve for x
reducing fractions
factor top and bottom and then cancel, or divide by factors that are common to both
* you can ONLY reduce across a multiplication sign
adding/subtracting
must have same denominator
bowtie method:
1) multiply the denominators together to get a common denominator
2) multiply the denominator of each fraction by the numerator of the other one
3) add/subtract those numbers
4) reduce if necessary
decimals–> %, %–> decimal
.01 = 1/100 = 1% .1 = 1/10 = 10% .2 = 1/5 = 20% .25 = 1/4 = 25% .333 = 1/3 = 33 1/3% .4 = 2/5 = 40% .5 = 1/2 = 50% .6 = 3/5 = 60% .66 = 2/3 = 66 2/3% .75 = 3/4 = 75% .8 = 4/5 = 80% 1.0 = 1/1 = 100% 2.0 = 2/1 = 200%
percentage change
percentage change = (difference/original) * 100
average
average = total/# of things
*as long as you have 2 out of 3 you can solve
standard deviation rule
68-96-100 - percentages of data points that fall 1, 2 and 3 standard deviations from the mean
rate
distance amount/(time*rate)
*as long as you have 2 out of 3 you can solve
the 3 angles inside a triangle add up to
180 degrees
the 4 angles inside any 4 sided figure add up to
360 degrees
a circle contains
360 degrees
when 2 lines are perpendicular to each other their intersection forms…
4 90 degree angles
vertical angles- angles across from each other when 2 lines intersect,
are ALWAYS equal
when 2 parallel lines are intersected by a 3rd line, big and small angles are formed
big angles = big angles, and small angles = small angles
any big + small angle = 180 degrees
equilateral triangle
all 3 sides equal, so all 3 angles = 60 degrees
isosceles triangle
2 of the 3 sides are equal, so 2 of the angles are equal
in any triangle the longest side is opposite…
the largest angle, and vice versa
3rd 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 other two sides (to solve add two sides, then subtract, the third side must lie between those two numbers, but not equal to either)
area of a triangle
A = 1/2bh (divide into 2 triangles to find h)
pythagorean theorm
in a right triangle the hypotenuse^2 = a^2 + b^2
a square divided in 1/2 makes 2 right isosceles triangles…the angles =?
a 90 degree angle, and two 45 degree angles
the side ratio of a right isosceles triangle is…
x : x : x*square root of 2
30-60-90 triangles
when you cut an equilateral triangle in half it has a 90 degree, a 60 degree, and a 30 degree angle
the side ratio of a 30-60-90 triangle is…
x : x*square root of 3 : 2x (if you see a triangle where one side is 2x, or there is a square root of 3 anywhere, you know it is a 30-60-90 triangle)
area of a parallelogram
a = bh (where h is a line drawn perpendicular to the base)
circumference of a circle
2pir or pi*d
area of a circle
pi*r^2
arc of a circle
an angle is formed by two radii, the number of degrees of that angle divided by 360 = fraction of the circumference that the arc is
slope
y=mx+b or rise/run
coordinate system quadrants
counterclockwise starting at top right - 1, 2, 3, 4
volume of a cylinder
(pi*r^2)(h)
surface area of a cube
total of the areas of each side (find each area and then add)
probability
of possible outcomes that satisfy conditions/# of total possible outcomes
probability of A AND B
prob A * prob B
probability of A OR B
prob A + prob B
factorial (ie. 6!)
equal to that number times every positive whole number smaller than that number down to 1
permutation
arrangement of things in a particular order (figure out how many slots you have, write down the number of options for each slot, multiply)
combination
a group where order doesn’t matter (figure out how many slots you have, fill in the slots like permutation, divide by the factorial of the # of slots, cancel out before solving)
groups
T = G1 + G2 - B + N where G1 and G2 are groups, B is members in both groups and N is members in neither group, T is total
