a collection of objects, the objects in the set are called elements.
natural number whose only divisors are 1 and itself, 1 is not a prime number.
all of the elements in one set are also contained in another set
the set of all objects in A or B or both
set of all objects contained in A and B.
complement of A is the set of elements in the universe but not in A.
1,2,3,4... N
0, natural numbers, negatives of the naturals. Z
can be expressed as a ratio of two integers, A/B where B cant be 0. Q
cant be expressed as a ratio of two integers. ex pi, square root of prime numbers, R\Q
all numbers on a number line. R. rationals take up 0% of R
every rational number can be written both as a ratio of integers and as a terminating or repeating decimal
A set has the density property if between any two numbers of the set there is another number that is also a member of the set. As a result, any two members of the set have infinitely many other members between them.
a value that a sequence approaches as the number of terms approaches infinity
any natural number greater than 1 can be written as a unique product of primes. ex) 6=2x3
cube roots can, square roots cannot
a²+a³ = a²+³
a³ / a² = a³-²
(a²)³ = a²³
(ab)³ = a³b³
(a/b)³ = a³/b³
a⁰ = 1
n√x is a symbol used to denote the nth root of some number x. there can be a coefficient, n is the index, x is the radicand. when there is no coefficient it is 1.
radical where the coefficient must be 1 or -1
radical where the coeffiecent cannot be 1 or -1
when the radicand does not contain any factors that are perfect powers of n
cartesian product of two sets A and B is denoted by AxB, is the set of all possible ordered pairs where the first element comes from A and the second from B. (a,b)
contians all possible pairings of the real numbers, if the cartesian plane represents RxR=R²
between two sets A and B is a subset of AxB, a set of ordered pairs
(a,b) first element
(a,b) second element
a relation is a function if each element in the domain is sent to only one element in the range. f(n)
infinitely many points
never meet, always have the same slope, no solution, different y intercept
meet at a right angle, slopes are negative reciprocals of each other, multiplying them gives -1
y=mx+b, m is slope b is y intercept
y-y1=m(x-x1)
x1 y1 is a point on the line, m is slope
Ax+By+C=0. a is a whole number and b and c are integers
draw line through each point in domain. if it intersects the graph at ony one point, it is a function
can take any value, no gaps in the graph. denoted by intervals.
can only take on certain values, gaps in the graph
A closed interval includes all real numbers between two given endpoints, and it also includes those two endpoints. [a,b], closed circles
An open interval contains all real numbers between two distinct real numbers, but excludes the two numbers themselves. (a,b). not the same as ordered pair, open circles
some functions are undefined at certain values. this is a discontinuity at zero, causes a hole in the graph, represented by an open circle
rise/run, vertical change/horizontal change
slope of zero and equation y=#
undefined slope and equation x=#
means to find the value of x that satisfies the equation
need to have two equations, a system of linear equations
graph lines, elimination, substitution
one solution, different slopes
infinite solutions, same slope and same y intercept, equations are related by multiplying or dividing by same number
r²=V/πh
r²=3V/πh
r²=SA/4π
r³=3V/4π
3πr² circular base on surface of hemisphere, so the surface area will be half the surface area of sphere, plus the area of circular base
multiply each term of the first binomial with each term of the second binomial, also called expanding
number multiplied by iteself, can also be expanded/foiled
result after binomial squared is expanded
when two binomials are multiplied together that are the same except with the opposite sign between them, middle terms cancel, so you are left with the first term of each binomial squared minus the second term of each binomial squared
reverse process of expanding
two binomials.
ax²+bx+c where a is equal to one
(x+_)(x+_)
with first blank being add to get b, second being mulitply two get c
ax²+bx+c
(_x+_)(_x+_)
with first and third blanks being factors of a, second and fourth blanks being factors of c, so that the outside and inside products combine to give the middle term bx
study of triangles. examines relationship between sides and angles. use capital letters to label angles, lowerscase to label sides, written opposite from their respective angles
hyp= longest, opposite from right angle
opp= opposite from reference angle
adj= adjacent to reference and right angle
reduced fractions or decimals (4 places)
determine measures of all unknown angles and lengths of all unkwon sides.
acute angle it makes with horizontal
angle between horizontal and line of sight. object above observer
angle between horizontal and line of sight. object below observer
king henry doesnt usually drink chocolate milk
kilo hecto deka unit(m) deci centi milli
each step is ten times as much as step on other side
1kilo=10hecto=100deka=1000m, etc
circular base and curved surface. base of cone is a circle, so the perimeter is the circumference of the circle (2πr), so surface area becomes πrs+πr²
Ah
1/3
1/3
V=2/3πr³ half the volume of a sphere
two or more distinct objects, to find volume, add volumes of distinct objects. to find surface area, you have to remove the bases and then add the surface area.
ex) hemi (no base) + cylinder (-1 circle)
2πr² + (πr²+2πrh)
3πr²+2πrh