Math Facts Flashcards
LCM definition and method
the smallest positive integer that is evenly divisible by both a and b
(factor trees for both #s, make sure # has all the ingredients of both)
gcf (gcd) definition and method
The largest positive integer that divides each of the integers.
(find factors of both #s, then the biggest one they have in common)
Partitive Division
We know the amount, but we want to know how many is in each group.
Quotative Division
We know the amount, but we want to know how many groups.
Commutative Property
The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division.
Associative Property
the way in which factors are grouped in a multiplication problem does not change the product.
Distributive Property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x(y+z) = xy + x*z
is always true in elementary algebra.
Identity Property
For 1, says that any number multiplied by 1 keeps its identity.
For 0, any number added to 0 keeps its identity.
PEMDAS reminder (specifically M and D)
Multiplication and division have equal priority, so do them left to right.
Relatively prime
Two integers are relatively prime if they share no common positive factors (divisors) except 1. ( if gcd(a,b) = 1 )
Additive Inverse
a number that when added to a given number gives zero
Integers
The set of positive/negative whole numbers and zero
Natural Numbers
The set {1,2,3,4…} Including zero according to ring theory
Rational Numbers
Numbers that can be expressed as a fraction p/q where p and q are integers.
Real Numbers
Overarching term for any number that can represent a distance (includes integers, rationals, and irrationals)
Irrational Numbers
All the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.
Polar Coordinates
Instead of real and imaginary axis, label coordinates with radius and an angle.
Polar form of complex # (4 separate formulas for: radius, theta, polar form, and powers of complex #)
With complex # a+bi r= sqrt (a^2 + b^2) theta= tan^-1 (b/a) polar form: z=r(cos(theta) + sin(theta)) power of complex #: (cos(theta) + sin(theta))^n = (cos(n(theta)) + i*sin(n(theta)))
Complex Conjugate
a+bi and a-bi (and any other like complex form)
Addition, Subtraction, Multiplication, Division of Complex #
Sum: (real + real) + (complex + complex)
Difference: (real - real) + (complex - complex)
Mult: (a+bi)(c+di) = (ac - bd) + (ad+bc)i or just foil normally and try your best
Divide: In fraction form, multiply num. and dem. by complex conjugate of dem.
Whole Numbers
The set of real numbers that includes zero and all positive counting numbers.
Exponent Properties (7)
- x^a*x^b=x^(a+b) –> when you multiply, add exponents
- (x^a)^b=x^ab –> parentheses, multiply exponents
- x^a/x^b=x^(a-b) –> divide, subtract exponents
- (xy)^a=x^a*y^a –> multiple terms in parentheses, distribute outside exponent to all terms
- (x/y)^a= x^a/y^a –> fraction in parentheses, distribute outside exponent to numerator and denominator
- x^(-a)=1/x^a –> term to a negative power is 1/term
- x^0=1 –> any term to the power of zero is one (easy money)
composite numbers
Numbers that have more than two factors
Division Algorithm
For any integers a and b, there exists unique integers q and r such that:
a=bq+r, where r is greater than or equal to 0 and less than 6.
Explain modulus (both formally and in Euclidean terms).
two integers a and b are congruent mod n if n is a divisor of their difference. (i.e. integer k such that a-b=kn). Basically, if a number is congruent to another mod n, they have the same remainder in the modulus.
Euclidean terms: a=pn+r and b=qn+r
Mersenne Primes
Primes 1 less than a power of 2: (2^n -1).
Perfect Number
A number that is equal to the sum of its factors (excluding itself).
-(primes can never be perfect :( )
Find a perfect # from a Mersenne prime.
Input Mersenne prime n into 2^(n-1) (2^n - 1).
Fermat’s Last Theorem
a^n + b^n = c^n cannot be true when n >2.
Arithmetic sequence and general formula
Arithmetic Sequence: Addition/Subtraction from one term to the next
a(subn)= d(subn)+a(sub0) where d is the common difference and a(sub0) is the starting - 1. (fancier verison is x(subn)=d(n-1) + a(sub1) where d is common difference, n is place value, and a(sub1) is the starting value).
Geometric Sequence definition and general formula
Geometric Sequence: Multiplication/Division from one term to the next
a(subn)= a(sub1) * r^(n-1) where a(sub1) is the starting value, ratio is the common ratio, and n is place value.
Fibonacci sequences converges… (explain why)
The fibonacci sequence converges phi (the golden ratio), which is an irrational # approx. 1.618. The fibonacci sequence converges phi in that ratios (a(sub2)/a(sub1), a(sub3)/a(sub2), ….) get closer and closer to irrationality.
Fibonacci Sequence
A sequence in which the next number is the sum of the two previous. Patterns in nature etc.
Pascal’s triangle
A triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression. The triangle can be filled out from the top by adding together the two numbers just above to the left and right of each position in the triangle. Ex: 0th row is 1, first row is 1 1, second row is 1 2 1, third is 1 3 3 1, etc
3 forms of parabolas
Standard: ax^2+bx+c
Intercept: a(x-p)(x-q) where p and q are the two roots of the function
Vertex: a(x-h)^2+k where h,k is the vertex
4 types of transformations/ what the heck do they mean
- Rotation(rotates on a degree),
- Dilation(shrink or stretch the function),
- Translation(left/right, up/down the axes),
- Reflection(reflect across the 45 degree line)
Rules for solving inequalities
When solving an inequality:
• you can add the same quantity to each side
• you can subtract the same quantity from each side
• you can multiply or divide each side by the same positive quantity
If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed
Absolute Value Equation
y = a|x-h| +k where a is the slope and (h,k) is the vertex (the point where the graph switches directions)
Direct Variation
Dumb way of saying a straight line that goes directly through the origin (direct cause and effect/direct proportion, you get it)
Arithmetic sequence Sum (finite) explain why tf it works
sum= (n/2)(a(sub1)+a(subn) because any arithmetic sum’s first and last numbers added are repeated the entire series (i.e 1,2,3…98,99,100 gives me 1+100=101, 2+99=101, 3+98=101, etc.). Sooo, if you take that sum and divide the number of terms by 2, you find the sum for the full finite sequence.
Geometric sum (finite)
Vector definition
Measurements that include both magnitude (length) and direction.
Dot Product definition and equations
Multiplication of two vectors that gives us a scalar.
Dot Product: A . B = A(subx)B(subx) + A(suby)B(suby) w/ given beginning and endpoints
OR: A . B = |A| * |B| cos (theta) w/ two magnitudes and an angle
Magnitude of a vector equation
Magnitude: |A| = sqrt(x^2 + y^2)
Cross Product definition and equation
Multiplication of two vectors that gives us a vector answer (sometimes called vector product).
a . b = |a| |b| (sin(theta)n), where a and b are magnitude (length), theta is the angle between the vectors, and n is the unit vector at right angles to a and b.
Scalar
A quantity described only by a magnitude, it has no direction (they ‘scale’ a vector up or down). Basically, they are just normal numbers.
Rules for changing Matrices (3)
- Swap any two rows
- Multiply rows by a constant
- A row can have a multiple of another row added to it
Addition, Subtraction, and Determinants of Matrices
Addition and Subtraction: As long as there are the same # of r and c, you’re all set.
Multiplication: # of c in M1 has to equal #of r in M2 then M1(r1)*2M(c1), add all answers and new matrix will have less spaces than og ones.
Identity Matrix and Inverse Matrix
Identity: Same (r*c), with 1s on the diagonal and 0s in the other spots.
Inverse: A * A^-1 = I (a matrix times its inverse gives us the identity matrix (1)).
Not all matrices have an inverse.
To find an inverse: [A | I ], then use operations to get to [ I | A^-1 ].
Determinant
A single # value that’s associated with a matrix (has to be a square matrix)
D = |M| Ex: the determinant of the identity matrix is 1.
Ring
A set that is closed under addition and multiplication (there are additive and multiplicative identities and additive inverses, addition is commutative, and the operations are associative and distributive).
Abelian Ring
if xy = yx for every x,y E G
Field
A field is a set in which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.
subgroup and how to determine
A group nested inside another group
To determine if a subset of a group is a subgroup, check if the subset is closed under operations and inverses.
Proper Subgroup
A group that does not contain all of the original group
Trivial Subgroup
A subgroup that only contains the identity (1 term).
Homomorphism
A map ‘f’ between two sets which preserves a particular algebraic structure.
What is completing the square?
Completing the square is converting a standard quadratic equation into vertex form. In order to complete the process, c must be equal to (b/2)^2.
Discriminant rules for b^2-4ac
If the Discriminant is:
- Negative: 2 complex solutions
- Positive: 2 real solutions
- Zero: 1 real solution
High or low point of a quadratic inequality
h = -b/2a is the x value, then sub in for y
Quadratic for the path of an object thrown over time
f(t) = -gt^2 + v(sub0) + h(sub0), where g is gravity (4.9m or 16ft), v(sub0) is initial velocity, and h(sub0) is initial height.
Rational Roots Theorem
In any polynomial, the possible roots are: p/q, where p is the factors of the constant, and q is the factors of the leading coefficient.
Remainder Theorem
If you divide polynomial f(x) by (x-h), the remainder will be f(h). If f(h)=0, then (x-h) is a factor of f(x).
Fundamental Theorem of Algebra
A polynomial function f(x) of degree n(n>0) has n complex solutions for f(x)=0.
In simpler terms, the number of roots in a polynomial is equal to the highest degree of the function.
note: if a+bi is a solution, then a-bi must be as well.
Descartes Rule of Signs
The number of positive real zeros in a polynomial equals the amount of signs changes that occur from one term to the next.
The number of negative real zeros in a polynomial equals the sign changes when P(x) is changed to P(-x).
greatest integer function (floor function)
Denoted as ⌊x⌋, any real # put into ⌊x⌋ gives us the greatest integer that is less than or equal to x. Ex: 11.5 = ⌊11⌋, -4.2 = ⌊-5⌋
least integer function (ceiling function)
Denoted as ⌈x⌉, any real # put into ⌈x⌉ gives us the integer that is greater than or equal to x. Ex: ⌈1.3⌉ = 2, ⌈-2.75⌉ = -2
Inverse Variation
Inverse of Direct Variation, it’s when y and x are inversely related. Equation is y= k/x, where k is the constant of variation. Asymptotes of the function are the x and y axis.
Vertical Asymptotes
Found when the denominator equals zero. If x=1 is a vertical asymptote, then (x-1) is a factor of the denominator. Function cannot cross.
Horizontal Asymptotes
Tells you the end behaviors of the graph. The function can cross the Horizontal Asymptote. If the degrees top and bottom are the same, the horizontal asymptote is a fraction of the two leading coefficients. If the bottom degree is higher than the top, then y=0 is the asymptote.
Slant Asymptote (oblique asymptote)
When the degree on top is one degree higher than on the bottom, divide the top polynomial by bottom polynomial to find the slant asymptote (eschewing remainder, since as x goes toward infinity it is obsolete).
Logarithms (equation, meaning, facts)
Equation: log(sub b)x = y, where b^y=x
Inverse of exponential function, reflected across y=x.
ln is log base e.
Change of base formula
log(sub b) a = (log(sub z) a)/(log(sub z) b)
Properties of logs/ln (product, quotient, power)
product: log(xy) = logx+logy
quotient: log(x/y) = logx-logy
power: logx^y = ylogx
Area of an ellipse
A=pi(ab) where a and b are the radii of the major and minor axes.
Area of a trapezoid
A=(a+b/2)h where a and b are the parallel side lengths and h is the height
Density Formula
Density = mass/volume where mass is kg and volume is meters cubed.
Circumference (perimeter) of a circle
2piR
Geometric Mean (formula and definition)
Used for patterns where the growth is multiplied instead of added.
nth root of (x1,x2,x3… xn)
