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.