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Algebra & Polynomials
Limits
Number Theory
Combinatorics
Geometry
Probability
Sequences & Series
Theorems related to Algebra and Polynomials:
Binomial Theorem: For real and complex values of a and b and a non-negative integer n, then
Theorems related to Algebra and Polynomials:
Binomial Theorem: For real and complex values of a and b and a non-negative integer n, then
Complex Conjugate Root Theorem: If P(x) is a polynomial is a polynomial with real coefficients, then a complex number is a root of P(x) if and only if its complex conjugate is also a root.
Theorems related to Limits:
Complex Conjugate Root Theorem: If P(x) is a polynomial is a polynomial with real coefficients, then a complex number is a root of P(x) if and only if its complex conjugate is also a root.
Theorems related to Limits:
Theorems related to Number Theory:
Fermat's Little Theorem: If a and p are relatively prime then aᵖ⁻¹ ≡ 1 (mod p)
Euclidean Algorithm: The GCD(a, b) = GCD(b, a mod b)
Binet's formula: If Fₙ is the nth Fibonacci number, then:
Theorems related to Number Theory:
Fermat's Little Theorem: If a and p are relatively prime then aᵖ⁻¹ ≡ 1 (mod p)
Euclidean Algorithm: The GCD(a, b) = GCD(b, a mod b)
Binet's formula: If Fₙ is the nth Fibonacci number, then:
Chinese Remainder Theorem: Let m be relatively prime to n. Then each residue class mod m is equal to the intersection of a unique residue class mod n and a unique residue class mod , and the intersection of each residue class mod with a residue class mod m is a residue class mod mn. This means that if we have b ≡ c (mod mn), we can deduce that b ≡ c (mod m) and b ≡ c (mod n). In other words, suppose you wish to find the least number x which leaves a remainder of y₁ when divided by d₁, y₂ when divided by d₂, y₃ when divided by d₃, ... yₙ when divided by dₙ such that d₁, d₂, d₃, ... , dₙ are all relatively prime. Let M = d₁,d₂,d₃,...,dₙ and bi = M/di. If the numbers ai satisfy ai * bi ≡ 1 mod di for every 1 ≤ i ≤ n, then a solution for x is sigma (n, i = 1) (ai * bi * yi mod n).
Euler's Totient Theorem: Let φ(n) be Euler's totient function. If n is a positive integer, φ(n) is the amount of integers in the set {1, 2, 3, ..., n} that are relatively prime to n. If a is an integer and m is a positive integer relatively prime to a, then
a ^ φ(m) ≡ 1 (mod m).
Theorems related to Combinatorics:
Principle of Inclusion-Exclusion: For a collection of n finite sets A₁, A₂, A₃ ..., Aₙ,
Theorems related to Combinatorics:
Principle of Inclusion-Exclusion: For a collection of n finite sets A₁, A₂, A₃ ..., Aₙ,
Pascal's Identity: n choose k = (n - 1 choose k - 1) + (n - 1 choose k)
Theorems related to Geometry
Angle Bisector Theorem: Given triangle ABC and angle bisector AD, where DC is on side BC, then
AB/BD = AC/CD.
Annular Steiner Chains: This theorem states that for n tangent externally tangent circles, with equal radii in the shape of an n-gon, the radius of the circle that is externally tangent to all the other circles can be written as r(1 - cos((90(n-2))/n)/(cos((90(n-2))/n)) and the radius of the circle that is internally tangent to all the other circles can be written as r(1 - cos((90(n-2))/n)/(cos((90(n-2))/n)) + 2r where r is the radius of one of the congruent circles and where n is the number of tangent circles.
Base Angle Theorem: In an isosceles triangle, the angles opposite the congruent sides are congruent.
Brahmagupta's Formula: This is a formula for determining the area of a cyclic quadrilateral given the four side lengths, give as follows: [ABCD] = √(s-a)(s-b)(s-c)(s-d) where a, b, c, d are the four side lengths and is
the semi perimeter which is (a + b + c + d)/2.
British Flag Theorem: In Euclidean geometry, if point P is chosen in a rectangle named ABCD, then AP²+PC² = BP²+PD².
Bretschneider's Formula: Suppose we have a quadrilateral with edges of length a, b, c, d (in that order) and diagonals of length p, q. Bretschneider's Formula states that the area
[ABCD] = (√(4p²q² - (b² + d² - a² - c²)²))/4.
Heron's Formula: Suppose we have a triangle with side lengths a, b, c, then its area will be √(s-a)(s-b)(s-c)
where s is the semiperimeter which is (a + b + c)/2.
Theorems related to Geometry
Angle Bisector Theorem: Given triangle ABC and angle bisector AD, where DC is on side BC, then
AB/BD = AC/CD.
Annular Steiner Chains: This theorem states that for n tangent externally tangent circles, with equal radii in the shape of an n-gon, the radius of the circle that is externally tangent to all the other circles can be written as r(1 - cos((90(n-2))/n)/(cos((90(n-2))/n)) and the radius of the circle that is internally tangent to all the other circles can be written as r(1 - cos((90(n-2))/n)/(cos((90(n-2))/n)) + 2r where r is the radius of one of the congruent circles and where n is the number of tangent circles.
Base Angle Theorem: In an isosceles triangle, the angles opposite the congruent sides are congruent.
Brahmagupta's Formula: This is a formula for determining the area of a cyclic quadrilateral given the four side lengths, give as follows: [ABCD] = √(s-a)(s-b)(s-c)(s-d) where a, b, c, d are the four side lengths and is
the semi perimeter which is (a + b + c + d)/2.
British Flag Theorem: In Euclidean geometry, if point P is chosen in a rectangle named ABCD, then AP²+PC² = BP²+PD².
Bretschneider's Formula: Suppose we have a quadrilateral with edges of length a, b, c, d (in that order) and diagonals of length p, q. Bretschneider's Formula states that the area
[ABCD] = (√(4p²q² - (b² + d² - a² - c²)²))/4.
Heron's Formula: Suppose we have a triangle with side lengths a, b, c, then its area will be √(s-a)(s-b)(s-c)
where s is the semiperimeter which is (a + b + c)/2.
Law of Cosines: For a triangle with edges of length a, b, and c opposites angles of measure A, B, and C, respectively, the Law of Cosines states:
c² = a² + b² - 2ab cos C
b² = a² + c² - 2bc cos B
a² = b² + c² - 2bc cos A
Pythagorean theorem: If a right triangle with legs a, b and a hypotenuse of c, then a² + b² = c².
Theorems related to Probability:
Theorems related to Sequences & Series:
Pythagorean theorem: If a right triangle with legs a, b and a hypotenuse of c, then a² + b² = c².
Theorems related to Probability:
Theorems related to Sequences & Series:
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