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• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I ...ath>, the power mean with exponent <math>t</math>, where <math>t\in\mathbb{R}</math>, is defined by
  3 KB (619 words) - 12:57, 23 August 2024
• Pigeonhole Principle
  ...ts, the principle may be referred to as the '''Dirichlet box principle'''. A common phrasing of the principle uses balls and boxes and is that if <math> ...nhole principle is as follows: suppose for contradiction that there exists a way to place <math>n</math> balls into <math>k</math> boxes where <math>n>k
  11 KB (1,986 words) - 18:13, 19 June 2024
• Combination
  ...generally concerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
  4 KB (615 words) - 10:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ...f and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of ...there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulati
  13 KB (2,048 words) - 14:28, 22 February 2024
• Ptolemy's theorem
  ...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 23:28, 19 September 2024
• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the
  5 KB (849 words) - 15:14, 18 May 2021
• Divisibility rules
  ...integer]]s are [[divisibility | divisible]] by particular other [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other b https://youtu.be/6xNkyDgIhEE?t=1699
  10 KB (1,572 words) - 21:11, 22 September 2024
• Jensen's Inequality
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jen ...[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n
  3 KB (623 words) - 12:10, 20 February 2024
• Combinatorial identity
  == Pascal's Identity == Pascal's Identity states that
  12 KB (1,993 words) - 22:47, 27 October 2024
• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
  9 KB (1,585 words) - 12:46, 2 September 2024
• Chain Rule
  ...to J</math> and <math>f:J \to \mathbb{R}</math>. Let <math>h:I \to \mathbb{R}</math> such that <math>h(x) = f(g(x)) \forall x \in I</math>. If <math>x_ ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.)
  12 KB (2,377 words) - 10:48, 22 July 2009
• Muirhead's Inequality
  ...>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n</math>: ...ath> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</math>,
  8 KB (1,346 words) - 11:53, 8 October 2023
• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe This section is for people who know what [[integral]]s are but don't know the Fundamental Theorem of Calculus yet, and would like to try to figu
  11 KB (2,082 words) - 14:23, 2 January 2022
• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  11 KB (1,876 words) - 18:01, 29 August 2024
• Logarithm
  ...would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ly 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing th
  4 KB (680 words) - 11:54, 16 October 2023
• Roots of unity
  ...ts of unity come up when we examine the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik
  3 KB (558 words) - 20:36, 11 December 2011
• Newton's Sums
  ...ficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|id Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
  4 KB (704 words) - 07:28, 24 November 2024
• Rational approximation
  ...n [[real number]] <math>x</math> can be approximated by [[rational number]]s. Of course, since the rationals are dense on the real line, we, surely, can ...th> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\fr
  7 KB (1,290 words) - 11:18, 30 May 2019
• Trigonometry
  ...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...[[Law of Sines]] and the [[Law of Cosines]]; many more, such as [[Stewart's Theorem]], are most easily proven using trigonometry. In algebra, expressio
  8 KB (1,217 words) - 19:15, 7 September 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...ric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says: ...g radicals because the 0th root of any number is undefined when the number's absolute value is greater than or equal to 1. This creates the indeterminat
  5 KB (913 words) - 14:44, 14 August 2024

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