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• Number theory/Advanced
  ...e can study the set of points where both coordinates lie in some subfield (like the reals or the rationals). One also needs to add a limit point, called th ...ve, they can be added in a way that satisfies the normal laws of addition, like associativity, commutativity and the existence of an identity and inverses.
  5 KB (849 words) - 16:14, 18 May 2021
• Ellipse
  ...an ellipse with semimajor and semiminor axes <math>a,b</math> is <math>ab\pi</math>. ...re relatively prime positive integers. Find <math>m+n</math>. ([[2001 AIME I Problems/Problem 5|Source]])
  5 KB (892 words) - 21:52, 1 May 2021
• Pi
  ...mber]], as proved by Lindemann in 1882) denoted by the Greek letter <math>\pi </math>. ...^2}=\frac{\pi^2}{6}</math>. Some common [[fraction]]al approximations for pi are <math>\frac{22}{7} \approx 3.14285</math> and <math>\frac{355}{113} \ap
  8 KB (1,469 words) - 21:11, 16 September 2022
• Logarithm
  ..., we have <math>\ln (-1)=i\pi</math>. Additionally, <math>\ln (-n)=\ln n+i\pi</math> for positive real <math>n</math>.
  4 KB (680 words) - 12:54, 16 October 2023
• Roots of unity
  ...as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik}</math>. ...math> ni\theta = 2\pi ik</math>. Solving this gives <math> \theta=\frac{2\pi k}n </math>. Additionally, we note that for each of <math> k=0,1,2,\ldots,
  3 KB (558 words) - 21:36, 11 December 2011
• Complex analysis
  <cmath> f^{(n)}(z_0)=\frac{n!}{2\pi i} \oint_\Gamma \frac{f(z)\; dz}{(z-z_0)^{n+1}}. </cmath> Let <math>f</math> be an [[entire]] function (i.e. holomorphic on the whole complex plane). If <math>\lvert f(z)\rvert \le
  2 KB (271 words) - 22:06, 12 April 2022
• Riemann Hypothesis
  ...th>. Then an equivalent statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>. ...tion]] to <math>\Re(s)>\frac{1}{2}</math>. Let <math>M(n)=\sum_{i=1}^n \mu(i)</math> be the [[Mertens function]]. It is easy to show that if <math>M(n)\
  2 KB (425 words) - 12:01, 20 October 2016
• Algebraic number
  ...ic is called a [[transcendental number]], such as <math>e</math> or <math>\pi</math>.
  1,006 bytes (151 words) - 21:56, 22 April 2022
• Prime Number Theorem
  specifically, it states that the functions <math>\pi(x)</math> and <math>x/\log x</math> are [[asymptotically equivalent]], where <math>\pi(x)</math> is the number
  10 KB (1,729 words) - 19:52, 21 October 2023
• LaTeX:Symbols
  ...texify] is an online application which allows you to draw the symbol you'd like and shows you the <math>\text{\LaTeX}</math> code for it! |<math>\pi</math>||\pi||<math>\varpi</math>||\varpi||<math>\rho</math>||\rho||<math>\varrho</math>
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2021 JMC 10
  ...can <math>|\pi - |e - | e - \pi|||</math> be expressed in terms of <math>\pi</math> and <math>e?</math> ...}2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</math>
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2004 AMC 12A Problems
  (i) <math>f(1) = 1</math>, and ...rt3}{2} \qquad \text {(D) } \frac {10\pi}{3} - \sqrt3 \qquad \text {(E) }4\pi - 2\sqrt3</math>
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2001 AMC 12 Problems
  ...that is, <math>A > B > C</math>, <math>D > E > F</math>, and <math>G > H > I > J</math>. Furthermore, ...</math> are consecutive even digits; <math>G</math>, <math>H</math>, <math>I</math>, and <math>J</math> are consecutive odd
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2004 AMC 12B Problems
  ...{C}) 75+100\pi \quad (\mathrm {D}) 100+100\pi \quad (\mathrm {E}) 100+125\pi</math> ...hrm{(C) \ } \pi c^2 \qquad \mathrm{(D) \ } \pi d^2 \qquad \mathrm{(E) \ } \pi e^2 </math>
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  \mathrm{(A)}\ 80-20\pi \qquad \mathrm{(B)}\ 60-10\pi \qquad
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2005 AMC 12B Problems/Problem 22
  ...z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</ma Since <math>|z_0|=1</math>, let <math>z_0=e^{i\theta_0}</math>, where <math>\theta_0</math> is an [[argument]] of <math>z_
  4 KB (660 words) - 17:40, 24 January 2021
• 2006 AMC 10A Problems
  MP('I', (8,-8), (0,0)); ...qquad\mathrm{(D)} \text{II, by}\ 8\pi\qquad\mathrm{(E)}\ \text{II, by}\ 10\pi</math>
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 12
  MP('I', (8,-8), (0,0)); ...{(D) } II,\,\textrm{ by }\,8\pi\qquad \textbf{(E) } II,\,\textrm{ by }\,10\pi </math>
  3 KB (424 words) - 10:14, 17 December 2021
• Newman's Tauberian Theorem
  <cmath> F(0)-F_T(0) = \frac{1}{2\pi i} \int_\Gamma K(z) (F(z)-F_T(z)) \le\int_{\Gamma_+} \frac{2M}{R^2} \lvert dz \rvert = \frac{2\pi M}{R} . </cmath>
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2005 AIME II Problems/Problem 9
  ...or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all real <math> t </math>? ...al number]]s <math>t</math> and all [[integer]]s <math>n</math>. So, we'd like to somehow convert our given expression into a form from which we can apply
  6 KB (1,154 words) - 03:30, 11 January 2024

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