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• 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]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• Complex plane
  ...point two units to the right and one unit down from the origin corresponds to the complex number <math>2 - i</math>.
  892 bytes (134 words) - 16:52, 3 September 2017
• Complex number
  The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial. (See the [[Fundamental Theorem of Algebra]] for more
  5 KB (860 words) - 15:36, 10 December 2023
• Cis
  ...plex number]]s. For all complex numbers <math>z</math>, we can write <math>z=r\mathrm{cis }(\theta)=r\cos \theta + ir\sin \theta</math>. Notice that <ma Once one gets used to the notation, it is almost always preferred to write <math>re^{i\theta}</math> rather than <math>r\mathrm{cis }(\theta)</m
  1 KB (171 words) - 20:59, 11 July 2023
• Polar form
  ...ows us to graph complex numbers given an [[angle]] <math>\theta</math> and a [[radius]] or [[magnitude]] <math>r</math>. ...u do not understand this notation.) This represents a complex number <math>z</math> that is <math>r</math> units away from the origin, and <math>\theta<
  633 bytes (105 words) - 13:35, 1 April 2022
• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we examine the [[complex numb ...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) - 21:36, 11 December 2011
• Angle bisector
  ...at the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>: pair A,B,C,D,E,F;
  3 KB (575 words) - 15:27, 19 March 2023
• Fermat's Last Theorem
  ...> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...marvelous demonstration of this proposition that this margin is too narrow to contain.''"
  3 KB (453 words) - 11:13, 9 June 2023
• 2005 USAMO Problems/Problem 4
  ...math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
  7 KB (1,276 words) - 20:51, 6 January 2024
• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or some ...antine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
  9 KB (1,434 words) - 13:10, 20 February 2024
• Area
  ...]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen
  6 KB (1,181 words) - 22:37, 22 January 2023
• Exponential form
  ...> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. So <math>z=re^{i\theta}</math> looks like:
  1 KB (238 words) - 22:51, 20 February 2022
• Floor function
  ...]</math>. The action of this function is the same as "rounding down." On a [[positive]] argument, this function is the same as "dropping everything af ...+b\rfloor\ge \lfloor a\rfloor+\lfloor b \rfloor</math> for all real <math>(a,b)</math>.
  3 KB (508 words) - 21:05, 26 February 2024
• Rational approximation
  ...and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how well can we approximate <math>x</math> by rationals with no ...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) - 12:18, 30 May 2019
• Complex analysis
  ...ctions of a real variable that cannot occur in complex variables. Here are a few spectacular results in complex analysis. ...ly connected]] [[domain]] ''D'', and let <math>\Gamma\subseteq D</math> be a [[simple closed Jordan curve]]. Then for any <math>z_0</math> in the interi
  2 KB (271 words) - 22:06, 12 April 2022
• Fundamental Theorem of Algebra
  ...tant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some ...is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bounded when <math>n=0</math>; we n
  5 KB (832 words) - 14:22, 11 January 2024
• Surjection
  ..., the range is a subset of the codomain.) In adjectival form, we say that a function is ''surjective'' or ''onto''. ...defined by <math>f(x) = x+1</math> is not surjective because there exists a [[natural number]] which is not one more than any other natural number.
  794 bytes (131 words) - 22:39, 13 May 2020
• Prime Number Theorem
  of primes less than or equal to <math>x</math>. In other words, it states <cmath> \lim_{x\to \infty} \frac{\pi(x) \log x}{x} = 1 . </cmath>
  10 KB (1,729 words) - 19:52, 21 October 2023
• Infinite
  ...h>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element
  1 KB (186 words) - 23:19, 16 August 2013
• Mock AIME
  A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent years, recurring competitio ** [https://artofproblemsolving.com/community/c5h2987334 Or(z)IME]
  8 KB (933 words) - 22:41, 23 May 2024

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