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- ...combinatorics, and number theory, along with sets of accompanying practice problems at the end of every section. ...88606&sprefix=after+school+maths+%2Caps%2C268&sr=8-2 100 Challenging Maths Problems]24 KB (3,198 words) - 19:44, 4 December 2024
- ...er sciences because it provides rigourous methods for developing models of complex phenomena. Such phenomena include the spread of computer viruses on a netwo ...\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up <br /> the base ten number system.}}6 KB (902 words) - 17:16, 22 October 2024
- ...are arguably a branch of [[elementary algebra]], and relate slightly to [[number theory]]. They deal with [[relations]] of [[variable]]s denoted by four sig For two [[number]]s <math>a</math> and <math>b</math>:12 KB (1,806 words) - 05:07, 19 June 2024
- ...re different topic tests ranging from trigonometry to analytic geometry to complex numbers. == Example Problems ==4 KB (632 words) - 16:09, 11 October 2020
- ...ntity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. Algebra can be used to solve equations as simple as 3x=9 but in some cases so complex that mathematicians have not figured how to solve the particular equation y3 KB (369 words) - 20:18, 18 June 2021
- == Algebraic Number Theory == ...hrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})</math>. Famous problems in algebraic number theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's5 KB (849 words) - 15:14, 18 May 2021
- ...n is the set <math>\{x:|x| \geq 3\}</math>, where <math>x</math> is a real number, because the square root is only defined when its argument is nonnegative. ...the pencil off the paper. To rigorously define continuous functions, more complex mathematics is necessary.10 KB (1,761 words) - 02:16, 12 May 2023
- ...acts that from the total number of possibilities. In problems that involve complex or tedious [[casework]], complementary counting is often a far simpler appr '''Solution''': We use a complementary approach. The total number of positive integers, with no restrictions, is <math>99</math> integers. Wh8 KB (1,192 words) - 16:20, 16 June 2023
- ...e with 5 possible answer choices. The remaining levels have tests with 30 problems, each multiple choice with 5 possible answer choices. ...thin the second one-third of the test is worth 4 points, and the remaining problems are worth 5 points. No penalty or partial credit is given to unanswered or6 KB (949 words) - 21:33, 17 November 2024
- == Problems == ...math> \ln(x) = \log_e(x).</math> However, in higher mathematics such as [[complex analysis]], the base 10 logarithm is typically disposed with entirely, the4 KB (680 words) - 11:54, 16 October 2023
- The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 ...th> i </math>, such that <math> i = \sqrt{-1} </math>. If we add this new number to the reals, we will have solutions to <math> x^2 = -1 </math>. It turns5 KB (860 words) - 14:36, 10 December 2023
- ...en as <math>1 \text{cis } \left(\frac{\pi}{2}\right)</math>. Any [[complex number]] can be expressed as <math>a+bi</math> for some real numbers <math>a</math ==Problems==2 KB (321 words) - 14:57, 5 September 2008
- ...ath> is the [[distance]] between <math>x</math> and [[zero]] on the real [[number line]]. The absolute value function exists among other contexts as well, including [[complex numbers]].2 KB (368 words) - 09:37, 5 January 2009
- ...th [[analytic geometry]], which is use of coordinates to solve geometrical problems. ...[[complex number]]s. (We can think of this as <math>n</math>-dimensional "complex Euclidean" space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_n]</math> be the [[p2 KB (361 words) - 00:59, 24 January 2020
- [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture [[complex number]]s except <math>s = 1</math>—see9 KB (1,547 words) - 02:04, 13 January 2021
- {{AMC12 Problems|year=2004|ab=A}} [[2004 AMC 12A Problems/Problem 1|Solution]]13 KB (1,953 words) - 23:31, 25 January 2023
- {{AMC12 Problems|year=2004|ab=B}} [[2004 AMC 12B Problems/Problem 1|Solution]]13 KB (2,049 words) - 12:03, 19 February 2020
- {{AMC12 Problems|year=2005|ab=B}} [[2005 AMC 12B Problems/Problem 1|Solution]]12 KB (1,781 words) - 13:59, 19 July 2024
- ...at <math>(\cos t + i \sin t)^n = \cos nt + i \sin nt</math> for all [[real number]]s <math>t</math> and all [[integer]]s <math>n</math>. So, we'd like to so ...i \sin u\right)^n = \cos nu + i\sin nu</math>. We know that two [[complex number]]s are equal if and only if both their [[real part]] and [[imaginary part]]6 KB (1,154 words) - 02:30, 11 January 2024
- === Solution 5 (Complex Numbers) === ...{e^{i \frac{\pi}{4}} \cdot (x + 450i)}{(x - 400) + 450i}</cmath> is a real number. Simplyfying using the fact that <math>e^{i \frac{\pi}{4}} = \dfrac{\sqrt{213 KB (2,080 words) - 12:14, 23 July 2024