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- ...of unity come up when we examine the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. .... The magnitude of the RHS is 1, making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>3 KB (558 words) - 20:36, 11 December 2011
- ...on of the plane and a dilation of the plane having the common center. The order in which the composition is taken is not important. ...the spiral similarity, and the argument <math>\arg(k)</math> is the angle of rotation.28 KB (4,853 words) - 22:23, 19 November 2024
- ...gments <math>PA</math>, <math>PB</math>, and <math>PC</math> are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes t ...math>\overrightarrow{AQ}</math>, and denote by <math>|q|</math> the length of <math>q</math>. The given conditions may be written as8 KB (1,470 words) - 21:24, 18 June 2022
- ...at <math>B</math> with radius <math>m</math>. If we find the minimum value of <math>a^2+b^2</math>, which is the only variable quantity, and prove that i ...a^2+b^2</math> in terms of <math>c,d,m,\theta,\phi</math>, using the [[Law of Cosines]]:6 KB (1,096 words) - 22:07, 26 August 2017
- Let <math>a_k = 7k + 4</math>. Find the number of perfect squares among <math>\{a_1, a_2, \ldots, a_{2010}\}</math>. Find the last three digits of the number of 7-tuples of positive integers <math>(a_1, a_2, a_3, a_4, a_5, a_6, a_7)</math> such tha8 KB (1,377 words) - 12:20, 2 August 2024
- Prove that for any pair of positive integers <math>k</math> and <math>n</math>, there exist <math>k</m ...ive step:''' Let <math>n</math> be arbitrary and fixed. Case on the parity of <math>n</math>:4 KB (654 words) - 00:30, 19 November 2023
- Let <math>P</math> units be the increase in circumference of a circle resulting from an increase in <math>\pi</math> units in the diamet The real value of <math>x</math> such that <math>64^{x-1}</math> divided by <math>4^{x-1}</ma16 KB (2,571 words) - 13:13, 20 February 2020
- ...x^{x^x}</math> with <math>.9<x<1.0</math>. Arranged in order of increasing magnitude, they are: .... However, the order of this set will be reversed again (back to the order of the original set) when we take the logarithm base <math>x</math> a second t2 KB (373 words) - 06:56, 18 July 2024
- Which one of the following points is not on the graph of <math>y=\dfrac{x}{x+1}</math>? If the reciprocal of <math>x+1</math> is <math>x-1</math>, then <math>x</math> equals:20 KB (3,122 words) - 13:17, 20 February 2020
- We will prove each part of the compound inequality in <math>(\bigstar):</math> Moreover, this part of <math>(\bigstar)</math> is independent of the precondition <math>bc-ad=1.</math>11 KB (1,937 words) - 23:18, 22 October 2023
- How many polynomials of the form <math>x^5 + ax^4 + bx^3 + cx^2 + dx + 2020</math>, where <math>a</ ...hat <math>\frac{-1+i\sqrt{3}}{2} = e^{2\pi i / 3}</math>. That is because of Euler's Formula : <math>e^{ix} = \cos(x) + i \cdot \sin(x)</math>. <math>\f5 KB (834 words) - 13:30, 11 October 2024
- Find the number of complex numbers <math>z</math> satisfying <math>|z|=1</math> and <math>z^{8 ...need the real parts to sum to <math>-1</math>. We check all the multiples of 15 that result in <math>cis(x)</math> being negative, and find that only tw3 KB (465 words) - 17:09, 6 February 2023
- What is the value of <cmath>3+\frac{1}{3+\frac{1}{3+\frac13}}?</cmath> ...is <math>40</math> less than the second number. What is the absolute value of the difference between the first and second numbers?15 KB (2,249 words) - 09:27, 25 September 2024
- The <b> Chebyshev polynomials of the first kind </b> are defined [[Recursion|recursively]] by <cmath>\begin{ ==Proof of equivalence of the two definitions==10 KB (1,919 words) - 14:24, 26 June 2023
- ...ble|variables]] is the [[derivative]] of that function with respect to one of its arguments. ...</cmath> where the value <math>n</math> is the arity (number of arguments) of the function in question.4 KB (734 words) - 19:51, 4 May 2022
- ...mbol can also be used to express that two numbers have the same [[order of magnitude]].640 bytes (100 words) - 20:40, 20 October 2024