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- * [[Diophantine equation | Diophantine equations]] ** [[Pell equation | Pell equations]]734 bytes (79 words) - 17:27, 25 April 2008
- * [[Diophantine equation | Diophantine equations]] ** [[Pell equation | Pell equations]]1,015 bytes (104 words) - 16:21, 1 September 2024
- In [[trigonometry]], '''trigonometric identities''' are equations involving trigonometric functions that are true for all input values. Trigo [[Euler's identity]] states that <math>e^{ix} = \cos (x) + i \sin(x)</math>. We hav8 KB (1,397 words) - 20:55, 20 January 2024
- '''Euler's number''' is a [[constant]] that appears in a variety of mathematical con An approximation for Euler's number is <math>e\approx 2.7182818284590452...</math>4 KB (764 words) - 20:09, 13 March 2022
- ...roblems, including finding solutions to [[Diophantine equation|Diophantine equations]], testing whether certain large numbers are prime, and even some problems * [[Euler's Totient Function]]14 KB (2,317 words) - 18:01, 29 October 2021
- ...oint | de Longchamps point]] <math>L</math>. It is named after [[Leonhard Euler]]. Its existence is a non-trivial fact of Euclidean [[geometry]]. Certain Euler line is the central line <math>L_{647}</math>.59 KB (10,203 words) - 03:47, 30 August 2023
- ...s. One of the problems Joshua and Alexis work on boils down to a system of equations: Let <math>\phi(n)</math> denote <math>\textit{Euler's phi function}</math>, the number of integers <math>1\leq i\leq n</math> t71 KB (11,749 words) - 11:39, 20 November 2024
- We can easily solve this system of equations to find <math>a_1 = 983</math>, <math>a_2 = 26</math>, <math>a_3 = 5</math> ...tangent to <math>\mathcal{C}</math>, and <math>IO^2 = R^2 - 2Rr</math> by Euler's distance formula, so <math>4rm + R^2 - 2Rr = R^2 - 2Rr + r^2</math>, yiel36 KB (6,214 words) - 19:22, 13 July 2023
- The '''Four-Square Identity''', credited to [[Leonhard Euler]], states that for any eight [[Number|numbers]] <math>x_1,x_2, x_3, x_4, y_ Then Euler's Four-Square Identity simply reads <math>|XY|^2 = |X|^2 |Y|^2</math>; i.e.1 KB (179 words) - 22:24, 5 November 2019
- ...ted, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means Remember from [[Euler's identity|Euler's Formula]] that <math> \cos\theta+i\sin\theta=e^{i\theta} </math>.1 KB (170 words) - 19:12, 27 May 2012
- *First order differential equations **Modeling exponential growth/decay with differential equations5 KB (665 words) - 14:27, 3 March 2016
- ...if <math>OI=d</math>, <math>DO+OI=R+d</math> and <math>R+d=DL+LI=5</math>. Euler gives us that <math>d^{2}=R(R-2r)</math>, and in this case, <math>r=LI=2</m ...gives <math>z=5</math> and <math>x=10/3</math>, when we rewrite the above equations in terms of <math>w</math>. Thus, our answer is <math>\boxed{013}</math> an16 KB (2,817 words) - 07:39, 5 January 2025
- ==Solution 1 (Euler's formula and Substitution)== ...ou would if the RHS of all equations were real. Multiply the first and 3rd equations out and then factor out <math>60</math> to find <math>x^2</math>, then use11 KB (2,077 words) - 19:15, 12 January 2024
- ...nt-line approximations are used in [[Differential equations#Approximations|Euler's method]] and [[Newton's method]].7 KB (1,280 words) - 11:39, 9 December 2022
- Differential equations are often presented along with an initial condition; that is, a given value ...s </b> is a convenient technique for solving certain types of differential equations. Essentially, the method involves rewriting the equation so that each side6 KB (969 words) - 19:33, 15 December 2023
- ...analysis is to fluid dynamics. In particular, one can describe the [[Euler equations]] via the methods of vector analysis.9 KB (1,488 words) - 23:07, 19 January 2025
- ...gle <math>ABC,</math> the point of concurrence <math>X_{22}</math> lies on Euler line of triangle <math>ABC, \vec {X_{22}} = \vec O + \frac {2}{J^2–3} (\v ...ga_B</math> point <math>D</math> lies on line <math>OG</math> which is the Euler line of <math>\triangle ABC</math> as desired.6 KB (1,002 words) - 08:24, 25 November 2022
- Note Euler's formula where <math>\text{Vertices}+\text{Faces}-\text{Edges}=2</math>. T Euler's formula states that, for all convex polyhedra, <math>V-E+F=2</math>. In o6 KB (1,040 words) - 11:39, 3 January 2025
- ===Solution 1 (Euler's Totient Theorem)=== where the last step of all 3 congruences hold by the [[Euler's Totient Theorem]].64 KB (10,594 words) - 21:50, 9 January 2025
- We can now plug back into our initial equations for <math>AB \cdot AC</math>: The problem now becomes very simple if one knows Euler's Formula for the distance between the incenter and the circumcenter of a t22 KB (3,760 words) - 01:37, 13 January 2025