Search results

  • ...[phinary]], which is base [[phi]]; others include "[[Fibonacci base]]" and base negative two. [[Binary]] is base 2. It's a favorite among computer programmers. It has just two digits: <ma
    2 KB (351 words) - 09:39, 1 October 2015
  • ...with entirely, the symbol <math>\log</math> is taken to mean the logarithm base <math>e</math> and the symbol <math>\ln</math> is not used at all. (This i All logarithms are undefined in nonpositive reals, as they are complex. From the identity <math>e^{i\pi}=-1</math>, we have <math>\ln (-1)=i\pi</m
    4 KB (680 words) - 11:54, 16 October 2023
  • ...e segment]]s through elementary means. One can find the area of even more complex regions via the use of [[calculus]]. ...ly find the area of a [[triangle]] by just noting that if our triangle has base <math>b</math> and height <math>h</math>, then the rectangle with length <m
    7 KB (1,185 words) - 19:48, 25 November 2024
  • ...cient]]s with <math>c_{2004}\not = 0</math> and <math>2004</math> distinct complex [[zero]]es <math>z_k = a_k + b_ki</math>, <math>1\leq k\leq 2004</math> wit For each integer <math>n\geq 4</math>, let <math>a_n</math> denote the base-<math>n</math> number <math>0.\overline{133}_n</math>. The product <math>a_
    13 KB (1,953 words) - 23:31, 25 January 2023
  • ...th>, where <math>i=\sqrt{-1}</math> and <math>\overline{z}</math> is the [[complex conjugate]] of <math>z</math>. How many values of <math>z</math> satisfy bo has three distinct positive roots. If the sum of the base-<math>2</math> [[logarithm]]s of the roots is <math>5</math>, what is the v
    13 KB (2,049 words) - 12:03, 19 February 2020
  • A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <
    5 KB (822 words) - 04:55, 3 January 2025
  • ...rem on triangles <math>AOB</math> twice, first using <math>E</math> as the base point and then <math>F</math>, we arrive at the equations <cmath>(450 \sqrt === Solution 5 (Complex Numbers) ===
    13 KB (2,080 words) - 12:14, 23 July 2024
  • ...from the base of <math>20</math> is <math>\sqrt{69}</math>. Therefore, the complex number <math>\sqrt{69}+10i</math> represents the bisection of <math>\angle .../math>. Therefore, we can add these two angles together by multiplying the complex numbers, finding
    12 KB (2,001 words) - 19:26, 23 July 2024
  • ...rface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid < The polynomial <math> P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} </math> has 34 complex roots of the form <math> z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2,
    9 KB (1,434 words) - 12:34, 29 December 2021
  • Suppose that the sum of the squares of two complex numbers <math>x</math> and <math>y</math> is <math>7</math> and the sum of ...uare base of side length <math>s</math>. The upper edge is parallel to the base and has length <math>2s</math>. All other edges have length <math>s</math>.
    7 KB (1,104 words) - 02:13, 27 May 2024
  • .../math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if ...h> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>0,1,2^{}_{},\ldots,n^2</ma
    7 KB (1,045 words) - 00:18, 5 January 2025
  • One base of a trapezoid is <math>100</math> units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid int Given that <math>z</math> is a complex number such that <math>z+\frac 1z=2\cos 3^\circ</math>, find the least inte
    6 KB (947 words) - 20:11, 19 February 2019
  • ...r</math> and <math>s</math> are integers, can be uniquely expressed in the base <math>-n+i</math> using the integers <math>0,1,2,\ldots,n^2</math> as digit to denote the base <math>-n+i</math> expansion of <math>r+si</math>. There are only finitely m
    2 KB (410 words) - 00:37, 25 August 2024
  • Consider the region <math>A</math> in the complex plane that consists of all points <math>z</math> such that both <math>\frac ...-circle with radius <math>20</math> minus an isosceles right triangle with base length <math>20</math>, and then doubled (to consider the entire overlapped
    2 KB (323 words) - 11:05, 16 July 2019
  • ...s. This [[function]] has the property that the image of each point in the complex plane is [[equidistant]] from that point and the [[origin]]. Given that <m Suppose we pick an arbitrary point on the [[complex plane]], say <math>(1,1)</math>. According to the definition of <math>f(z)<
    6 KB (1,010 words) - 18:01, 24 May 2023
  • ...ath>\angle FAB = 120^\circ</math>, we can use rewrite <math>F</math> using complex numbers: ...h>) and a parallelogram (<math>ABDE</math>, with height <math>8</math> and base <math>\frac{10}{\sqrt{3}}</math>).
    9 KB (1,472 words) - 14:24, 29 December 2024
  • ...1</math>cm is given. <math>P</math> is a point on the circumference of the base and the shortest path from <math>P</math> around the cone and back is drawn Find the sum of the squares of the roots, real or complex, of the system of simultaneous equations
    5 KB (848 words) - 22:49, 25 February 2017
  • Let <math>ABC</math> and <math>DBC</math> be isosceles triangle with the base <math>BC</math>. We know that <math>\angle ABD = \frac{\pi}{2}</math>. Let Let <math>A</math> be a <math>n\times n</math> matrix with complex elements and let <math>A^\star</math> be the classical adjoint of <math>A</
    11 KB (1,779 words) - 13:57, 7 May 2012
  • ...k</math>, where <math>d_i</math> denotes the <math>i</math>th digit in the base-<math>10</math> representation of <math>n</math> for <math>i = 1,2, \ldots, Find the number of distinct complex roots of <math>P_1 \cdot P_2 \cdot P_3</math>.
    8 KB (1,370 words) - 20:52, 27 February 2007
  • <math>\zeta_1, \zeta_2,</math> and <math>\zeta_3</math> are complex numbers such that <math>ABC</math> is an isosceles triangle with base <math>\overline{AB}</math>. <math>D</math> is a point on <math>\overline{AC
    7 KB (1,135 words) - 22:53, 24 March 2019

View (previous 20 | next 20) (20 | 50 | 100 | 250 | 500)