Create the page "If only" on this wiki! See also the search results found.

• Mill's Constant
  ...we are not sure) and Mill's Constant is the smallest element in that set. If the [[Riemann Hypothesis]] is true, Mill's constant is approximately <math>
  794 bytes (105 words) - 01:59, 15 January 2022
• 2020 CIME II Problems/Problem 10
  ...ath>2c+6</math>. It follows that <math>a+b\in\{c,2c,3c,4c,5c\}</math>, for if <math>a+b\geq 6c</math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Right
  2 KB (332 words) - 09:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ..._i</math> for all <math>1 \leq i \leq n</math>, or if one list consists of only zeroes. Along with the [[AM-GM Inequality]], Cauchy-Schwarz forms the found ...math> such that <math>\overrightarrow{v} = t \overrightarrow{w}</math>, or if one of the vectors is zero. This formulation comes in handy in linear algeb
  13 KB (2,048 words) - 15:28, 22 February 2024
• Discriminant
  ...the discriminant is negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. *If <math>D=0</math>, then at least two of the roots are equal;
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  If line <math>PQ</math> intersecting <math>AB</math> on <math>\triangle ABC</m The converse of Menelaus' theorem is also true. If <math>\frac{BP}{PC} \cdot \frac{CQ}{QA} \cdot \frac{AR}{RB} = 1</math> in t
  5 KB (804 words) - 03:01, 12 June 2023
• Number theory
  Some branches of number theory may only deal with a certain subset of the real numbers, such as [[integer]]s, [[pos ...p://www.amazon.com/exec/obidos/ASIN/0262680289 (details)] Out of print but if you can find it in a library or used, you might love it and learn a lot. Wr
  3 KB (399 words) - 23:08, 8 January 2024
• Fermat's Little Theorem
  ...r arithmetic]] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is If <math>{a}</math> is an [[integer]], <math>{p}</math> is a [[prime number]]
  16 KB (2,658 words) - 16:02, 8 May 2024
• Parabola
  ...p) upwards/right if <math>a</math> is positive, and will be downwards/left if <math>a</math> is negative.
  3 KB (551 words) - 16:22, 13 September 2023
• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if On the other hand, if <math> a_1\geq a_2\geq ... \geq a_n </math> and
  1 KB (214 words) - 20:32, 13 March 2022
• Euler's Totient Theorem
  ...h>\{1,2,3\cdots{,n}\}</math> which are relatively prime to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[r .... All elements of <math>B</math> are relatively prime to <math>m</math> so if all elements of <math>B</math> are distinct, then <math>B</math> has the sa
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  The [[Isoperimetric Inequality]] states that if a figure in the plane has [[area]] <math>A</math> and [[perimeter]] <math>P ...e the total value of the expression. To prove this, all we need to show is if <math>0<A,B<180</math>, then <math>\sin(A+B)+\sin(A-B)<2\sin A</math>. This
  7 KB (1,296 words) - 14:22, 22 October 2023
• Brahmagupta's Formula
  ...formula]] for determining the [[area]] of a [[cyclic quadrilateral]] given only the four side [[length]]s, given as follows: <cmath>[ABCD] = \sqrt{(s-a)(s- If we draw <math>AC</math>, we find that <math>[ABCD]=\frac{ab\sin B}{2}+\frac
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...> be the regular heptagon. Consider the quadrilateral <math> ABCE </math>. If <math> a </math>, <math> b </math>, and <math> c </math> represent the leng
  7 KB (1,198 words) - 20:39, 9 March 2024
• Elementary symmetric sum
  ...th>k</math> of those numbers (<math>1 \leq k \leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>,
  2 KB (275 words) - 12:51, 26 July 2023
• Cyclic quadrilateral
  ...rty is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
  1 KB (162 words) - 20:39, 9 March 2024
• Algebra
  As if to add to the confusion, "[[algebra (structure) |algebra]]" is the name for
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...mmarize the group law by saying <math>P+Q+R=\mathcal{O}</math> if and only if <math>P,Q</math> and <math>R</math> lie on a line. ...ssociated to <math>z</math> and <math>z'</math> are isomorphic if and only if there is a relation
  5 KB (849 words) - 16:14, 18 May 2021
• 1959 IMO Problems/Problem 2
  ...<math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots? '''Case I:''' If <math>x \le 1</math>, then <math>|x-1| = 1 - x</math>, and the equation red
  3 KB (466 words) - 12:04, 12 April 2024
• Harmonic mean
  ...to positive numbers is necessary to avoid division by zero. For instance, if we tried to take the harmonic mean of the set <math>\{-2, 3, 6\}</math> we
  1 KB (196 words) - 00:49, 6 January 2021
• Euler's totient function
  ...an or equal to <math> n </math> that are divisible by <math> p_1 </math>. If we do the same for each <math> p_i </math> and add these up, we get (d) If <math>p</math> is prime and <math>n\ge{1},</math> then <math>\phi(p^n)=p^n-
  5 KB (898 words) - 19:12, 28 January 2024

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