Mersenne Prime/Current Status

Mersenne Prime/Current Status 0 26909 484342 446311 2020-08-31T11:04:21Z Prime.mover 59 wikitext text/x-wiki == Currently known Mersenne Primes == The list of all known Mersenne primes is as follows: <onlyinclude> {{Begin-Mersenne-Primes}} {{Mersenne-Prime |  1 | 2 | 3 | 1 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  2 | 3 | 7 | 1 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  3 | 5 | 31 | 2 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  4 | 7 | 127 | 3 | Known to {{AuthorRef|Euclid}} | }} {{Mersenne-Prime |  5 | 13 | 8191 | 4 | 1456 | }} {{Mersenne-Prime |  6 | 17 | 131 \, 071 | 6 | 1588 | {{AuthorRef|Pietro Antonio Cataldi}} }} {{Mersenne-Prime |  7 | 19 | 524 \, 287 | 6 | 1588 | {{AuthorRef|Pietro Antonio Cataldi}} }} {{Mersenne-Prime |  8 | 31 | 2 \, 147 \, 483 \, 647 | 10 | 1772 | {{AuthorRef|Leonhard Paul Euler|}} }} {{Mersenne-Prime |  9 | 61 | 2 \cdotp 305 \times 10^{18} | 19 | 1883 | {{AuthorRef|Ivan Mikheevich Pervushin}} }} {{Mersenne-Prime | 10 | 89 | 6 \cdotp 189 \times 10^{26} | 27 | 1911 | {{AuthorRef|R.E. Powers}} }} {{Mersenne-Prime | 11 | 107 | 1 \cdotp 622 \times 10^{32} | 33 | 1914 | {{AuthorRef|R.E. Powers}} }} {{Mersenne-Prime | 12 | 127 | 1 \cdotp 701 \times 10^{38} | 39 | 1876 | {{AuthorRef|François Édouard Anatole Lucas}} }} {{Mersenne-Prime | 13 | 521 | 6 \cdotp 865 \times 10^{156} | 157 | 30 Jan 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 14 | 607 | 5 \cdotp 311 \times 10^{182} | 183 | 30 Jan 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 15 | 1279 | 1 \cdotp 041 \times 10^{385} | 386 | 25 Jun 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 16 | 2203 | 1 \cdotp 476 \times 10^{663} | 664 | 7 Oct 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 17 | 2281 | 4 \cdotp 461 \times 10^{686} | 687 | 9 Oct 1952 | {{AuthorRef|Raphael Mitchel Robinson}} }} {{Mersenne-Prime | 18 | 3217 | 2 \cdotp 591 \times 10^{968} | 969 | 8 Sept 1957 | {{AuthorRef|Hans Ivar Riesel}} }} {{Mersenne-Prime | 19 | 4253 | 1 \cdotp 908 \times 10^{1280} | 1281 | 3 Nov 1961 | {{AuthorRef|Alexander Hurwitz}} }} {{Mersenne-Prime | 20 | 4423 | 2 \cdotp 855 \times 10^{1331} | 1332 | 3 Nov 1961 | {{AuthorRef|Alexander Hurwitz}} }} {{Mersenne-Prime | 21 | 9689 | 4 \cdotp 782 \times 10^{2916} | 2917 | 11 May 1963 | {{AuthorRef|Donald Bruce Gillies}} }} {{Mersenne-Prime | 22 | 9941 | 3 \cdotp 461 \times 10^{2992} | 2993 | 16 May 1963 | {{AuthorRef|Donald Bruce Gillies}} }} {{Mersenne-Prime | 23 | 11 \, 213 | 2 \cdotp 814 \times 10^{3375} | 3376 | 2 Jun 1963 | {{AuthorRef|Donald Bruce Gillies}} }} {{Mersenne-Prime | 24 | 19 \, 937 | 4 \cdotp 315 \times 10^{6001} | 6002 | 4 Mar 1971 | {{AuthorRef|Bryant Tuckerman}} }} {{Mersenne-Prime | 25 | 21 \, 701 | 4 \cdotp 487 \times 10^{6532} | 6533 | 30 Oct 1978 | {{AuthorRef|Landon Curt Noll}} and {{AuthorRef|Ariel Nickel}} }} {{Mersenne-Prime | 26 | 23 \, 209 | 4 \cdotp 029 \times 10^{6986} | 6987 | 9 Feb 1979 | {{AuthorRef|Landon Curt Noll}} }} {{Mersenne-Prime | 27 | 44 \, 497 | 8 \cdotp 545 \times 10^{13 \, 394} | 13 \, 395 | 8 Apr 1979 | {{AuthorRef|Harry Lewis Nelson}} and {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 28 | 86 \, 243 | 5 \cdotp 369 \times 10^{25 \, 961} | 25 \, 962 | 25 Sept 1982 | {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 29 | 110 \, 503 | 5 \cdotp 219 \times 10^{33 \, 264} | 33 \, 265 | 28 Jan 1988 | {{AuthorRef|Walt Colquitt}} and {{AuthorRef|Luke Welsh}} }} {{Mersenne-Prime | 30 | 132 \, 049 | 5 \cdotp 127 \times 10^{39 \, 750} | 39 \, 751 | 19 Sept 1983 | {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 31 | 216 \, 091 | 7 \cdotp 461 \times 10^{65 \, 049} | 65 \, 050 | 1 Sept 1985 | {{AuthorRef|David Slowinski}} }} {{Mersenne-Prime | 32 | 756 \, 839 | 1 \cdotp 741 \times 10^{227 \, 831} | 227 \, 832 | 19 Feb 1992 | {{AuthorRef|David Slowinski}} and {{AuthorRef|Paul Gage}} }} {{Mersenne-Prime | 33 | 859 \, 433 | 1 \cdotp 295 \times 10^{258 \, 715} | 258 \, 716 | 4 Jan 1994 | {{AuthorRef|David Slowinski}} and {{AuthorRef|Paul Gage}} }} {{Mersenne-Prime | 34 | 1 \, 257 \, 787 | 4 \cdotp 122 \times 10^{378 \, 631} | 378 \, 632 | 3 Sept 1996 | {{AuthorRef|David Slowinski}} and {{AuthorRef|Paul Gage}} }} {{Mersenne-Prime | 35 | 1 \, 398 \, 269 | 8 \cdotp 147 \times 10^{420 \, 920} | 420 \, 921 | 13 Nov 1996 | GIMPS / {{AuthorRef|Joel Armengaud}} }} {{Mersenne-Prime | 36 | 2 \, 976 \, 221 | 6 \cdotp 233 \times 10^{895 \, 931} | 895 \, 932 | 24 Aug 1997 | GIMPS / {{AuthorRef|Gordon Spence}} }} {{Mersenne-Prime | 37 | 3 \, 021 \, 377 | 1 \cdotp 274 \times 10^{909 \, 525} | 909 \, 526 | 27 Jan 1998 | GIMPS / {{AuthorRef|Roland Clarkson}} }} {{Mersenne-Prime | 38 | 6 \, 972 \, 593 | 4 \cdotp 371 \times 10^{2 \, 098 \, 959} | 2 \, 098 \, 960 | 1 Jun 1999 | GIMPS / {{AuthorRef|Nayan Hajratwala}} }} {{Mersenne-Prime | 39 | 13 \, 466 \, 917 | 9 \cdotp 249 \times 10^{4 \, 053 \, 945} | 4 \, 053 \, 946 | 14 Nov 2001 | GIMPS / {{AuthorRef|Michael Cameron}} }} {{Mersenne-Prime | 40 | 20 \, 996 \, 011 | 1 \cdotp 260 \times 10^{6 \, 320 \, 429} | 6 \, 320 \, 430 | 17 Nov 2003 | GIMPS / {{AuthorRef|Michael Shafer}} }} {{Mersenne-Prime | 41 | 24 \, 036 \, 583 | 2 \cdotp 994 \times 10^{7 \, 235 \, 732} | 7 \, 235 \, 733 | 15 May 2004 | GIMPS / {{AuthorRef|Josh Findley}} }} {{Mersenne-Prime | 42 | 25 \, 964 \, 951 | 1 \cdotp 222 \times 10^{7 \, 816 \, 229} | 7 \, 816 \, 230 | 18 Feb 2005 | GIMPS / {{AuthorRef|Martin Nowak}} }} {{Mersenne-Prime | 43 | 30 \, 402 \, 457 | 3 \cdotp 154 \times 10^{9 \, 152 \, 051} | 9 \, 152 \, 052 | 15 Dec 2005 | GIMPS / {{AuthorRef|Curtis Cooper}} and {{AuthorRef|Steven Boone}} }} {{Mersenne-Prime | 44 | 32 \, 582 \, 657 | 1 \cdotp 246 \times 10^{9 \, 808 \, 358} | 9 \, 808 \, 358 | 4 Sept 2006 | GIMPS / {{AuthorRef|Curtis Cooper}} and {{AuthorRef|Steven Boone}} }} {{Mersenne-Prime | 45 | 37 \, 156 \, 667 | 2 \cdotp 023 \times 10^{11 \, 185 \, 271} | 11 \, 185 \, 272 | 6 Sept 2008 | GIMPS / {{AuthorRef|Hans-Michael Elvenich}} }} {{Mersenne-Prime | 46 | 42 \, 643 \, 801 | 1 \cdotp 699 \times 10^{12 \, 837 \, 063} | 12 \, 837 \, 064 | 12 Apr 2009 | GIMPS / {{AuthorRef|Odd Magnar Strindmo}} }} {{Mersenne-Prime | 47 | 43 \, 112 \, 609 | 3 \cdotp 165 \times 10^{12 \, 978 \, 188} | 12 \, 978 \, 189 | 23 Aug 2008 | GIMPS / {{AuthorRef|Edson Smith}} }} {{Mersenne-Prime |  | 57 \, 885 \, 161 | 5 \cdotp 818 \times 10^{17 \, 425 \, 169} | 17 \, 425 \, 170 | 25 Jan 2013 | GIMPS / {{AuthorRef|Curtis Cooper}} }} {{Mersenne-Prime |  | 74 \, 207 \, 281 | 3 \cdotp 003 \times 10^{22 \, 338 \, 617} | 22 \, 338 \, 618 | 07 Jan 2016 | GIMPS / {{AuthorRef|Curtis Cooper}} }} {{Mersenne-Prime |  | 77 \, 232 \, 917 | 4 \cdotp 673 \times 10^{23 \, 249 \, 424} | 23 \, 249 \, 425 | 26 Dec 2017 | GIMPS / {{AuthorRef|Jon Pace}} }} {{Mersenne-Prime |  | 82 \, 589 \, 933 | 1 \cdotp 488 \times 10^{24 \, 862 \, 047} | 24 \, 862 \, 048 | 07 Dec 2018 | GIMPS / {{AuthorRef|Patrick Laroche}} }} {{End-Mersenne-Primes}} Note that the index numbers of Mersenne primes after no. $47$ are uncertain, as there may still be undiscovered Mersenne primes between the $47$th and $51$st. Not all numbers in that range have been explored yet. </onlyinclude> == Also see == {{OEIS|A000668}} {{OEIS|A000043|order = index}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Barlow's Prediction|next = Do Mersenne Primes Go On For Ever?}}: $28$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Definition:Mersenne Prime/Historical Note|next = Mathematician:David Slowinski}}: Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Barlow's Prediction|next = Do Mersenne Primes Go On For Ever?}}: $28$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Mersenne Prime/Historical Note|next = Theorem of Even Perfect Numbers|entry = Mersenne numbers}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Mersenne Prime/Historical Note|next = Theorem of Even Perfect Numbers|entry = Mersenne numbers}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Definition:Mersenne Prime/Historical Note|next = Mathematician:Diophantus of Alexandria}}: Chapter $7$: Patterns in Numbers: Euclid * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Mersenne Prime|next = 1,257,787/Historical Note|entry = Mersenne prime}} Category:Mersenne Primes regie24mutkxbdw88rl7k3c450bvvcg


Divisibility by 9/Corollary

Divisibility by 9/Corollary 0 27285 380675 376067 2018-12-08T15:51:59Z Prime.mover 59 wikitext text/x-wiki == Corollary to Divisibility by 9 == <onlyinclude> A number expressed in decimal notation is divisible by $3$ {{iff}} the sum of its digits is divisible by $3$. That is: :$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $3$  {{iff}}: :$a_0 + a_1 + \ldots + a_n$ is divisible by $3$. </onlyinclude> == Proof == From Divisibility by 9 we have that: :$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $3^2$  {{iff}}: :$a_0 + a_1 + \ldots + a_n$ is divisible by $3^2$. So: {{begin-eqn}} {{eqn | l = \paren {a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n}       | o = \equiv       | r = \paren {a_0 + a_1 + a_2 + \cdots + a_n}       | rr= \pmod {3^2}       | c =  }} {{eqn | ll= \leadstoandfrom       | l = \paren {a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n}       | o = \equiv       | r = \paren {a_0 + a_1 + a_2 + \cdots + a_n}       | rr= \pmod 3       | c = Congruence by Divisor of Modulus }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Divisibility by 9/Proof 1|next = Binomial Theorem/Integral Index}}: $\S 2.6$. Algebra of congruences: Example $41$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Integer as Sum of Three Squares|next = Vinogradov's Theorem/Corollary 2}}: $3$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Integer as Sum of Three Squares|next = Vinogradov's Theorem/Corollary 2}}: $3$ Category:Divisibility Tests Category:3 c6f43kipzt6rfbqz66egca9gwvlk9ua


Square Root/Examples/2

Square Root/Examples/2 0 27339 447779 435327 2020-02-09T15:40:28Z Prime.mover 59 wikitext text/x-wiki == Decimal Expansion == <onlyinclude> The decimal expansion of the square root of $2$ starts: :$\sqrt 2 \approx 1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$ </onlyinclude> {{OEIS|A002193}} == Also known as == :$\sqrt 2$ is sometimes known as '''{{AuthorRef|Pythagoras of Samos|Pythagoras}}'s constant''', for {{AuthorRef|Pythagoras of Samos}}. == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Euler's Number/Limit of Sequence|next = Square Root/Examples/3}}: $\S 1$: Special Constants: $1.3$ * {{BookReference|Theory and Problems of Differential and Integral Calculus|1972|Frank Ayres, Jr.|author2 = J.C. Ault|ed = SI|edpage = SI Edition|prev = Definition:Real Number|next = Definition:Pi/Decimal Expansion}}: Chapter $1$: Variables and Functions: The Set of Real Numbers * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Square Root of 2 is Irrational|next = Pi is Irrational}}: $1$: Complex Numbers: The Real Number System: $4$ * {{BookReference|Les Nombres Remarquables|1983|François Le Lionnais|author2 = Jean Brette|prev = Number of Primes of Form n^2 + 1/Historical Note|next = Length of Diagonal of Unit Square}}: $1,41421 35623 73 \ldots$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Doubling the Cube/Cissoid of Diocles/Historical Note|next = Length of Diagonal of Unit Square}}: $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Doubling the Cube/Cissoid of Diocles/Historical Note|next = Length of Diagonal of Unit Square}}: $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$ Category:Square Root of 2 Category:Examples of Square Roots Category:Square Root of 2 is Irrational paqwv8ns78wqocfd6f2v6mctkqevyn0


Square Root/Examples/5

Square Root/Examples/5 0 27342 433313 393211 2019-11-01T09:07:52Z Prime.mover 59 wikitext text/x-wiki == Decimal Expansion == <onlyinclude> The decimal expansion of $\sqrt 5$ starts: :$\sqrt 5 \approx 2 \cdotp 23606 \, 79774 \, 99789 \, 6964 \ldots$ </onlyinclude> {{OEIS|A002163}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Square Root of 3|next = Cube Root of 2}}: $\S 1$: Special Constants: $1.5$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Definition:Wallis's Number/Historical Note|next = Natural Logarithm of 10}}: $2 \cdotp 236 \, 067 \ldots$ Category:Examples of Square Roots oeqnocm2i451v28jlvpueviyb2j3u6a


Axiom:Euclid's Second Postulate/Production 100 27568 446523 424515 2020-02-05T12:19:33Z Prime.mover 59 wikitext text/x-wiki == Definition == <onlyinclude> The process of extending a straight line segment is called '''production'''. Such a straight line segment is said to be '''produced'''. The line segment can be extended in either direction, thus producing an infinite half-line, or in both directions, thus producing an infinite straight line. </onlyinclude> {{EuclidPostulateStatement|Second|To produce a finite straight line continuously in a straight line.}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Proclus Lycaeus|next = Definition:Product|entry = produce}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Proclus Lycaeus|next = Definition:Producer's Risk|entry = produce}} Category:Definitions/Geometry r4zwvubgslksrb9h401ckzxm1rfn7yx


Sine and Cosine are Periodic on Reals/Corollary 0 27870 441347 400163 2019-12-29T08:52:42Z Prime.mover 59 wikitext text/x-wiki {{mergeto|Cosine of Angle plus Straight Angle]] and [[Sine of Angle plus Straight Angle}} == Corollary to Sine and Cosine are Periodic on Reals == <onlyinclude> :$\map \cos {x + \pi} = -\cos x$ :$\map \sin {x + \pi} = -\sin x$ :$\cos x$ is strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$ and strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$ :$\sin x$ is strictly positive on the interval $\openint 0 \pi$ and strictly negative on the interval $\openint \pi {2 \pi}$ </onlyinclude> == Proof == From Sine and Cosine are Periodic on Reals: :$\map \sin {x + \dfrac \pi 2} = \cos x$ :$\map \cos {x + \dfrac \pi 2} = -\sin x$ Thus: :$\map \sin {x + \pi} = \map \cos {x + \dfrac \pi 2} = -\sin x$ :$\map \cos {x + \pi} = -\map \sin {x + \dfrac \pi 2} = -\cos x$ It follows directly that: :$\forall x \in \closedint {-\dfrac \pi 2} {\dfrac \pi 2}: \cos x \ge 0$ Hence: :$\forall x \in \closedint {\dfrac \pi 2} {\dfrac {3 \pi} 2}: \cos x \le 0$ The result for $\sin x$ follows similarly, or we can use: :$\map \sin {x + \dfrac \pi 2} = \cos x$ {{qed}} Category:Sine Function Category:Cosine Function qgvtqrzg1h4bt2cizdu44meq4qeausj


Sum of Squares of Sine and Cosine/Corollary 2 0 28010 396161 310516 2019-03-18T21:07:09Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Squares of Sine and Cosine == For all $x \in \C$: <onlyinclude> :$\csc^2 x - \cot^2 x = 1 \quad \text {(when $\sin x \ne 0$)}$ </onlyinclude> where $\csc$, $\cot$ and $\sin$ are cosecant, cotangent and sine respectively. == Proof  == When $\sin x \ne 0$: {{begin-eqn}} {{eqn | l = \sin^2 x + \cos^2 x       | r = 1       | c = Sum of Squares of Sine and Cosine }} {{eqn | ll= \leadsto       | l = 1 + \frac {\cos^2 x} {\sin^2 x}       | r = \frac 1 {\sin^2 x}       | c = dividing both sides by $\sin^2 x$, as $\sin x \ne 0$ }} {{eqn | ll= \leadsto       | l = 1 + \cot^2 x       | r = \csc^2 x       | c = {{Defof|Cotangent}} and {{Defof|Cosecant}} }} {{eqn | ll= \leadsto       | l = \csc^2 x - \cot^2 x       | r = 1       | c = rearranging }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\csc^2 x = 1 + \cot^2 x \quad \text{(when $\sin x \ne 0$)}$ or: :$\cot^2 x = \csc^2 x - 1 \quad \text{(when $\sin x \ne 0$)}$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Difference of Squares of Secant and Tangent|next = Shape of Sine Function}}: $\S 5$: Trigonometric Functions: $5.21$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Difference of Squares of Secant and Tangent|next = Sine Function is Odd}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ Category:Cosecant Function Category:Cotangent Function Category:Sum of Squares of Sine and Cosine 9itjzv9n2qejn4wh2opxqmb342lp1o1


Kernel of Linear Transformation is Orthocomplement of Range of Adjoint/Corollary 0 28057 477161 414675 2020-07-06T05:42:36Z Prime.mover 59 wikitext text/x-wiki == Corollary to Kernel of Linear Transformation is Orthocomplement of Range of Adjoint == Let $H$ be a Hilbert space. <onlyinclude> Let $A \in \map B H$ be a normal operator. Then: :$\ker A = \paren {\Rng A}^\perp$ </onlyinclude> where: :$A^*$ denotes the adjoint of $A$ :$\ker A$ is the kernel of $A$ :$\Rng A$ is the range of $A$ :$\perp$ signifies orthocomplementation == Proof == {{begin-eqn}} {{eqn | l = \ker A       | r = \paren {\Rng {A^*} }^\perp       | c = Kernel of Linear Transformation is Orthocomplement of Range of Adjoint }} {{eqn | ll= \leadsto       | l = \ker A^*       | r = \paren {\Rng {A^*} }^\perp       | c = Kernel of Normal Operator is Kernel of Adjoint }} {{eqn | ll= \leadsto       | l = \ker A^{**}       | r = \paren {\Rng {A^{**} } }^\perp       | c = substituting $A^*$ for $A$ }} {{eqn | ll= \leadsto       | l = \ker A       | r = \paren {\Rng A}^\perp       | c = Double Adjoint is Itself }} {{end-eqn}} {{qed}} Category:Adjoints Category:Linear Transformations on Hilbert Spaces jcdx00qq1bfueds6rc7iciu6120mx4t


Bisection of Angle in Cartesian Plane/Corollary 0 28065 429534 154560 2019-10-04T22:39:12Z Prime.mover 59 wikitext text/x-wiki == Corollary to Bisection of Angle in Cartesian Plane == Let $\theta$ be the angular coordinate of a point $P$ in a polar coordinate plane. Let $QOR$ be a straight line that bisects the angle $\theta$. <onlyinclude> If $\theta$ is in quadrant I or quadrant II, then the angular coordinates of $Q$ and $R$ are in quadrant I and quadrant III. If $\theta$ is in quadrant III or quadrant IV, then the angular coordinates of $Q$ and $R$ are in quadrant II and quadrant IV. </onlyinclude> == Proof == :500px From Bisection of Angle in Cartesian Plane, the angular coordinates of $Q$ and $R$ are $\dfrac \theta 2$ and $\pi + \dfrac \theta 2$. {{WLOG}}, let $\angle Q = \dfrac \theta 2$ and $\angle R = \pi + \dfrac \theta 2$. Let $\theta$ be in quadrant I or quadrant II. Then $0 < \theta < \pi$. Dividing each term in the inequality by $2$: :$0 < \dfrac \theta 2 < \dfrac \pi 2$ and so $Q$ lies in quadrant I. Adding $\pi$ to each expression in the inequality: :$\pi < \pi + \dfrac \theta 2 < \dfrac {3 \pi} 2$ and so $R$ lies in quadrant I. {{qed|lemma}} Let $\theta$ be in quadrant III or quadrant IV. Then $\pi < \theta < 2 \pi$. Dividing each term in the inequality by $2$: :$\dfrac \pi 2 < \dfrac \theta 2 < \pi$ and so $Q$ lies in quadrant II. Adding $\pi$ to each expression in the inequality: :$\dfrac {3 \pi} 2 < \pi + \dfrac \theta 2 < 2 \pi$ and so $R$ lies in quadrant IV. {{qed}} Category:Analytic Geometry 5baopujzi1710fuimqg1baff7pfmldl


Shape of Cosecant Function/Graph

Shape of Cosecant Function/Graph 0 28287 155043 2013-08-05T21:22:52Z Prime.mover 59 Created page with "== Theorem ==  The graph of the cosecant function appears as:  <onlyinclude> :800px </onlyinclude..." wikitext text/x-wiki == Theorem == The graph of the cosecant function appears as: <onlyinclude> :800px </onlyinclude> == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Graph of Secant Function|next=Sine Function is Odd}}: $\S 5$: Trigonometric Functions: $5.27$ Category:Cosecant Function 7wxbvz76udx7omx41aox29v4y85hk3f


Tangent of Sum/Corollary

Tangent of Sum/Corollary 0 28295 451135 396176 2020-02-28T16:13:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Tangent of Sum == <onlyinclude> :$\map \tan {a - b} = \dfrac {\tan a - \tan b} {1 + \tan a \tan b}$ </onlyinclude> where $\tan$ is tangent. == Proof == {{begin-eqn}} {{eqn | l = \map \tan {a - b}       | r = \frac {\tan a + \map \tan {-b} } {1 - \tan a \, \map \tan {-b} }       | c = Tangent of Sum }} {{eqn | r = \frac {\tan a - \tan b} {1 + \tan a \tan b}       | c = Tangent Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Tangent of Sum|next = Cotangent of Sum}}: $\S 5$: Trigonometric Functions: $5.36$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Tangent of Sum|next = Definition:Hyperbolic Sine}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent of Sum|next = Prosthaphaeresis Formulas|entry = compound angle formulae|subentry = in trigonometry}} Category:Tangent Function Category:Compound Angle Formulas mefu39i4ew9frgauq5j67uwkbdxd4v2


Cotangent of Sum/Corollary

Cotangent of Sum/Corollary 0 28297 405404 405403 2019-05-21T06:22:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to Cotangent of Sum == <onlyinclude> :$\map \cot {a - b} = \dfrac {\cot a \cot b + 1} {\cot b - \cot a}$ </onlyinclude> where $\cot $ is cotangent. == Proof == {{begin-eqn}} {{eqn | l = \map \cot {a - b}       | r = \frac {\cot a \, \map \cot {-b} - 1} {\cot a + \map \cot {-b} }       | c = Cotangent of Sum }} {{eqn | r = \frac {-\cot a \cot b - 1} {\cot a - \cot b}       | c = Cotangent Function is Odd }} {{eqn | r = \frac {\cot a \cot b + 1} {\cot b - \cot a}       | c = multiplying numerator and denominator by $-1$ and rearranging }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cotangent of Sum|next = Trigonometric Functions of Negative Angle}}: $\S 5$: Trigonometric Functions: $5.37$ Category:Cotangent Function 5k1x25al0ub4ob134b1ahvttpe3llqk


Double Angle Formulas/Cosine/Corollary 1 0 28372 428443 363776 2019-09-29T10:11:48Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Cosine  == <onlyinclude> :$\cos 2 \theta = 2 \cos^2 \theta - 1$ </onlyinclude> where $\cos$ denotes cosine. == Proof  == {{begin-eqn}} {{eqn | l = \cos 2 \theta       | r = \cos^2 \theta - \sin^2 \theta       | c = Double Angle Formula for Cosine }} {{eqn | r = \cos^2 \theta - \paren {1 - \cos^2 \theta}       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 2 \cos^2 \theta - 1 }} {{end-eqn}} {{qed}} == Also see == * Square of Cosine: $\cos^2 \theta = \dfrac {1 + \cos 2 \theta} 2$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Double Angle Formulas/Cosine/Corollary 2|next = Double Angle Formula for Tangent}}: $\S 5$: Trigonometric Functions: $5.39$ Category:Cosine Function Category:Double Angle Formula for Cosine d84haju1i992an6yk6pc6k6hgfn61ig


Double Angle Formulas/Cosine/Corollary 2 0 28373 428428 314404 2019-09-29T09:44:53Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Cosine == <onlyinclude> :$\cos 2 \theta = 1 - 2 \sin^2 \theta$ </onlyinclude> where $\cos$ and $\sin$ denote cosine and sine respectively. == Proof == {{begin-eqn}} {{eqn | l = \cos 2 \theta       | r = \cos^2 \theta - \sin^2 \theta       | c = Double Angle Formula for Cosine }} {{eqn | r = \paren {1 - \sin^2 \theta} - \sin^2 \theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 1 - 2 \sin^2 \theta }} {{end-eqn}} {{qed}} == Also see == * Square of Sine: $\sin^2 \theta = \dfrac {1 - \cos 2 \theta} 2$ == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Double Angle Formula for Cosine|next = Double Angle Formulas/Cosine/Corollary 1}}: $\S 5$: Trigonometric Functions: $5.39$ Category:Cosine Function Category:Double Angle Formula for Cosine 8jjxrseaf5s53xdzmyowd2wikpxx2v1


Double Angle Formulas/Sine/Corollary

Double Angle Formulas/Sine/Corollary 0 28375 452789 419747 2020-03-05T21:49:19Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Sine == <onlyinclude> :$\sin 2 \theta = \dfrac {2 \tan \theta} {1 + \tan^2 \theta}$ </onlyinclude> where $\sin$ and $\tan$ denote sine and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \sin 2 \theta       | r = 2 \sin \theta \cos \theta       | c = Double Angle Formula for Sine }} {{eqn | r = 2 \sin \theta \cos \theta \frac {\cos \theta} {\cos \theta}       | c =  }} {{eqn | r = 2 \tan \theta \cos^2 \theta       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {2 \tan \theta} {\sec^2 \theta}       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \frac {2 \tan \theta} {1 + \tan^2 \theta}       | c = Difference of Squares of Secant and Tangent }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Half Angle Formula for Sine|next = Tangent Half-Angle Substitution for Cosine|entry = half-angle formula}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent Half-Angle Substitution Formulas|next = Tangent Half-Angle Substitution for Cosine|entry = $t$-formulae}} Category:Sine Function Category:Tangent Function Category:Double Angle Formula for Sine c0yrzrs3aqrnzj45c2snqu25i1eukvq


Tangent Half-Angle Substitution for Cosine 0 28376 452788 419753 2020-03-05T21:48:24Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Cosine == <onlyinclude> :$\cos 2 \theta = \dfrac {1 - \tan^2 \theta} {1 + \tan^2 \theta}$ </onlyinclude> where $\cos$ and $\tan$ denote cosine and tangent respectively. == Proof == {{begin-eqn}} {{eqn | l = \cos 2 \theta       | r = \cos^2 \theta - \sin^2 \theta       | c = Double Angle Formula for Cosine }} {{eqn | r = \paren {\cos^2 \theta - \sin^2 \theta} \frac {\cos^2 \theta}{\cos^2 \theta}       | c =  }} {{eqn | r = \paren {1 - \tan^2 \theta} \cos^2 \theta       | c = Tangent is Sine divided by Cosine }} {{eqn | r = \frac {1 - \tan^2 \theta} {\sec^2 \theta}       | c = Secant is Reciprocal of Cosine }} {{eqn | r = \frac {1 - \tan^2 \theta} {1 + \tan^2 \theta}       | c = Difference of Squares of Secant and Tangent }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent Half-Angle Substitution for Sine|next = Double Angle Formula for Tangent|entry = half-angle formula}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Tangent Half-Angle Substitution for Sine|next = Mathematician:Thales of Miletus|entry = $t$-formulae}} Category:Cosine Function Category:Tangent Function Category:Double Angle Formula for Cosine cmell4vk6px0vbhnzeyqdcz456yc505


Quintuple Angle Formulas

Quintuple Angle Formulas 0 28391 155470 2013-08-07T19:50:49Z Prime.mover 59 Created page with "== Theorem == <onlyinclude> === Quintuple Angle Formula for Sine === {{:Quintuple Angle Formulas/Sine}}  === Quintuple Angle Formulas/Cosin..." wikitext text/x-wiki == Theorem == <onlyinclude> === Quintuple Angle Formula for Sine === {{:Quintuple Angle Formulas/Sine}} === Quintuple Angle Formula for Cosine === {{:Quintuple Angle Formulas/Cosine}} === Quintuple Angle Formula for Tangent === {{:Quintuple Angle Formulas/Tangent}} </onlyinclude> where $\sin, \cos, \tan$ denote sine, cosine and tangent respectively. Category:Trigonometric Identities nok0n01h6jkfbx2i6u2owqr41zd7l9b


Sum of Geometric Sequence/Proof 4/Lemma

Sum of Geometric Sequence/Proof 4/Lemma 0 28621 456250 456109 2020-03-19T10:04:09Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $n \in \N_{>0}$. Then: :$\displaystyle \paren {1 - x} \sum_{i \mathop = 0}^{n - 1} x^i = 1 - x^n$ </onlyinclude> == Proof == Proof by induction on $n$: For all $n \in \N_{> 0}$, let $\map P n$ be the proposition: :$\displaystyle \paren {1 - x} \sum_{i \mathop = 0}^{n - 1} x^i = 1 - x^n$ === Basis for the Induction === $\map P 1$ is the case: {{begin-eqn}} {{eqn | l = \paren {1 - x} \sum_{i \mathop = 1}^{1 - 1} x^i       | r = 1 \paren {1 - x} }} {{eqn | r = 1 - x }} {{eqn | r = 1 - x^1 }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true. So this is our induction hypothesis: :$\displaystyle \paren {1 - x} \sum_{i \mathop = 0}^{k - 1} x^i = 1 - x^k$ Then we need to show: :$\displaystyle \paren {1 - x} \sum_{i \mathop = 0}^k x^i = 1 - x^{k + 1}$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = \paren {1 - x} \sum_{i \mathop = 0}^k x^i       | r = \paren {1 - x} \paren {x^k + \sum_{i \mathop = 0}^{k - 1} x^i} }} {{eqn | r = x^k - x^{k + 1} + \paren {1 - x} \sum_{i \mathop = 0}^{k - 1} x^i }} {{eqn | r = x^k - x^{k + 1} + 1 - x^k       | c = from the induction hypothesis }} {{eqn | r = 1 - x^{k + 1}       | c = gathering terms }} {{end-eqn}} So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\displaystyle \paren {1 - x} \sum_{i \mathop = 0}^{n - 1} x^i = 1 - x^n$ {{qed}} Category:Sum of Geometric Sequence r79zvqlj9xu8id3oy0keckf5iabeyqf


General Distributivity Theorem/Lemma 1

General Distributivity Theorem/Lemma 1 0 28638 353098 353078 2018-05-02T21:19:51Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\left({R, \circ, *}\right)$ be a ringoid. Then for every sequence $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ of elements of $R$, and for every $b \in R$: :$\displaystyle \left({\sum_{j \mathop = 1}^n a_j}\right) * b = \sum_{j \mathop = 1}^n \left({a_j * b}\right)$ </onlyinclude> where: :$\displaystyle \sum_{j \mathop = 1}^n a_j$ is the composite $a_1 \circ a_2 \circ \cdots \circ a_n$ :$n$ is a strictly positive integer: $n \in \Z_{> 0}$. == Proof == The proof proceeds by the Principle of Mathematical Induction. Recall that as $\left({R, \circ, *}\right)$ is a ringoid, $*$ is distributive over $\circ$: :$\forall a, b, c \in S: \left({a \circ b}\right) * c = \left({a * c}\right) \circ \left({b * c}\right)$ For all $n \in \Z_{> 0}$, let $P \left({n}\right)$ be the proposition: :$\displaystyle \left({\sum_{j \mathop = 1}^n a_j}\right) * b = \sum_{j \mathop = 1}^n \left({a_j * b}\right)$ === Basis for the Induction === $P(1)$ is true, as this just says: :$a_1 * b = a_1 * b$ $P(2)$ is the case: {{begin-eqn}} {{eqn | l = \left({\sum_{j \mathop = 1}^2 a_j}\right) * b       | r = \left({a_1 \circ a_2}\right) * b       | c = Definition of Composite }} {{eqn | r = \left({a_1 * b}\right) \circ \left({a_2 * b}\right)       | c = $*$ is distributive over $\circ$ as $\left({R, \circ, *}\right)$ is a ringoid }} {{eqn | r = \sum_{j \mathop = 1}^2 \left({a_j * b}\right)       | c = Definition of Composite }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true. So this is our induction hypothesis: :$\displaystyle \left({\sum_{j \mathop = 1}^k a_j}\right) * b = \sum_{j \mathop = 1}^k \left({a_j * b}\right)$ Then we need to show: :$\displaystyle \left({\sum_{j \mathop = 1}^{k + 1} a_j}\right) * b = \sum_{j \mathop = 1}^{k + 1} \left({a_j * b}\right)$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = \left({\sum_{j \mathop = 1}^{k + 1} a_j}\right) * b       | r = \left({\left({\sum_{j \mathop = 1}^k a_j}\right) \circ a_{k+1} }\right) * b       | c = Definition of Composite }} {{eqn | r = \left({\left({\sum_{j \mathop = 1}^k a_j}\right) * b}\right) \circ \left({a_{k+1} * b}\right)       | c = Basis for the Induction }} {{eqn | r = \left({\sum_{j \mathop = 1}^k \left({a_j * b}\right)}\right) \circ \left({a_{k+1} * b}\right)       | c = Induction Hypothesis }} {{eqn | r = \sum_{j \mathop = 1}^{k + 1} \left({a_j * b}\right)       | c = Associativity of $\circ$ in $\left({R, \circ, *}\right)$ }} {{end-eqn}} So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\displaystyle \forall n \in \Z_{> 0}: \left({\sum_{j \mathop = 1}^k a_j}\right) * b = \sum_{j \mathop = 1}^k \left({a_j * b}\right)$ {{qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = General Commutativity Theorem|next = General Distributivity Theorem/Lemma 2}}: $\S 18$: Theorem $18.8$ Category:General Distributivity Theorem qr2k266ad5c6kwabp79q8cn5pv5s9j5


General Distributivity Theorem/Lemma 2

General Distributivity Theorem/Lemma 2 0 28639 411216 401085 2019-07-02T07:24:13Z Prime.mover 59 wikitext text/x-wiki == Lemma == <onlyinclude> Let $\struct {R, \circ, *}$ be a ringoid. Then for every sequence $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$ of elements of $R$, and for every $b \in R$: :$\displaystyle b * \paren {\sum_{j \mathop = 1}^n a_j} = \sum_{j \mathop = 1}^n \paren {b * a_j}$ </onlyinclude> where: :$\displaystyle \sum_{j \mathop = 1}^n a_j$ is the summation $a_1 \circ a_2 \circ \cdots \circ a_n$ :$n$ is a strictly positive integer: $n \in \Z_{> 0}$. == Proof == The proof proceeds by the Principle of Mathematical Induction. Recall that as $\struct {R, \circ, *}$ is a ringoid, $*$ is distributive over $\circ$: :$\forall a, b, c \in R: a * \paren {b \circ c} = \paren {a * b} \circ \paren {a * c}$ For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition: :$\displaystyle b * \paren {\sum_{j \mathop = 1}^n a_j} = \sum_{j \mathop = 1}^n \paren {b * a_j}$ We have that $\struct {R, \circ, *}$ is a ringoid, and so: :$\forall a, b, c \in R: a * \paren {b \circ c} = \paren {a * b} \circ \paren {a * c}$ === Basis for the Induction === $\map P 1$ is true, as this just says: :$b * a_1 = b * a_1$ $\map P 2$ is the case: {{begin-eqn}} {{eqn | l = b * \paren {\sum_{j \mathop = 1}^2 a_j}       | r = b * \paren {a_1 \circ a_2}       | c = {{Defof|Composite (Abstract Algebra)|Composite}} }} {{eqn | r = \paren {b * a_1} \circ \paren {b * a_2}       | c = $*$ is distributive over $\circ$ as $\paren {R, \circ, *}$ is a ringoid }} {{eqn | r = \sum_{j \mathop = 1}^2 \paren {b * a_j}       | c = {{Defof|Composite (Abstract Algebra)|Composite}} }} {{end-eqn}} This is our basis for the induction. === Induction Hypothesis === Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true. So this is our induction hypothesis: :$\displaystyle b * \paren {\sum_{j \mathop = 1}^k a_j} = \sum_{j \mathop = 1}^k \paren {b * a_j}$ Then we need to show: :$\displaystyle b * \paren {\sum_{j \mathop = 1}^{k + 1} a_j} = \sum_{j \mathop = 1}^{k + 1} \paren {b * a_j}$ === Induction Step === This is our induction step: {{begin-eqn}} {{eqn | l = b * \paren {\sum_{j \mathop = 1}^{k + 1} a_j}       | r = b * \paren {\paren {\sum_{j \mathop = 1}^k a_j} \circ a_{k + 1} }       | c =  }} {{eqn | r = \paren {b * \paren {\sum_{j \mathop = 1}^k a_j} } \circ \paren {b * a_{k + 1} }       | c = Basis for the Induction }} {{eqn | r = \paren {\sum_{j \mathop = 1}^k \paren {a_j * b} } \circ \paren {b * a_{k + 1} }       | c = Induction Hypothesis }} {{eqn | r = \sum_{j \mathop = 1}^{k + 1} \paren {b * a_j}       | c = Associativity of $\circ$ in $\paren {R, \circ, *}$ }} {{end-eqn}} So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction. Therefore: :$\displaystyle \forall n \in \Z_{> 0}: b * \paren {\sum_{j \mathop = 1}^k a_j} = \sum_{j \mathop = 1}^k \paren {b * a_j}$ {{qed}} == Sources == * {{BookReference|Modern Algebra|1965|Seth Warner|prev = General Distributivity Theorem/Lemma 1|next = Definition:Indexing Set}}: $\S 18$: Theorem $18.8$ * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Principle of Mathematical Induction/Zero-Based|next = General Distributivity Theorem}}: Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: Exercise $2$ Category:General Distributivity Theorem h7knb9uy7erih2y17j9y5ug27s476gt


Integer Multiplication Distributes over Addition/Corollary 0 28652 473551 353134 2020-06-12T07:06:00Z Prime.mover 59 wikitext text/x-wiki == Corollary to Integer Multiplication Distributes over Addition == <onlyinclude> The operation of multiplication on the set of integers $\Z$ is distributive over subtraction: : $\forall x, y, z \in \Z: x \times \left({y - z}\right) = \left({x \times y}\right) - \left({x \times z}\right)$ : $\forall x, y, z \in \Z: \left({y - z}\right) \times x = \left({y \times x}\right) - \left({z \times x}\right)$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = x \times \left({y - z}\right)       | r = x \times \left({y + \left({- z}\right)}\right)       | c = {{Defof|Integer Subtraction}} }} {{eqn | r = x \times y + x \times \left({- z}\right)       | c =  }} {{eqn | r = x \times y + \left({- \left({x \times z}\right)}\right)       | c = Product with Ring Negative }} {{eqn | r = x \times y - x \times z       | c = {{Defof|Integer Subtraction}} }} {{end-eqn}} {{qed|lemma}} {{begin-eqn}} {{eqn | l = \left({y - z}\right) \times x       | r = x \times \left({y - z}\right)       | c = Integer Multiplication is Commutative }} {{eqn | r = x \times y - x \times z       | c = from above }} {{eqn | r = y \times z - z \times x       | c = Integer Multiplication is Commutative }} {{end-eqn}} {{qed}} Category:Integer Multiplication Category:Subtraction Category:Distributive Operations 57cvjbx161m0pv7sdlm5h1a05u5icz9


Divisors of Product of Coprime Integers/Corollary 0 28770 437926 265344 2019-12-06T16:57:40Z Prime.mover 59 wikitext text/x-wiki == Corollary to Divisors of Product of Coprime Integers == <onlyinclude> Let $p$ be a prime. Let $p \divides b c$, where $b \perp c$. Then $p \divides b$ or $p \divides c$, but not both. </onlyinclude> == Proof == From the main result, $p = r s$, where $r \divides b$ and $s \divides c$. But as $p$ is prime, either: :$r = 1$ and $s = p$, or: :$r = p$ and $s = 1$. So $p \divides b$ or $p \divides c$. But $p$ can not divide both, as $b \perp c$. {{qed}} Category:Coprime Integers Category:Prime Numbers jl8ilgejcoz1uv9o814h5jm23hh6oew


Common Factor Cancelling in Congruence/Corollary 1 0 28800 392620 392607 2019-02-17T23:23:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Common Factor Cancelling in Congruence == Let $a, b, x, y, m \in \Z$. Let: :$a x \equiv b y \pmod m$ and $a \equiv b \pmod m$ where $a \equiv b \pmod m$ denotes that $a$ is congruent modulo $m$ to $b$. <onlyinclude> If $a$ is coprime to $m$, then: :$x \equiv y \pmod m$ </onlyinclude> == Proof == Let $a \perp m$. Then by definition of coprime: :$\gcd \set {a, m} = 1$ The result follows immediately from Common Factor Cancelling in Congruence. {{qed}} == Warning == {{:Common Factor Cancelling in Congruence/Corollary 1/Warning}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Modulo Multiplication is Well-Defined|next = Congruence by Product of Moduli}}: $\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\text{B}$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Law of Inverses (Modulo Arithmetic)/Corollary 1|next = Fundamental Theorem of Arithmetic}}: $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $20$ Category:Common Factor Cancelling in Congruence 5l5a3qhfgplllpmlq3j5e75o0h2xfm6


Sequence of Powers of Reciprocals is Null Sequence/Corollary 0 28878 408133 408132 2019-06-15T12:41:08Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sequence of Powers of Reciprocals is Null Sequence == <onlyinclude> Let $\sequence {x_n}$ be the sequence in $\R$ defined as: :$x_n = \dfrac 1 n$ Then $\sequence {x_n}$ is a null sequence. </onlyinclude> == Proof == $n = n^1$ from the definition of power. As $1 \in \Q_{>0}$ the result follows from Sequence of Powers of Reciprocals is Null Sequence. {{qed}} Category:Sequence of Powers of Reciprocals is Null Sequence 7yp45kvd2ebzco481calh7q6exmwqpe


Integer Less One divides Power Less One/Corollary 0 29499 454188 164538 2020-03-13T09:02:32Z Prime.mover 59 wikitext text/x-wiki == Corollary to Integer Less One divides Power Less One == <onlyinclude> Let $m, n, q \in \Z_{>0}$. Let: :$m \divides n$ where $\divides$ denotes divisibility. Then: :$\paren {q^m - 1} \divides \paren {q^n - 1}$ </onlyinclude> === Converse to Corollary === {{:Integer Less One divides Power Less One/Corollary/Converse}} == Proof == By hypothesis: :$m \divides n$ By definition of divisibility: :$\exists k \in \Z: k m = n$ Thus: :$q^n = q^{k m} = \paren {q^m}^k$ {{explain|Find the exponent combination laws to justify the above}} Then by Integer Less One divides Power Less One: :$\paren {q^m - 1} \divides \paren {\paren {q^m}^k - 1}$ Hence the result. {{qed}} == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Integer Less One divides Power Less One|next = Integer Less One divides Power Less One/Corollary/Converse}}: $\S 2.4$: The rational numbers and some finite fields: Further Exercises $5$ Category:Number Theory b69gx4718d7m09c0l3o10ufnr355dm6


Axiom:Axiom of Choice/Formulation 1

Axiom:Axiom of Choice/Formulation 1 100 30420 491830 491825 2020-10-01T07:04:09Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set. :$\displaystyle \forall s: \paren {\O \notin s \implies \exists \paren {f: s \to \bigcup s}: \forall t \in s: \map f t \in t}$  That is, one can always create a choice function for selecting one element from each element of the set. </onlyinclude> == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Axiom:Axiom of Choice/Formulation 2|next = Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1}}: $\S 15$: The Axiom of Choice * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Definition:Choice Function|next = Axiom:Axiom of Choice/Historical Note}}: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents * {{BookReference|The Axiom of Choice|1973|Thomas J. Jech|next = Definition:Choice Function}}: $1.1$ The Axiom of Choice Choice 34dmh7pd26iip1t65a6xykrb6fqsnr2


Axiom:Axiom of Choice/Formulation 2

Axiom:Axiom of Choice/Formulation 2 100 30421 444060 444022 2020-01-18T02:02:50Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty. Then there exists an indexed family $\family {x_i}_{i \mathop \in I}$ such that: :$\forall i \in I: x_i \in X_i$ That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty. </onlyinclude> == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Lexicographic Order|next = Axiom:Axiom of Choice/Formulation 1}}: $\S 15$: The Axiom of Choice * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Well-Ordering Theorem|next = Equivalence of Versions of Axiom of Choice}}: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents Choice s59iex6vzqt7v2z3rdl05apu6t2kaw0


Axiom:Axiom of Choice/Formulation 3

Axiom:Axiom of Choice/Formulation 3 100 30422 450952 446516 2020-02-27T23:48:41Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> Let $\SS$ be a set of non-empty pairwise disjoint sets. Then there is a set $C$ such that for all $S \in \SS$, $C \cap S$ has exactly one element. Symbolically: :$\forall s: \paren {\paren {\O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O} \implies \exists c: \forall t \in s: \exists x: t \cap c = \set x}$ </onlyinclude> == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{citation|date = 1908|title = Neuer Beweis für die Möglichkeit einer Wohlordnung|trans = A new proof of the possibility of well-ordering|journal = Mathematische Annalen|abbr = Math. Ann.|volume = 65|startpage = 107|endpage = 128|author = Ernst Zermelo}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Axiomatic System|next = Definition:Axis|entry = axiom of choice}} Choice aa2hjgpjv581bfh9pq3d6z8ccjnxs4g


Image of Empty Set is Empty Set/Corollary 0 30445 487397 374945 2020-09-13T12:44:23Z Prime.mover 59 wikitext text/x-wiki == Corollary of Image of Empty Set is Empty Set == <onlyinclude> Let $f: S \to T$ be a mapping. The image of the empty set is the empty set: :$f \sqbrk \O = \O$ </onlyinclude> == Proof == By definition, a mapping is a relation. Thus Image of Empty Set is Empty Set applies. {{qed}} == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Definition:Image of Subset under Mapping/Definition 1|next = Image of Domain of Mapping is Image Set}}: $\S 1.10$: Functions: Remark $10.8 \ \text{(a)}$ Category:Mapping Theory Category:Empty Set 7xavt25fvkht9z61ptq1ge391e36cn5


Axiom:Propositions of Incidence/Line in Plane 100 30593 486759 181304 2020-09-10T07:22:08Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> A line defined by any two distinct points in a plane lies entirely within that plane. </onlyinclude> == Sources == * {{BookReference|Projective Geometry|1952|T. Ewan Faulkner|ed = 2nd|edpage = Second Edition|prev = Axiom:Propositions of Incidence/Plane|next = Plane contains Infinite Number of Lines}}: Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence Category:Axioms/Projective Geometry 0vbntmbzl40iteejpqipw7xojoqesbg


Axiom:Propositions of Incidence/Plane and Line 100 30599 486768 181300 2020-09-10T07:33:20Z Prime.mover 59 wikitext text/x-wiki == Axiom == <onlyinclude> A plane and a (straight) line which does not lie in that plane have exactly one point in common. </onlyinclude> That is, a point is completely determined by any two of the infinite number of distinct (straight) lines which intersect at that point. == Sources == * {{BookReference|Projective Geometry|1952|T. Ewan Faulkner|ed = 2nd|edpage = Second Edition|prev = Definition:Coplanar Points|next = Two Planes have Line in Common}}: Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence Category:Axioms/Projective Geometry h68vlrk0phnfre2d5wb5p5w1pn0frga


Primitive of Reciprocal/Corollary 2

Primitive of Reciprocal/Corollary 2 0 30760 407575 387660 2019-06-12T19:52:43Z Prime.mover 59 wikitext text/x-wiki {{mergeto|Derivative of Natural Logarithm Function}} {{refactor|What would probably be best would be for this to be a corollary of the above page.}} == Corollary to Primitive of Reciprocal == <onlyinclude> :$\displaystyle \frac {\d} {\d x} \ln \size x = \frac 1 x$ for $x \ne 0$. </onlyinclude> == Proof == Follows directly from Primitive of Reciprocal. {{qed}} == Sources == * {{BookReference|Calculus|2005|Roland E. Larson|author2 = Robert P. Hostetler|author3 = Bruce H. Edwards|ed = 8th|edpage = Eighth Edition}}: $\S 5.1$ Category:Primitive of Reciprocal 6jdjjj4dmtd0kkhoz4e9550qkatoy1z


Comparison Test/Corollary

Comparison Test/Corollary 0 30788 261392 172357 2016-07-02T10:40:21Z Prime.mover 59 wikitext text/x-wiki == Corollary to Comparison Test == <onlyinclude> Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers. Let $\left \langle {a_n} \right \rangle$ be a sequence in sequence $\R$ or sequence in $\C$. Let $H \in \R$. Let $\exists M: \forall n > M: \left\vert {a_n}\right\vert \le H b_n$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges. </onlyinclude> == Proof == Let $\epsilon > 0$. Then $\dfrac \epsilon H > 0$. As $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converges, its tail tends to zero. So: : $\displaystyle \exists N: \forall n > N: \sum_{k \mathop = n+1}^\infty b_k < \frac \epsilon H$ Let $\left \langle s_n \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$. Then $\forall n > m > \max \left\{{M, N}\right\}$: {{begin-eqn}} {{eqn | l = \left\vert{s_n - s_m}\right\vert       | r = \left\vert{\left({a_1 + a_2 + \cdots + a_n}\right) - \left({a_1 + a_2 + \cdots + a_m}\right)}\right\vert       | c =  }} {{eqn | r = \left\vert{a_{m+1} + a_{m+2} + \cdots + a_n}\right\vert       | c =  }} {{eqn | o = \le       | r = \left\vert { a_{m+1} } \right\vert + \left\vert { a_{m+2} } \right\vert + \cdots + \left\vert { a_n } \right\vert       | c = Triangle Inequality }} {{eqn | o = \le       | r = H b_{m+1} + H b_{m+2} + \cdots + H b_n       | c =  }} {{eqn | o = \le       | r = H \sum_{k \mathop = n+1}^\infty b_k }} {{eqn | o = <       | r = H \frac \epsilon H }} {{eqn | r = \epsilon       | c =  }} {{end-eqn}} So $\left \langle s_n \right \rangle$ is a Cauchy sequence and the result follows from: : Real Number Line is Complete Metric Space or: : Complex Plane is Complete Metric Space. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Comparison Test|next = Ratio Test}}: $\S 6.15$ Category:Series Category:Convergence Tests Category:Named Theorems l0v7nb5242yzp18zkm48ik0e8dsm1q6


Sum of Sequence of Products of Consecutive Reciprocals/Corollary 0 30789 492870 406446 2020-10-06T09:45:56Z Prime.mover 59 wikitext text/x-wiki == Corollary to Sum of Sequence of Products of Consecutive Reciprocals == <onlyinclude> :$\displaystyle \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} } = 1$ </onlyinclude> == Proof == From Sum of Sequence of Products of Consecutive Reciprocals: :$\displaystyle \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} } = \frac n {n + 1} = 1 - \frac 1 {n + 1}$ We have that: :$\dfrac 1 {n + 1} < \dfrac 1 n$ and that $\sequence {\dfrac 1 n}$ is a basic null sequence. Thus by the Squeeze Theorem: :$\displaystyle \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} } = 1$ {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Sum of Sequence of Products of Consecutive Reciprocals/Proof 2|next = Harmonic Series is Divergent/Proof 1}}: $\S 6.3$ * {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition|prev = Definition:Divergent Series|next = Divergent Sequence may be Bounded}}: $\S 1.2$: Infinite Series of Constants: Example $1$ Category:Limits of Series Category:Sum of Sequence of Products of Consecutive Reciprocals pjh79wjetqqj9wjlalfrvf0z5lnkxm0


Newton's Method

Newton's Method 0 30799 421737 421732 2019-08-29T07:05:45Z Prime.mover 59 wikitext text/x-wiki {{MissingLinks}} == Proof Technique == '''Newton's method''' is a method of solving an equation expressed as a real function for which there may be no convenient closed form solution. The derivative of the function has to be known in order to use '''Newton's method'''. Let the equation to be solved be of the form: :$y = \map f x$ Let the value of $x$ be required for a given $y$. Then an iterative improvement on an initial guess is of the form :$x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$ where $\map {f'} {x_1}$ is the derivative of $f$ {{WRT|Differentiation}} $x$ evaluated at $x_1$. == Proof == The function $\map f x$ can be expanded using Taylor's Theorem: :$\map f {x_2} = \map f {x_1} + \map {f'} {x_1} \paren {x_2 - x_1} + \dfrac 1 2 \map {f''} {x_1} \paren {x_2 - x_1}^2 + \dotsb$ As $x_2$ gets closer to $x_1$, this series can be truncated to: :$\map f {x_2} = \map f {x_1} + \map {f'} {x_1} \paren {x_2 - x_1}$ {{explain|Not sure but that assumptions about convergence need to be made?}} Let $x_\infty$ be the exact solution where: :$\map f {x_\infty} = y$ Let $\epsilon \in \R_{>0}$ be the difference from the new estimate to the solution: :$x_\infty = x_2 + \epsilon$ Then the function expanded around the new estimate is: :$y = \map f {x_2} + \map {f'} {x_2} \epsilon$ Solving for $x_2$ produces: :$x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} } - \epsilon \dfrac {\map {f'} {x_2} } {\map {f'} {x_1} }$ For $\epsilon$ small enough the final term can be neglected: :$x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$ {{explain|Greater linguistic and mathematical precision may be needed in the above statement.}} {{qed}} == Example == {{:Newton's Method/Example}} {{Namedfor|Isaac Newton|cat = Newton}} == Sources == {{SourceReview|Expressed in the below in a slightly different form. Need to check whether it is equivalent to the one given here.}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Third Law of Motion|next = Mathematician:Jerzy Neyman|entry = Newton's method}} Category:Calculus Category:Numerical Analysis Category:Newton's Method 6zkz90kzd5dpfo2n27n7pnf026sstc8


Negated Upper Index of Binomial Coefficient/Corollary 1 0 30863 492913 265088 2020-10-06T10:05:26Z Prime.mover 59 wikitext text/x-wiki == Corollary to Negated Upper Index of Binomial Coefficient == <onlyinclude> Let $r \in \R, k \in \Z$. Then: :$\dbinom {-r} k = \paren {-1}^k \dbinom {r + k - 1} k$ where $\dbinom {-r} k$ is a binomial coefficient. </onlyinclude> == Proof 1 == {{:Negated Upper Index of Binomial Coefficient/Corollary 1/Proof 1}} == Proof 2 == {{:Negated Upper Index of Binomial Coefficient/Corollary 1/Proof 2}} Category:Binomial Coefficients Category:Negated Upper Index of Binomial Coefficient rn42o7kdzo196kyc77gdmuvaiakf4vk


Set of Integer Combinations includes Zero 0 31285 490828 175187 2020-09-25T21:35:34Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $a, b \in \Z$ be integers. Let $S = \set {a x + b y: x, y \in \Z}$ be the set of integer combinations of $a$ and $b$. Then $0 \in S$. == Proof == By setting $x = 0$ and $y = 0$: :$a \cdot 0 + b \cdot 0 = 0$ {{qed}} == Sources == * {{BookReference|Computability and Unsolvability|1982|Martin Davis|ed = 2nd|edpage = Second Edition|prev = Definition:Coprime Integers|next = Set of Integer Combinations includes those Integers}}: Appendix $1$: Some Results from the Elementary Theory of Numbers: Lemma $1$ Category:Integer Combinations jg8bt0yar901ccdz6zqpjclfj6rdoyw


Euler's Formula/Real Domain/Corollary

Euler's Formula/Real Domain/Corollary 0 31506 365724 365148 2018-09-14T22:06:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Euler's Formula: Real Domain == Let $\theta \in \R$ be a real number. Then: <onlyinclude> :$e^{-i \theta} = \cos \theta - i \sin \theta$ </onlyinclude> where: : $e^{-i \theta}$ denotes the complex exponential function : $\cos \theta$ denotes the real cosine function : $\sin \theta$ denotes the real sine function : $i$ denotes the imaginary unit. == Proof == {{begin-eqn}} {{eqn | l = e^{-i \theta}       | r = \cos \paren {-\theta} + i \sin \paren {-\theta}       | c = Euler's Formula }} {{eqn | r = \cos \theta + i \sin \paren {-\theta}       | c = Cosine Function is Even }} {{eqn | r = \cos \theta - i \sin \theta       | c = Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Definition:Exponential Form of Complex Number|next = Cosine Exponential Formulation/Real Domain/Proof 3}}: $\S 2$. Geometrical Representations * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Euler's Formula/Real Domain|next = Sine Exponential Formulation/Real Domain}}: $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.16$ Category:Euler's Formula oo9fbxooo0c061wwaf5h54uu2rztlkg


Hyperbolic Sine of Sum/Corollary

Hyperbolic Sine of Sum/Corollary 0 31573 493512 396199 2020-10-10T09:03:49Z Prime.mover 59 wikitext text/x-wiki == Corollary of Hyperbolic Sine of Sum == <onlyinclude> :$\map \sinh {a - b} = \sinh a \cosh b - \cosh a \sinh b$ </onlyinclude> where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn | l = \map \sinh {a - b}       | r = \sinh a \map \cosh {-b} + \cosh a \map \sinh {-b}       | c = Hyperbolic Sine of Sum }} {{eqn | r = \sinh a \cosh b - \cosh a \sinh b       | c = Hyperbolic Cosine Function is Even and Hyperbolic Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Hyperbolic Sine of Sum|next = Hyperbolic Cosine of Sum}}: $\S 8$: Hyperbolic Functions: $8.20$: Addition Formulas * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Hyperbolic Sine of Sum|next = Hyperbolic Cosine of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $5$ Category:Hyperbolic Sine Function e5cgmmolsrp6zpkc8l6envukvzfavbg


Hyperbolic Cosine of Sum/Corollary

Hyperbolic Cosine of Sum/Corollary 0 31576 495816 396201 2020-10-21T21:29:31Z Prime.mover 59 wikitext text/x-wiki == Corollary of Hyperbolic Cosine of Sum == <onlyinclude> :$\map \cosh {a - b} = \cosh a \cosh b - \sinh a \sinh b$ </onlyinclude> where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn | l = \map \cosh {a - b}       | r = \cosh a \map \cosh {-b} + \sinh a \map \sinh {-b}       | c = Hyperbolic Sine of Sum }} {{eqn | r = \cosh a \cosh b - \sinh a \sinh b       | c = Hyperbolic Cosine Function is Even and Hyperbolic Sine Function is Odd }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Hyperbolic Cosine of Sum|next = Hyperbolic Tangent of Sum}}: $\S 8$: Hyperbolic Functions: $8.21$: Addition Formulas * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Hyperbolic Cosine of Sum|next = Hyperbolic Tangent of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $5$ Category:Hyperbolic Cosine Function miqq02aaj5mfsudrc7w3i8yhxecz1bh


Double Angle Formulas/Hyperbolic Cosine/Corollary 1 0 31585 191810 176650 2014-08-19T05:52:09Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Hyperbolic Cosine == <onlyinclude> : $\cosh 2 x = 2 \cosh^2 x - 1$ </onlyinclude> where $\cosh$ denotes hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn|l = \cosh 2 x      |r = \cosh^2 x + \sinh^2 x      |c = Double Angle Formula for Hyperbolic Cosine }} {{eqn|r = \cosh^2 x + \left({\cosh^2 x - 1}\right)      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 2 \ \cosh^2 x - 1 }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Double Angle Formula for Hyperbolic Cosine|next=Double Angle Formulas/Hyperbolic Cosine/Corollary 2}}: $\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas Category:Hyperbolic Cosine Function rtooyulhdl9gowv7kklh74yqkpzyv4l


Double Angle Formulas/Hyperbolic Cosine/Corollary 2 0 31586 191463 176652 2014-08-13T20:16:07Z Prime.mover 59 wikitext text/x-wiki == Corollary to Double Angle Formula for Hyperbolic Cosine == <onlyinclude> : $\cosh 2 x = 1 + 2 \sinh^2 x$ </onlyinclude> where $\cosh$ and $\sinh$ denote hyperbolic cosine and hyperbolic sine respectively. == Proof == {{begin-eqn}} {{eqn|l = \cosh 2 x      |r = \cosh^2 x + \sinh^2 x      |c = Double Angle Formula for Hyperbolic Cosine }} {{eqn|r = \left({1 + \sinh^2 x}\right) + \sinh^2 \theta      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 1 + 2 \sinh^2 x }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Double Angle Formulas/Hyperbolic Cosine/Corollary 1|next=Double Angle Formula for Hyperbolic Tangent}}: $\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas Category:Hyperbolic Cosine Function mcdukzs104yk3brgaticzbby3hyrfy6


Complex Multiplication as Geometrical Transformation/Corollary 0 32293 394698 362679 2019-03-04T20:27:08Z Prime.mover 59 wikitext text/x-wiki == Corollary to Complex Multiplication as Geometrical Transformation == <onlyinclude> Let $z = \polar {r, \theta}$ be a complex number expressed in polar form. Let $z$ be represented on the complex plane $\C$ in vector form. The effect of multiplying $z$ by $e^{i \alpha}$ is to rotate it about the origin of $\C$ by $\alpha$ in the positive direction </onlyinclude> == Proof == From Complex Multiplication as Geometrical Transformation, the effect of multiplying a complex number by $r e^{i \alpha}$ is: :to rotate it about the origin of $\C$ by $\alpha$ in the positive direction :to multiply its modulus by $r$. In this instance $r = 1$. Hence the result. {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Power Reduction Formulas/Cosine to 4th/Proof 2|next = Periodicity of Complex Exponential Function}}: $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $24$ Category:Complex Multiplication as Geometrical Transformation cjjo033pavj4kfjh0i8k5yqyeu4im4e


Semantic Tableau Algorithm

Semantic Tableau Algorithm 0 32295 181022 180972 2014-03-26T11:54:06Z Lord Farin 560 wikitext text/x-wiki == Algorithm == <onlyinclude> Let $\mathbf A$ be a WFF of propositional logic. The purpose of this algorithm is to create a completed semantic tableau $T$ for $\mathbf A$. :'''Step 1''': Start with a labeled tree $T$ comprising only a root node labeled with the singleton $\left\{{\mathbf A}\right\}$. :'''Step 2''': Choose a leaf node $t$ of $T$ that has not yet been marked. Let $U \left({t}\right)$ be the set that labels $t$. :'''Step 3''': If $U \left({t}\right)$ is a set of literals, mark $t$ as follows: ::'''a)''': If $U \left({t}\right)$ contains a complementary pair, mark it closed, $\times$, and go to '''Step 2'''. ::'''b)''': Otherwise, mark it open, $\odot$, and go to '''Step 2'''. :'''Step 4''': Choose a formula $\mathbf B \in U \left({t}\right)$ that is not a literal. Classify $\mathbf B$ as an $\alpha$-formula or a $\beta$-formula, and determine $\mathbf B_1, \mathbf B_2$ accordingly. ::'''a)''': If $\mathbf B$ is an $\alpha$-formula, add a child $t'$ to $t$, labeled $U \left({t'}\right) = \left({U \left({t}\right) \setminus \left\{{\mathbf B}\right\}}\right) \cup \left\{{\mathbf B_1, \mathbf B_2}\right\}$. ::'''b)''': If $\mathbf B$ is a $\beta$-formula, add two children $t', t''$ to $t$ with labels: ::::$U \left({t'}\right) = \left({U \left({t}\right) \setminus \left\{{\mathbf B}\right\}}\right) \cup \left\{{\mathbf B_1}\right\}$ ::::$U \left({t''}\right) = \left({U \left({t}\right) \setminus \left\{{\mathbf B}\right\}}\right) \cup \left\{{\mathbf B_2}\right\}$ :'''Step 5''': Go to '''Step 2''' if there remain leaves that are not marked. Otherwise, '''stop'''. </onlyinclude> === Finiteness === The finiteness of this algorithm is demonstrated on Semantic Tableau Algorithm Terminates. {{expand}} === Heuristics === {{:Semantic Tableau Algorithm/Heuristics}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition|prev=Definition:Beta-Formula|next=Definition:Marked Leaf}}: $\S 2.6.2$: Algorithm $2.64$ Category:Propositional Logic bkeg0ynhcyu8zstorn9rkmzkyf9ulgn


Quadruple Angle Formulas/Sine/Corollary 1 0 32365 395682 395680 2019-03-15T21:52:47Z Prime.mover 59 wikitext text/x-wiki == Corollary to Quadruple Angle Formula for Sine == <onlyinclude> For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$ : $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 \cos \theta$ where $\sin$ denotes sine and $\cos$ denotes cosine. </onlyinclude> == Proof == First note that when $\theta = 0, \pm \pi, \pm 2 \pi \ldots$: :$\sin \theta = 0$ so $\dfrac {\sin 4 \theta} {\sin \theta}$ is undefined. Therefore for the rest of the proof it is assumed that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$ {{begin-eqn}} {{eqn | l = \sin 4 \theta       | r = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta       | c = Quadruple Angle Formula for Sine }} {{eqn | ll= \leadsto       | l = \dfrac {\sin 4 \theta} {\sin \theta}       | r = 4 \cos \theta - 8 \paren {1 - \cos^2 \theta} \cos \theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 8 \cos^3 \theta - 4 \cos \theta       | c = multiplying out and gathering terms }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Cosine of 72 Degrees|next = Quadruple Angle Formulas/Sine/Corollary 2}}: $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $93 \ \text{(a)}$ ::(although see Quadruple Angle Formulas/Sine/Mistake for analysis of an error in that work) Category:Quadruple Angle Formula for Sine nvp3xlkcgwbsgfy6s6kjwe4gbhsohh1


Soundness and Completeness of Semantic Tableaus/Corollary 2 0 32422 424258 181030 2019-09-09T13:48:45Z Prime.mover 59 wikitext text/x-wiki == Corollary to Soundness and Completeness of Semantic Tableaus == Let $\mathbf A$ be a WFF of propositional logic. Then <onlyinclude>$\mathbf A$ is a tautology {{iff}} $\neg \mathbf A$ has a closed tableau. </onlyinclude> == Proof == By Tautology iff Negation is Unsatisfiable, $\mathbf A$ is a tautology {{iff}} $\neg \mathbf A$ is unsatisfiable. By the Soundness and Completeness of Semantic Tableaus, this amounts to the existence of a closed tableau for $\neg \mathbf A$. {{qed}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Soundness and Completeness of Semantic Tableaus/Corollary 1|next = Semantic Tableau Algorithm is Decision Procedure for Tautologies}}: $\S 2.7$: Corollary $2.69$ Category:Propositional Logic jd2m34fes6e32zzz1jlk10v28ul1oz3


Exponential of Sum/Real Numbers/Corollary 0 32532 396500 396464 2019-03-21T07:17:34Z Prime.mover 59 wikitext text/x-wiki == Corollary to Exponential of Sum: Real Numbers == <onlyinclude> Let $x, y \in \R$ be real numbers. Let $\exp x$ be the exponential of $x$. Then: :$\map \exp {x - y} = \dfrac {\exp x} {\exp y}$ </onlyinclude> == Proof == By Exponential of Sum: Real Numbers: :$\map \exp {x - y} = \exp x \, \map \exp {-y}$ {{improve|No real counterpart exists to link to}} By Reciprocal of Complex Exponential: :$\dfrac 1 {\exp y} = \map \exp {-y}$ Combining these two, we obtain the result: :$\map \exp {x - y} = \dfrac {\exp x} {\exp y}$ {{qed}} Category:Exponential of Sum mxtd2q3sd9p8g2l9a9hrt0wwaz5otr9


True Statement is implied by Every Statement/Formulation 2/Proof 2 0 32742 182453 2014-04-07T11:54:46Z Lord Farin 560 Created page with "== Theorem ==  : $\vdash q \implies \left({p \implies q}\right)$   == Proof == <onlyinclude> We apply the Method of Truth Tables.  As can be seen by inspection, the Defi..." wikitext text/x-wiki == Theorem == : $\vdash q \implies \left({p \implies q}\right)$ == Proof == <onlyinclude> We apply the Method of Truth Tables. As can be seen by inspection, the truth value under the main connective, the first instance of $\implies$, is $T$ for each boolean interpretation. $\begin{array}{|ccccc|} \hline q & \implies & ( p & \implies & q ) \\ \hline F & T & T & F & F \\ F & T & F & T & F \\ T & T & T & T & T \\ T & T & F & T & T \\ \hline \end{array}$ {{qed}} </onlyinclude> Category:True Statement is implied by Every Statement Category:Truth Table Proofs 3dihouk51t4v9u4gxoo2ky98iaarh25


Axiom:Peano's Axioms

Axiom:Peano's Axioms 100 32855 488567 488564 2020-09-17T07:44:22Z Prime.mover 59 wikitext text/x-wiki == Axioms == '''Peano's Axioms''' are a set of properties which can be used to serve as a basis for logical deduction of the properties of the natural numbers. <onlyinclude> '''Peano's Axioms''' are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ and $0$ as an element of $\N$. Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$. Historically, the existence of $s$ and the existence of $0$ were considered the first two of '''Peano's Axioms''': {{begin-axiom}} {{axiom | n = \text P 1         | q =          | m = 0 \in P         | t = $0$ is an element of $P$ }} {{axiom | n = \text P 2         | q = \forall n \in P         | m = \map s n \in P         | t = For all $n \in P$, its successor $\map s n$ is also in $P$ }} {{end-axiom}} The other three are as follows: </onlyinclude> === Formulation 1 === <onlyinclude>{{:Axiom:Peano's Axioms/Formulation 1}}</onlyinclude> === Formulation 2 === {{:Axiom:Peano's Axioms/Formulation 2}} == Terminology == === Successor Mapping === {{:Definition:Successor Mapping}} === Non-Successor Element === {{:Definition:Non-Successor Element}} === Peano Structure === Such a set $P$, together with the successor mapping $s$ and non-successor element $0$ as defined above, is known as a '''Peano structure'''. == Also presented as == Some sources present the axioms in a different order. For example, {{BookReference|Number Theory|1964|J. Hunter}} places the induction axiom $(\text P 5)$ as axiom $(3)$ and moves $(\text P 3)$ and $(\text P 4)$ down to be $(4)$ and $(5)$ respectively. == Also defined as == {{:Axiom:Peano's Axioms/Also defined as}} {{expand|subpage with Peano's original exposition/axioms}} == Also known as == {{:Axiom:Peano's Axioms/Also known as}} == Also see == * Equivalence of Formulations of Peano's Axioms * Minimal Infinite Successor Set forms Peano Structure * Principle of Mathematical Induction for Peano Structure {{LinkToCategory|Peano's Axioms|Peano's axioms}} {{NamedforAxiom|Giuseppe Peano|author2 = Richard Dedekind|cat = Peano|cat2 = Dedekind}} == Historical Note == {{:Axiom:Peano's Axioms/Historical Note}} == Sources == * {{BookReference|Numbers, Sets and Axioms|1982|Alan G. Hamilton|prev = Successor Mapping on Natural Numbers has no Fixed Element|next = Natural Number Multiplication is Commutative}}: $\S 1$: Numbers: $1.1$ Natural Numbers and Integers: Examples $1.1 \ \text {(e)}$ Category:Axioms/Number Theory Category:Axioms/Abstract Algebra Category:Axioms/Peano's Axioms h7wez3z3cfirxfptwzhjfbnjv7y8428


Stirling's Formula/Proof 2/Lemma 1

Stirling's Formula/Proof 2/Lemma 1 0 32930 349891 183945 2018-04-06T17:49:48Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $f \left({x}\right)$ be the real function defined on the open interval $\left({-1 \,.\,.\, 1}\right)$ as: :$f \left({x}\right) := \dfrac 1 {2 x} \ln \left({\dfrac {1 + x} {1 - x} }\right) - 1$ Then: :$\displaystyle f \left({x}\right) = \sum_{k \mathop = 1}^\infty \dfrac {x^{2 n} } {2n + 1}$ == Proof == {{begin-eqn}} {{eqn | l = f \left({x}\right)       | o = :=       | r = \frac 1 {2 x} \ln \left({\frac {1 + x} {1 - x} }\right) - 1       | c = for $\left\vert{x}\right\vert < 1$ }} {{eqn | r = \frac 1 {2 x} \left({\ln \left({1 + x}\right) - \ln \left({1 - x}\right)}\right) - 1       | c = Difference of Logarithms }} {{eqn | r = \frac 1 {2 x} \left({\sum_{k \mathop = 1}^\infty \left({\left({-1}\right)^{k-1} \frac {x^k} k}\right) - \sum_{k \mathop = 1}^\infty \left({-\frac {x^k} k}\right)}\right) - 1       | c = Power Series Expansion for $\ln \left({1 + x}\right)$ }} {{eqn | r = \sum_{k \mathop = 1}^\infty \frac {x^{2n} } {2n + 1}       | c = simplifying }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev=Stirling's Formula/Proof 2|next=Stirling's Formula/Proof 2/Lemma 2}}: $\S 17.2$ Category:Stirling's Formula n3vjrngd7oz373utxaven1ltvu78x5c


Stirling's Formula/Proof 2/Lemma 3

Stirling's Formula/Proof 2/Lemma 3 0 32936 456533 357722 2020-03-19T22:52:07Z Prime.mover 59 wikitext text/x-wiki == Lemma == Let $\sequence {d_n}$ be the sequence defined as: :$d_n = \map \ln {n!} - \paren {n + \dfrac 1 2} \ln n + n$ Then the sequence: :$\sequence {d_n - \dfrac 1 {12 n} }$ is increasing. == Proof == We have: {{begin-eqn}} {{eqn | l = d_n - d_{n + 1}       | r = \map \ln {n!} - \paren {n + \frac 1 2} \ln n + n       | c =  }} {{eqn | o =        | ro= -       | r = \paren {\map \ln {\paren {n + 1}!} - \paren {n + 1 + \frac 1 2} \map \ln {n + 1} + n + 1}       | c =  }} {{eqn | r = -\map \ln {n + 1} - \paren {n + \frac 1 2} \ln n + \paren {n + \frac 3 2} \map \ln {n + 1} - 1       | c = (as $\map \ln {\paren {n + 1}!} = \map \ln {n + 1} + \map \ln {n!}$) }} {{eqn | r = \paren {n + \frac 1 2} \map \ln {\frac {n + 1} n} - 1       | c = }} {{eqn | n = 1       | r = \frac {2n + 1} 2 \map \ln {\frac {1 + \paren {2 n + 1}^{-1} } {1 - \paren {2 n + 1}^{-1} } } - 1       | c = }} {{end-eqn}} Let: :$\map f x := \dfrac 1 {2 x} \map \ln {\dfrac {1 + x} {1 - x} } - 1$ for $\size x < 1$. Then: {{begin-eqn}} {{eqn | l = \map f x       | r = \sum_{k \mathop = 1}^\infty \frac {x^{2 n} } {2 n + 1}       | c = Lemma 1 }} {{eqn | o = <       | r = \frac {x^2} 3 \sum_{k \mathop = 0}^\infty x^{2 n}       | c =  }} {{eqn | n = 1       | o = <       | r = \frac {x^2} {3 \paren {1 - x^2} }        | c = Sum of Infinite Geometric Sequence }} {{end-eqn}} As $-1 < \dfrac 1 {2 n + 1} < 1$ it can be substituted for $x$ in $(1)$: {{begin-eqn}} {{eqn | l = d_n - d_{n - 1}       | o = \le       | r = \frac 1 {3 \paren {\paren {2 n + 1}^2 - 1} }       | c = simplifying }} {{eqn | r = \frac 1 {12 n} - \frac 1 {12 \paren {n + 1} }       | c =  }} {{eqn | ll= \leadsto       | l = d_n - \frac 1 {12 n}        | o = \le       | r = d_{n-1} - \frac 1 {12 \paren {n + 1} } }} {{end-eqn}} Thus the sequence: :$\sequence {d_n - \dfrac 1 {12 n} }$ is increasing. {{qed}} == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Stirling's Formula/Proof 2/Lemma 2|next = Definition:Integral Form of Gamma Function}}: $\S 17.2$ Category:Stirling's Formula fwgijw7rkjuf1ymytkgtu8tr1ss7f48


Definite Integral of Even Function/Corollary 0 33081 457202 457198 2020-03-23T11:40:02Z Prime.mover 59 wikitext text/x-wiki == Corollary to Definite Integral of Even Function == <onlyinclude> Let $f$ be an even function with a primitive on the open interval $\openint {-a} a$, where $a > 0$. Then the improper integral of $f$ on $\openint {-a} a$ is: :$\displaystyle \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x = 2 \int_0^{\mathop \to a} \map f x \rd x$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \int_{\mathop \to -a}^{\mathop \to a} \map f x \rd x       | r = \lim_{y \to a} \int_{-y}^y \map f x \rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{eqn | r = \lim_{y \mathop \to a} 2 \int_0^y \map f x \rd x       | c = Definite Integral of Even Function }} {{eqn | r = 2 \lim_{y \mathop \to a} \int_0^y \map f x \rd x       | c = Multiple Rule for Limits of Functions }} {{eqn | r = 2 \int_0^{\mathop \to a} \map f x \rd x       | c = {{Defof|Improper Integral over Open Interval}} }} {{end-eqn}} {{qed}} Category:Integral Calculus Category:Even Functions Category:Improper Integrals k0k6hi21scfqdada75v9gd7i54hsbqj


Negative of Absolute Value/Corollary 2

Negative of Absolute Value/Corollary 2 0 33091 459712 459711 2020-04-06T12:21:05Z Prime.mover 59 wikitext text/x-wiki == Corollary to Negative of Absolute Value == Let $x, y \in \R$ be a real numbers. Let $\size x$ be the absolute value of $x$. Then: <onlyinclude> :$\size x \le y \iff -y \le x \le y$ that is: :$\size x \le y \iff \begin {cases} x & \le y \\ -x & \le y \end {cases}$ </onlyinclude> == Proof == === Necessary Condition === Let $\size x \le y$. If $\size x < y$ then from Corollary 1: :$-y < x < y$ Thus: :$-y \le x \le y$ Otherwise, if $\size x = y$ then either $x = y$ or $-x = y$. Hence the result. {{qed|lemma}} === Sufficient Condition === Let $-y \le x \le y$. If $-y < x < y$ then from Corollary 1: :$\size x < y$ Hence: :$\size x \le y$ Otherwise, if either $-y = x$ or $x = y$ then: :$\size x = y$ Hence the result. {{qed}} Category:Negative of Absolute Value ak12cdl05u79k5whelfxgro09qcjliy


Derivative of Identity Function/Corollary 0 33458 466596 274640 2020-05-08T07:57:01Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Identity Function == <onlyinclude> :$\map {\dfrac {\d} {\d x} } {c x} = c$ where $c$ is a constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \map {\frac \d {\d x} } {c x}       | r = c \frac \d {\d x} x       | c = Derivative of Constant Multiple }} {{eqn | r = c \times 1       | c = Derivative of Identity Function }} {{eqn | r = c       | c =  }} {{end-eqn}} {{qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Derivative of Constant|next = Power Rule for Derivatives/Corollary}}: $\S 13$: General Rules of Differentiation: $13.3$ Category:Derivatives Category:Identity Mappings el09hmuomqzx0kmw0u8p3o825e3wlx9


Derivative of Constant Multiple/Real/Corollary 0 33461 485979 351652 2020-09-06T22:20:46Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Constant Multiple == Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then: <onlyinclude> :$\map {\dfrac {\d^n} {\d x^n} } {c \map f x} = c \map {\dfrac {\d^n} {\d x^n} } {\map f x}$ </onlyinclude> == Proof == {{refactor|Expand this into the formal {{ProofWiki}} structure for induction proofs.}} By induction: the base case is for $n = 1$ and is proved in Derivative of Constant Multiple. Now consider $\map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}$, assuming the induction hypothesis $\map {\dfrac {\d^k} {\d x^k} } {c \map f x} = c \map {\dfrac {\d^k} {\d x^k} } {\map f x}$: {{begin-eqn}} {{eqn | l = \map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {c \map f x}       | r = \map {\dfrac \d {\d x} } {\map {\dfrac {\d^k} {\d x^k} } {c \map f x} }       | c = {{Defof|Higher Derivative}} }} {{eqn | r = \map {\dfrac \d {\d x} } {c \map {\dfrac {\d^k} {\d x^k} } {\map f x} }       | c = Induction Hypothesis }} {{eqn | r = c \map {\dfrac \d {\d x} } {\map {\dfrac {\d^k} {\d x^k} } {\map f x} }       | c = Basis for the Induction }} {{eqn | r = c \map {\dfrac {\d^{k + 1} } {\d x^{k + 1} } } {\map f x}       | c = {{Defof|Higher Derivative}} }} {{end-eqn}} Hence the result by induction. {{qed}} Category:Differential Calculus 6hfezikkcl9kwfjgeu7oorglrv9re5v


Axiom:Axiomatization of 1-Based Natural Numbers 100 33519 493106 400696 2020-10-07T21:59:07Z Prime.mover 59 wikitext text/x-wiki {{rename|Hardly elegant}} == Axioms == The following axioms are intended to capture the behaviour of the ($1$-based) natural numbers $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations of addition $+$ and multiplication $\times$ as they pertain to $\N_{>0}$: <onlyinclude> {{begin-axiom}} {{axiom | n = \text A         | q = \exists_1 1 \in \N_{> 0}         | m = a \times 1 = a = 1 \times a }} {{axiom | n = \text B         | q = \forall a, b \in \N_{> 0}         | m = a \times \paren {b + 1} = \paren {a \times b} + a }} {{axiom | n = \text C         | q = \forall a, b \in \N_{> 0}         | m = a + \paren {b + 1} = \paren {a + b} + 1 }} {{axiom | n = \text D         | q = \forall a \in \N_{> 0}, a \ne 1         | m = \exists_1 b \in \N_{> 0}: a = b + 1 }} {{axiom | n = \text E         | q = \forall a, b \in \N_{> 0}         | m = \)Exactly one of these three holds:\( }} {{axiom | m = a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} }} {{axiom | n = \text F         | q = \forall A \subseteq \N_{> 0}         | m = \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} }} {{end-axiom}} </onlyinclude> == Note == The above axiom schema specifies the old-fashioned definition of the natural numbers as: :$\text{The set of natural numbers} = \set {1, 2, 3, \ldots}$ as opposed to the more modern approach which defines them as: :$\text{The set of natural numbers} = \set {0, 1, 2, 3, \ldots}$ In order to eliminate confusion, on {{ProofWiki}} the set $\set {1, 2, 3, \ldots}$ will be denoted as $\N_{> 0}$ or $\N_{\ne 0}$ or $\N_{\ge 1}$. When $\N$ is used, $\N = \set {0, 1, 2, 3, \ldots}$ is to be understood. == Sources == * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Definition:Natural Numbers|next = Principle of Mathematical Induction/Predicate/One-Based/Proof 2}}: $\S 2.1$: Definition $2.1$ Category:Axioms/Number Theory Category:Axioms/Abstract Algebra 52o5lgnwu3yvkvn2zs2xjk7ww3xqi3g


Primitive of Square of Cosine Function/Corollary 0 33567 450907 350620 2020-02-27T16:25:10Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Square of Cosine Function == <onlyinclude> :$\displaystyle \int \cos^2 x \rd x = \frac {x + \sin x \cos x} 2 + C$ where $C$ is an arbitrary constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \int \sin^2 x \rd x       | r = \frac x 2 + \frac {\sin 2 x} 4 + C       | c = Primitive of Square of Cosine Function }} {{eqn | r = \frac x 2 + \frac {2 \sin x \cos x} 4 + C       | c = Double Angle Formula for Sine }} {{eqn | r = \frac {x + \sin x \cos x} 2 + C       | c =  }} {{end-eqn}} {{Qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Cosine Function|next = Primitive of Product of Secant and Tangent}}: $\S 14$: General Rules of Integration: $14.22$ Category:Primitive of Square of Cosine Function 98map9s8trcnvivvlb9kw3pk4xr5ylq


Primitive of Square of Hyperbolic Sine Function/Corollary 0 33613 450896 386012 2020-02-27T16:06:34Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Square of Hyperbolic Sine Function == <onlyinclude> :$\displaystyle \int \sinh^2 x \rd x = \frac {\sinh x \cosh x - x} 2 + C$ where $C$ is an arbitrary constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \int \sinh^2 x \rd x       | r = \frac {\sinh 2 x} 4 - \frac x 2 + C       | c = Primitive of Square of Hyperbolic Sine Function }} {{eqn | r = \frac {2 \sinh x \cosh x} 4 - \frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \frac {\sinh x \cosh x - x} 2 + C       | c = rearranging }} {{end-eqn}} {{Qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Hyperbolic Sine Function|next = Primitive of Square of Hyperbolic Cosine Function}}: $\S 14$: General Rules of Integration: $14.35$ Category:Primitives of Hyperbolic Functions Category:Hyperbolic Sine Function 3yt56sthwj4k60tsifyp5769ossyy77


Primitive of Square of Hyperbolic Cosine Function/Corollary 0 33615 450895 187106 2020-02-27T16:06:15Z Prime.mover 59 wikitext text/x-wiki == Corollary to Primitive of Square of Hyperbolic Cosine Function == <onlyinclude> :$\displaystyle \int \cosh^2 x \rd x = \frac {\sinh x \cosh x + x} 2 + C$ where $C$ is an arbitrary constant. </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \int \cosh^2 x \rd x       | r = \frac {\sinh 2 x} 4 + \frac x 2 + C       | c = Primitive of Square of Hyperbolic Cosine Function }} {{eqn | r = \frac {2 \sinh x \cosh x} 4 + \frac x 2 + C       | c = Double Angle Formula for Hyperbolic Sine }} {{eqn | r = \frac {\sinh x \cosh x + x} 2 + C       | c = rearranging }} {{end-eqn}} {{Qed}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Primitive of Square of Hyperbolic Cosine Function|next = Primitive of Product of Hyperbolic Secant and Tangent}}: $\S 14$: General Rules of Integration: $14.36$ Category:Primitives of Hyperbolic Functions Category:Hyperbolic Cosine Function rl2pzj3o9d8bfq71uvr5eota4bng132


Derivative of Arcsine Function/Corollary 0 33685 490935 490933 2020-09-26T10:56:08Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arcsine Function == Let $a \in \R$ be a constant Let $x \in \R$ be a real number such that $x^2 < a^2$. Let $\map \arcsin {\dfrac x a}$ be the arcsine of $\dfrac x a$. Then: <onlyinclude> :$\dfrac {\map \d {\map \arcsin {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {a^2 - x^2} }$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arcsin {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {\sqrt {1 - \paren {\frac x a}^2} }       | c = Derivative of Arcsine Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\sqrt {\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \frac 1 a \frac a {\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = \frac 1 {\sqrt {a^2 - x^2} }       | c =  }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arcsec \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Square of Hyperbolic Cosecant Function|next = Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form}}: Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $12$. Category:Derivatives of Inverse Trigonometric Functions Category:Arcsine Function bxzuxbn8fq0adzva1h8iktngpzrewj6


Derivative of Arccosine Function/Corollary 0 33688 490936 415995 2020-09-26T10:56:54Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arccosine Function == Let $a \in \R$ be a constant Let $x \in \R$ be a real number such that $x^2 < a^2$. Let $\map \arccos {\dfrac x a}$ be the arccosine of $\dfrac x a$. Then: <onlyinclude> :$\dfrac {\map \d {\map \arccos {\frac x a} } } {\d x} = \dfrac {-1} {\sqrt {a^2 - x^2} }$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arccos {\frac x a} } } {\d x}       | r = \frac 1 a \frac {-1} {\sqrt {1 - \paren {\frac x a}^2} }       | c = Derivative of Arccosine Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-1} {\sqrt {\frac {a^2 - x^2} {a^2} } }       | c =  }} {{eqn | r = \frac 1 a \frac {-a} {\sqrt {a^2 - x^2} }       | c =  }} {{eqn | r = \frac {-1} {\sqrt {a^2 - x^2} }       | c =  }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arcsec \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form|next = Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form}}: Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $12$. Category:Derivatives of Inverse Trigonometric Functions Category:Arccosine Function ihf059sfr876p92yzwofnydsnwomu26


Derivative of Arctangent Function/Corollary 0 33691 490938 345418 2020-09-26T11:03:31Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arctangent Function == Let $x \in \R$. Let $\map \arctan {\dfrac x a}$ denote the arctangent of $\dfrac x a$. Then: <onlyinclude> :$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {a^2 + x^2}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arctan {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {1 + \paren {\frac x a}^2}       | c = Derivative of Arctangent Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\frac {a^2 + x^2} {a^2} }       | c =  }} {{eqn | r = \frac 1 a \frac {a^2} {a^2 + x^2}       | c =  }} {{eqn | r = \frac a {a^2 + x^2}       | c =  }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {x^2 + a^2}$ == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arcsec \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ == Sources == * {{BookReference|Integration|1944|R.P. Gillespie|ed = 2nd|edpage = Second Edition|prev = Primitive of Reciprocal of Root of a squared minus x squared/Arccosine Form|next = Primitive of Reciprocal of x squared plus a squared/Arctangent Form}}: Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $13$. Category:Derivative of Arctangent Function 9m2dw8z5a9aarqiw97351fdcd94r5o0


Derivative of Arccotangent Function/Corollary 0 33693 345414 190993 2018-03-02T17:59:22Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arccotangent Function == Let $x \in \R$. Let $\operatorname{arccot} \left({\dfrac x a}\right)$ be the arccotangent of $\dfrac x a$. Then: <onlyinclude> :$\dfrac {\mathrm d \left({\operatorname{arccot} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac {-a} {a^2 + x^2}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{arccot} \left({\frac x a}\right)}\right)} {\mathrm d x}       | r = \frac 1 a \frac {-1} {1 + \left({\frac x a}\right)^2}       | c = Derivative of Arccotangent Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-1} {\frac {a^2 + x^2} {a^2} }       | c =  }} {{eqn | r = \frac 1 a \frac {-a^2} {a^2 + x^2}       | c =  }} {{eqn | r = \frac {-a} {a^2 + x^2}       | c =  }} {{end-eqn}} {{qed}} == Also defined as == This result can also be reported as: :$\dfrac {\mathrm d \left({\operatorname{arccot} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac {-a} {x^2 + a^2}$ == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\operatorname{arcsec} \dfrac x a$ * Derivative of $\operatorname{arccsc} \dfrac x a$ Category:Derivative of Arccotangent Function 158l17r4ursuysne6d8w58ey3dgfzqc


Derivative of Arcsecant Function/Corollary 1 0 33697 485936 345409 2020-09-06T20:52:05Z Prime.mover 59 wikitext text/x-wiki == Corollary to Derivative of Arcsecant Function == Let $x \in \R$. Let $\map \arcsec {\dfrac x a}$ be the arcsecant of $\dfrac x a$. Then: <onlyinclude> :$\dfrac {\map \d {\map \arcsec {\frac x a} } } {\d x} = \dfrac a {\size x \sqrt {x^2 - a^2} } = \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $x > a$})  \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $x < -a$}) \\ \end{cases}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arcsec {\frac x a} } } {\d x}       | r = \frac 1 a \frac 1 {\size {\frac x a} \sqrt {\paren {\frac x a}^2 - 1} }       | c = Derivative of Arcsecant Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac 1 {\size {\frac x a} \frac {\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \frac 1 a \frac {a^2} {\size x {\sqrt {x^2 - a^2} } }       | c =  }} {{eqn | r = \frac a {\size x {\sqrt {x^2 - a^2} } }       | c =  }} {{end-eqn}} {{qed|lemma}} Similarly: {{begin-eqn}} {{eqn | l = \frac {\map \d {\map \arcsec {\frac x a} } } {\d x}       | r = \begin {cases} \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \ (\text {that is: $\dfrac x a > 1$}) \\ \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\paren {\frac x a}^2 - 1} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \ (\text {that is: $\dfrac x a < -1$}) \\ \end{cases}       | c = Derivative of Arcsecant Function <br/>and Derivative of Function of Constant Multiple	 }} {{eqn | r = \begin {cases} \dfrac a {x \sqrt {x^2 - a^2} } & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\ \dfrac {-a} {x \sqrt {x^2 - a^2} } & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end{cases}       | c = simplifying as above }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\arccot \dfrac x a$ * Derivative of $\arccsc \dfrac x a$ Category:Derivative of Arcsecant Function g9q7rhstacx1hjaz0jzh26nr4utcfp3


Derivative of Arccosecant Function/Corollary 0 33700 415975 190995 2019-08-02T19:58:00Z Caliburn 3218 wikitext text/x-wiki == Corollary to Derivative of Arccosecant Function == Let $x \in \R$. Let $\operatorname{arccsc} \left({\dfrac x a}\right)$ be the arccosecant of $\dfrac x a$. Then: <onlyinclude> :$\dfrac {\mathrm d \left({\operatorname{arccsc} \left({\frac x a}\right) }\right)} {\mathrm d x} = \dfrac {-a} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } } = \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}$ </onlyinclude> == Proof == {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{arccsc} \left({\frac x a}\right)}\right)} {\mathrm d x}       | r = \frac 1 a \frac {-1} {\left\vert{\frac x a}\right\vert \sqrt {\left({\frac x a}\right)^2 - 1} }       | c = Derivative of Arccosecant Function and Derivative of Function of Constant Multiple	 }} {{eqn | r = \frac 1 a \frac {-1} {\left\vert{\frac x a}\right\vert \frac {\sqrt {x^2 - a^2} } a}       | c =  }} {{eqn | r = \frac 1 a \frac {-a^2} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } }       | c =  }} {{eqn | r = \frac {-a} {\left\vert{x}\right\vert {\sqrt {x^2 - a^2} } }       | c =  }} {{end-eqn}} {{qed|lemma}} Similarly: {{begin-eqn}} {{eqn | l = \frac {\mathrm d \left({\operatorname{arccsc} \left({\frac x a}\right)}\right)} {\mathrm d x}       | r = \begin{cases} \dfrac 1 a \dfrac {-1} {\frac x a \sqrt {\left({\frac x a}\right)^2 - 1} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac 1 a \dfrac {+1} {\frac x a \sqrt {\left({\frac x a}\right)^2 - 1} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}       | c = Derivative of Arccosecant Function <br/>and Derivative of Function of Constant Multiple	 }} {{eqn | r = \begin{cases} \dfrac {-a} {x \sqrt {x^2 - a^2} } & : 0 < \operatorname{arccsc} \dfrac x a < \dfrac \pi 2 \\ \dfrac a {x \sqrt {x^2 - a^2} } & : -\dfrac \pi 2 < \operatorname{arccsc} \dfrac x a < 0 \\ \end{cases}       | c = simplifying as above }} {{end-eqn}} {{qed}} == Also see == * Derivative of $\arcsin \dfrac x a$ * Derivative of $\arccos \dfrac x a$ * Derivative of $\arctan \dfrac x a$ * Derivative of $\operatorname{arccot} \dfrac x a$ * Derivative of $\operatorname{arcsec} \dfrac x a$ Category:Derivatives of Inverse Trigonometric Functions Category:Arccosecant Function rha9282refmkf3hxesaj45e2azyhcus