Brahmagupta-Fibonacci two-square identity - Revision history

Vipul: /* In terms of complex numbers */

2012-01-16T04:58:32Z

In terms of complex numbers

← Older revision		Revision as of 04:58, 16 January 2012
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	* <math>a^2 + b^2</math> is the modulus-squared of the complex number <math>a + ib</math>		* <math>a^2 + b^2</math> is the modulus-squared of the complex number <math>a + ib</math>
	* <math>c^2 + d^2</math> is the modulus-squared of the complex number <math>c + id<math>.		* <math>c^2 + d^2</math> is the modulus-squared of the complex number <math>c + id</math>.
	* <math>(ac - bd)^2 + (ad + bc)^2</math> is the modulus-squared of the complex number <math>(a + ib)(c + id)</math>.		* <math>(ac - bd)^2 + (ad + bc)^2</math> is the modulus-squared of the complex number <math>(a + ib)(c + id)</math>.

Vipul: moved Product of sums of two squares is sum of two squares to Brahmagupta-Fibonacci two-square identity

2010-08-13T22:04:21Z

moved Product of sums of two squares is sum of two squares to Brahmagupta-Fibonacci two-square identity

← Older revision	Revision as of 22:04, 13 August 2010
(No difference)

Vipul: /* Related facts */

2010-08-13T21:59:57Z

Related facts

← Older revision		Revision as of 21:59, 13 August 2010
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	This is based on the interpretation in terms of complex numbers, and relates the formula to the angle sum formulas for sine and cosine. {{fillin}}		This is based on the interpretation in terms of complex numbers, and relates the formula to the angle sum formulas for sine and cosine. {{fillin}}
	==Related facts==		==Related facts==

			* [[Euler's four-square identity]]
			* [[Degen's eight-square identity]]

	===Applications===		===Applications===

	* [[Positive integer is a sum of two squares iff it has no prime divisor that is 3 mod 4 with odd multiplicity]]		* [[Positive integer is a sum of two squares iff it has no prime divisor that is 3 mod 4 with odd multiplicity]]

Vipul: /* Statement */

2010-08-13T21:59:16Z

Statement

← Older revision		Revision as of 21:59, 13 August 2010
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	==Statement==		==Statement==

			===As an identity===

			For all <math>a,b,c,d</math> in a commutative ring:

			<math>\! (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2</math>

	===For products two at a time===		===For products two at a time===
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	In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[fact about::sum of two squares]], so can <math>xy</math>.		In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[fact about::sum of two squares]], so can <math>xy</math>.

	More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. ~~In other words, for all <math>a,b,c,d</math> in a commutative ring:~~		More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>.

	~~<math>\! (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2</math>~~

	===For products more than two at a time===		===For products more than two at a time===

Vipul at 21:57, 13 August 2010

2010-08-13T21:57:05Z

← Older revision		Revision as of 21:57, 13 August 2010
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	==Statement==		==Statement==

			===For products two at a time===

	In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[fact about::sum of two squares]], so can <math>xy</math>.		In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[fact about::sum of two squares]], so can <math>xy</math>.
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	<math>\! (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2</math>		<math>\! (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2</math>

			===For products more than two at a time===

			In any [[commutative unital ring]], if <math>x_1, x_2, \dots, x_n</math> are elements each of which can be written as a [[sum of two squares]], so can the product <math>x_1x_2 \dots x_n</math>.

			Note that this follows by induction from the statement for products two at a time.

	==Interpretations==		==Interpretations==

Vipul at 21:55, 13 August 2010

2010-08-13T21:55:38Z

← Older revision		Revision as of 21:55, 13 August 2010
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	More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring:		More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring:

	<math>\! (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2</math>		<math>\! (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2</math>

			==Interpretations==

			===In terms of complex numbers===

			This identity is motivated by complex numbers. In fact, it is precisely the statement that the modulus-squared operation for complex numbers is multiplicative. To see this note that:

			* <math>a^2 + b^2</math> is the modulus-squared of the complex number <math>a + ib</math>
			* <math>c^2 + d^2</math> is the modulus-squared of the complex number <math>c + id<math>.
			* <math>(ac - bd)^2 + (ad + bc)^2</math> is the modulus-squared of the complex number <math>(a + ib)(c + id)</math>.

			===In terms of trigonometry===

			This is based on the interpretation in terms of complex numbers, and relates the formula to the angle sum formulas for sine and cosine. {{fillin}}
	==Related facts==		==Related facts==

Vipul at 21:51, 13 August 2010

2010-08-13T21:51:43Z

← Older revision		Revision as of 21:51, 13 August 2010
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	==Statement==		==Statement==

	In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[sum of two squares]], so can <math>xy</math>.		In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[fact about::sum of two squares]], so can <math>xy</math>.

	More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring:		More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring:

	<math>\! (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2</math>		<math>\! (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2</math>

			==Related facts==

			===Applications===

			* [[Positive integer is a sum of two squares iff it has no prime divisor that is 3 mod 4 with odd multiplicity]]

Vipul: Created page with "==Statement== In any commutative unital ring, if $x$ and $y$ can each be written as a sum of two squares, so can $xy$ . More concretely,..."

2010-08-13T21:50:26Z

Created page with "==Statement== In any commutative unital ring, if <math>x</math> and <math>y</math> can each be written as a sum of two squares, so can <math>xy</math>. More concretely,..."

New page

==Statement==

In any [[commutative unital ring]], if <math>x</math> and <math>y</math> can each be written as a [[sum of two squares]], so can <math>xy</math>.

More concretely, if <math>x = a^2 + b^2</math> and <math>y = c^2 + d^2</math>, then <math>xy = (ac + bd)^2 + (ad - bc)^2</math>. In other words, for all <math>a,b,c,d</math> in a commutative ring:

<math>\! (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2</math>