Difference between revisions of "Euler's Four-Square Identity"
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==Identity== | ==Identity== | ||
| − | '' | + | The '''Four-Square Identity''', credited to [[Leonhard Euler]], states that for any eight [[Number|numbers]] <math>x_1,x_2, x_3, x_4, y_1, y_2, y_3, y_4</math>, we must have <cmath>(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)</cmath> <cmath>=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2</cmath> <cmath>+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2</cmath> <cmath>+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2</cmath> <cmath>+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.</cmath> |
| + | (This statement can be easily verified by expansion.) | ||
| + | In other words, ''the product of the sums of four squares is itself the sum of four squares''. | ||
| − | + | ==Quaternionic interpretation== | |
| + | Define <math>X := x_1 + x_2 i + x_3 j + x_4 k</math> and <math>Y: = y_1 + y_2 i + y_3 j + y_4 k</math>. Recall that the quaternion norm of a number <math>a + bi + cj + dk</math>, written as <math>|a + bi + cj + dk|^2</math>, is simply <math>a^2 + b^2 + c^2 + d^2</math>. | ||
| + | |||
| + | Then Euler's Four-Square Identity simply reads <math>|XY|^2 = |X|^2 |Y|^2</math>; i.e. the quaternion norm is multiplicative. | ||
| + | |||
| + | [[Category: Equations]] | ||
Latest revision as of 22:24, 5 November 2019
Identity
The Four-Square Identity, credited to Leonhard Euler, states that for any eight numbers 
, we must have ![\[(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)\]](http://latex.artofproblemsolving.com/6/b/6/6b6fc940cccd9e26036d3f56a9b7842087b693e3.png)
![\[=(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2\]](http://latex.artofproblemsolving.com/7/d/3/7d3e8fc99dee75a54c48f5ab4044d84c7bdf9180.png)
![\[+(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2\]](http://latex.artofproblemsolving.com/d/3/f/d3fe8021c88eaea0a2d2df6f02d536fea296da26.png)
![\[+(x_1y_3-x_3y_1+x_4y_2-x_2y_4)^2\]](http://latex.artofproblemsolving.com/e/4/8/e48a7385a25ce27a5c9bf7c8f2c455ea15429307.png)
![\[+(x_1y_4-x_4y_1 + x_2y_3 - x_3y_2)^2.\]](http://latex.artofproblemsolving.com/2/6/8/268b1ee217541ac04dc2ef23023e64b2659a20c8.png)
(This statement can be easily verified by expansion.)
In other words, the product of the sums of four squares is itself the sum of four squares.
Quaternionic interpretation
Define 
and 
. Recall that the quaternion norm of a number 
, written as 
, is simply 
.
Then Euler's Four-Square Identity simply reads 
; i.e. the quaternion norm is multiplicative.
