Difference between revisions of "Euler's Four-Square Identity"
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''The product of the sum of the four squares is itself, the sum of four squares.'' | ''The product of the sum of the four squares is itself, the sum of four squares.'' | ||
| − | Mathematically, for any eight [[Complex numbers|complex numbers]] <math>x_1,x_2, x_3, x_4, y_1, y_2, y_3, y_4</math>, we must have | + | Mathematically, for any eight [[Complex numbers|complex 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)=(x_1y_1+x_2y_2+x_3y_3+x^4y^4)^2</cmath> <cmath>\text{ }+hi</cmath> |
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Revision as of 11:37, 11 August 2018
Identity
The product of the sum of the four squares is itself, the sum of four squares.
Mathematically, for any eight complex numbers 
, we must have ![\[(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)=(x_1y_1+x_2y_2+x_3y_3+x^4y^4)^2\]](http://latex.artofproblemsolving.com/1/c/c/1cc04c349196fc677c3db6de5de06b312453a9cc.png)
![\[\text{ }+hi\]](http://latex.artofproblemsolving.com/c/9/4/c9473befbea95e253308417ae08015954dbf50e4.png)
