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| | ==Identity== | | ==Identity== |
| − | The '''Four-Square Identity''', credited to [[Leonhard Euler]], states that 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)</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> | + | 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''. |
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| − | ==Proof== | + | ==Quaternionic interpretation== |
| − | First, let us expand the left-hand side of the identity: <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_1^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_2^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_3^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_4^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2)</cmath> <cmath>= x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2 + x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2 + x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2 + x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2.</cmath> Thus, we 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_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2</cmath> <cmath>+ x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2</cmath> <cmath>+ x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2</cmath> <cmath>+ x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2. \text{ (1)}</cmath>
| + | 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>. |
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| − | Now, let us expand the first square of the right-hand side of the identity, <math>(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2</math>: <cmath>(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)^2 = (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)</cmath> <cmath>= x_1y_1 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_2 y_2 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_3 y_3 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_4 y_4 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)</cmath> <cmath>= x_1^2 y_1^2 + x_1x_2y_1y_2 + x_1x_3y_1y_3 + x_1x_4y_1y_4 + x_1x_2y_1y_2 + x_2^2 y_2^2 + x_2x_3y_2y_3 + x_2x_4y_2y_4 + x_1x_3y_1y_3 + x_2x_3y_2y_3 + x_3^2 y_3^2 + x_3x_4y_3y_4 + x_1x_4y_1y_4 + x_2x_4y_2y_4 + x_3x_4y_3y_4 + x_4^2 y_4^2</cmath> <cmath>= (x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 + x_4^2 y_4^2) + (2x_1x_2y_1y_2 + 2x_1x_3y_1y_3 + 2x_1x_4y_1y_4 + 2x_2x_3y_2y_3 + 2x_2x_4y_2y_4 + 2x_3x_4y_3y_4).</cmath> Therefore, we now also have <cmath>(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2 = (x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 + x_4^2 y_4^2) + (2x_1x_2y_1y_2 + 2x_1x_3y_1y_3 + 2x_1x_4y_1y_4 + 2x_2x_3y_2y_3 + 2x_2x_4y_2y_4 + 2x_3x_4y_3y_4). \text{ (2)}</cmath>
| + | Then Euler's Four-Square Identity simply reads <math>|XY|^2 = |X|^2 |Y|^2</math>; i.e. the quaternion norm is multiplicative. |
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| − | Next, let us expand the second square of the right-hand side of the identity, <math>(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2</math>: <cmath>(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2 = (x_1y_2-x_2y_1+x_3y_4-x_4y_3)(x_1y_2-x_2y_1+x_3y_4-x_4y_3)</cmath> <cmath>= x_1y_2 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) - x_2y_1 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) + x_3y_4 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) - x_4y_3 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3)</cmath> <cmath> = x_1^2 y_2^2 - x_1x_2y_1y_2 + x_1x_3y_2y_4 - x_1x_4y_2y_3 - x_1x_2y_1y_2 + x_2^2y_1^2 - x_2x_3y_1y_4 + x_2x_4y_1y_3 + x_1x_3y_2y_4 - x_2x_3y_1y_4 + x_3^2y_4^2 - x_3x_4y_3y_4 - x_1x_4y_2y_3 + x_2x_4y_1y_3 - x_3x_4y_3y_4 + x_4^2 y_3^2</cmath>
| + | [[Category: Equations]] |
, 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.
and 
. Recall that the quaternion norm of a number 
, written as 
, is simply 
.
; i.e. the quaternion norm is multiplicative.
