Difference between revisions of "Euler's Four-Square Identity"
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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> | 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> | ||
| − | 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> | + | 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{ (1)}</cmath> |
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_2y_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> | 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_2y_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> | ||
Revision as of 08:35, 29 March 2019
Identity
The Four-Square Identity, credited to Leonhard Euler, states that for any eight complex numbers 
, we must have ![\[(x_1^2+ x_2^2 + x_3 ^2 + x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2)\]](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)
Proof
First, let us expand the left-hand side of the identity: ![\[(x_1^2+x_2^2+x_3^2+x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2) = x_1^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_2^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_3^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2) + x_4^2 \cdot (y_1^2+y_2^2+y_3^2+y_4^2)\]](http://latex.artofproblemsolving.com/7/7/e/77e8c69805418fe82050c4c2f22ff27ac65b4fe8.png)
![\[= x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2 + x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2 + x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2 + x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2.\]](http://latex.artofproblemsolving.com/b/5/0/b507c2728a218499131d4083713b3ef8d0dc48ed.png)
Thus, we have ![\[(x_1^2+x_2^2+x_3^2+x_4^2)(y_1^2+y_2^2+y_3^2+y_4^2) = x_1^2 y_1^2 + x_1^2 y_2^2 + x_1^2 y_3^2 + x_1^2 y_4^2\]](http://latex.artofproblemsolving.com/d/b/2/db2c6de8c4286f0edaa413d6866281078f5b27db.png)
![\[+ x_2^2 y_1^2 + x_2^2 y_2^2 + x_2^2 y_3^2 + x_2^2 y_4^2\]](http://latex.artofproblemsolving.com/b/3/4/b34721ae634891a90a39c0adae5f988262377d9a.png)
![\[+ x_3^2 y_1^2 + x_3^2 y_2^2 + x_3^2 y_3^2 + x_3^2 y_4^2\]](http://latex.artofproblemsolving.com/8/c/6/8c693a66ac907feeb00ac7bdb4300cbe0749f895.png)
![\[+ x_4^2 y_1^2 + x_4^2 y_2^2 + x_4^2 y_3^2 + x_4^2 y_4^2. \text{ (1)}\]](http://latex.artofproblemsolving.com/0/1/9/019ce590756e918d5f8e35cbceafd7b700b9bcab.png)
Now, let us expand the first square of the right-hand side of the identity, 
: ![\[(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)^2 = (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)(x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)\]](http://latex.artofproblemsolving.com/7/d/e/7deeb1281816a88371e493dc9761d5229dc48d60.png)
![\[= x_1y_1 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_2 y_2 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_3 y_3 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4) + x_4 y_4 (x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4)\]](http://latex.artofproblemsolving.com/2/3/4/234fa8a691f69b78ce16b70d3b0f00a32fbb7a23.png)
![\[= x_1^2 y_1^2 + x_1x_2y_1y_2 + x_1x_3y_1y_3 + x_1x_4y_1y_4 + x_1x_2y_1y_2 + x_2^2 y_2^2 + x_2x_3y_2y_3 + x_2x_4y_2y_4 + x_1x_3y_1y_3 + x_2x_3y_2y_3 + x_3^2 y_3^2 + x_3x_4y_3y_4 + x_1x_4y_1y_4 + x_2x_4y_2y_4 + x_3x_4y_3y_4 + x_4^2 y_4^2\]](http://latex.artofproblemsolving.com/e/c/c/eccd81543541dc1a6a7d3ee5c6df93f0b1c78c54.png)
![\[= (x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 + x_4^2 y_4^2) + (2x_1x_2y_1y_2 + 2x_1x_3y_1y_3 + 2x_1x_4y_1y_4 + 2x_2x_3y_2y_3 + 2x_2x_4y_2y_4 + 2x_3x_4y_3y_4).\]](http://latex.artofproblemsolving.com/b/7/1/b71de415d78539d39fd8388bee9fb3b76dd0d1fe.png)
Therefore, we now also have ![\[(x_1y_1+x_2y_2+x_3y_3+x_4y_4)^2 = (x_1^2 y_1^2 + x_2^2 y_2^2 + x_3^2 y_3^2 + x_4^2 y_4^2) + (2x_1x_2y_1y_2 + 2x_1x_3y_1y_3 + 2x_1x_4y_1y_4 + 2x_2x_3y_2y_3 + 2x_2x_4y_2y_4 + 2x_3x_4y_3y_4). \text{ (1)}\]](http://latex.artofproblemsolving.com/4/c/b/4cb437d8f6b2f9fe577d0918dbef55cd168d31d6.png)
Next, let us expand the second square of the right-hand side of the identity, 
: ![\[(x_1y_2-x_2y_1+x_3y_4-x_4y_3)^2 = (x_1y_2-x_2y_1+x_3y_4-x_4y_3)(x_1y_2-x_2y_1+x_3y_4-x_4y_3)\]](http://latex.artofproblemsolving.com/3/b/a/3ba759ce733badf6d63f4f80ce90687a525627d0.png)
![\[= x_1y_2 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) - x_2y_1 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) + x_3y_4 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3) - x_4y_3 \cdot (x_1y_2-x_2y_1+x_3y_4-x_4y_3)\]](http://latex.artofproblemsolving.com/5/0/0/500d6126254a4fe9cabd79b665980edfbd69f88a.png)
![\[= x_1^2 y_2^2 - x_1x_2y_1y_2 + x_1x_3y_2y_4 - x_1x_4y_2y_3 - x_1x_2y_1y_2 + x_2^2y_1^2 - x_2x_3y_1y_4 + x_2x_4y_1y_3 + x_1x_3y_2y_4 - x_2x_3y_2y_4 + x_3^2y_4^2 - x_3x_4y_3y_4 - x_1x_4y_2y_3 + x_2x_4y_1y_3 - x_3x_4y_3y_4 + x_4^2 y_3^2\]](http://latex.artofproblemsolving.com/f/d/4/fd42ff5c067602d6268e7f68dc1a5ab11b7cd6e9.png)
