Euler's four-square identity
Hi there, buddy! Do you know what numbers are? Yes, they are like counting things. Let's say you have 1 cookie, then you get another, how many do you have now? Exactly, 2 cookies!
Now, have you ever heard of the mathematician called Euler? He came up with something really cool called Euler's four-square identity. It's a way to add four whole numbers together in a special way.
Let's say you have four numbers: A, B, C, and D. Euler found out that if you take the square of A and add it to the square of B, and add the square of C and the square of D, that equals something really special.
That special thing is that if you take the square of the sum of A and B, and the square of the difference of A and B, and add them together, and do the same thing with C and D, and add those answers together too, then you will get the same answer as if you added the squares of A, B, C, and D together.
That might sound a little tricky, but let's try an example. Let's say A is 1, B is 2, C is 3, and D is 4.
The first thing we do is we square each of those numbers. So A squared is 1, B squared is 4, C squared is 9, and D squared is 16.
Now we add those together: 1 + 4 + 9 + 16 = 30.
That's the answer to the left side of Euler's four-square identity.
Next, we take the sum of A and B, which is 1 + 2 = 3, and we square it, which gives us 9.
Then we take the difference of A and B, which is 1 - 2 = -1, and we square it, which gives us 1.
We add those two answers together: 9 + 1 = 10.
We do the same thing with C and D. So we take the sum of C and D, which is 3 + 4 = 7, and we square it, which gives us 49.
Then we take the difference of C and D, which is 3 - 4 = -1, and we square it, which gives us 1.
We add those two answers together: 49 + 1 = 50.
Finally, we add those two numbers together: 10 + 50 = 60.
And that is also equal to 30, which was the answer we got for adding the squares of A, B, C, and D together!
So, that's what Euler's four-square identity is all about. It's a way to add up four whole numbers in a special way and get the same answer as if you added up their squares. Pretty cool, huh?
Now, have you ever heard of the mathematician called Euler? He came up with something really cool called Euler's four-square identity. It's a way to add four whole numbers together in a special way.
Let's say you have four numbers: A, B, C, and D. Euler found out that if you take the square of A and add it to the square of B, and add the square of C and the square of D, that equals something really special.
That special thing is that if you take the square of the sum of A and B, and the square of the difference of A and B, and add them together, and do the same thing with C and D, and add those answers together too, then you will get the same answer as if you added the squares of A, B, C, and D together.
That might sound a little tricky, but let's try an example. Let's say A is 1, B is 2, C is 3, and D is 4.
The first thing we do is we square each of those numbers. So A squared is 1, B squared is 4, C squared is 9, and D squared is 16.
Now we add those together: 1 + 4 + 9 + 16 = 30.
That's the answer to the left side of Euler's four-square identity.
Next, we take the sum of A and B, which is 1 + 2 = 3, and we square it, which gives us 9.
Then we take the difference of A and B, which is 1 - 2 = -1, and we square it, which gives us 1.
We add those two answers together: 9 + 1 = 10.
We do the same thing with C and D. So we take the sum of C and D, which is 3 + 4 = 7, and we square it, which gives us 49.
Then we take the difference of C and D, which is 3 - 4 = -1, and we square it, which gives us 1.
We add those two answers together: 49 + 1 = 50.
Finally, we add those two numbers together: 10 + 50 = 60.
And that is also equal to 30, which was the answer we got for adding the squares of A, B, C, and D together!
So, that's what Euler's four-square identity is all about. It's a way to add up four whole numbers in a special way and get the same answer as if you added up their squares. Pretty cool, huh?
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