Because you are squaring numbers, the only way to get double is by using the square root of 2 which is 1.414.
A good example of this is paper sizes.
A4 is 210x297
A3 is double the size of A4 but to get that you need to multiply the sides by 1.414. That gives 297x420
But if you double each side of A4 you get 420x594 which is A2 and that is 4 times as big as A4 not 2 times as big.
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Take a look at the corresponding square roots (of the square numbers). If n^2 is a perfect square (the square of "n" - note: using "^" for exponentiation), then twice that number is the 2n^2. The square root of that is root(2) x n. And since root(2) (the square root of 2) is irrational, it follows that root(2) x n is not a whole number.
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You cannot in real numbers because the square of any number, negative or positive, is positive. However, -25 does have a square root in complex numbers: it is + or - 5i where i is the imaginary square root of -1.
it means that a square is not a rectangle
0 is nothing, you cannot divide a number by nothing, it would be the same as putting the number then the divide symbol and not putting another number in.
10 = 2*5 (prime decomposition)So for a power of 10 to be a square means that the powers its factors must be even. A power of 10 is a square if it contains an even number of factors of 2 and 5. Since 503 is an odd number, 10^503 has an odd number of powers of 2 and 5, so it can't be the square of an integer.