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de:Quadratzahl ja:平方数 sl:kvadratno število zh:平方数
In mathematics, a square number, sometimes also called a perfect square, is a positive integer that can be written as the square of some other integer. So for example, 9 is a square number since it can be written as 3×3. By convention, the first square number is 1. The number m is a square number if and only if one can arrange m points in a square:
1:
+ x
4:
x + x x
+ + x x
9:
x x + x x x
x x + x x x
+ + + x x x
16:
x x x + x x x x
x x x + x x x x
x x x + x x x x
+ + + + x x x x
25:
x x x x + x x x x x
x x x x + x x x x x
x x x x + x x x x x
x x x x + x x x x x
+ + + + + x x x x x
The first 50 squares are:
1 4 9 16 25 36 49 64 81 100
121 144 169 196 225 256 289 324 361 400
441 484 529 576 625 676 729 784 841 900
961 1024 1089 1156 1225 1296 1369 1444 1521 1600
1681 1764 1849 1936 2025 2116 2209 2304 2401 2500
The formula for the nth square number is n2. This is also equal to the sum of the first n odd numbers, as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (marked as '+').
So for example, 52 = 25 = 1 + 3 + 5 + 7 + 9.
A square number is also the sum of two consecutive triangular numbers.
Lagrange's four-square theorem states that any positive integer can be written as the sum of at most 4 perfect squares. 3 squares are not sufficient for numbers of the form 4k(8l + 7). A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4k+3.
This is generalized by Waring's problem.
A positive integer that has no perfect square divisors except 1 is called square-free.
Since the product of two negative numbers is positive, and the product of two positive numbers is also positive, it follows that no square number is negative. This has important consequences. It follows, in particular, that no square root can be taken of a negative number within the system of real numbers. This leaves a gap in the real number system that mathematicians fill by postulating imaginary numbers, beginning with i, which by convention is the square root of -1.
Squaring is also useful for statisticians in determining the standard deviation of a population or sample from its mean. Each datum is subtracted from the mean, and the result is squared. Then an average is taken of the new set of numbers (each of which is positive). This average is the variance, and its square root is the standard deviation -- in finance, the volatility.
See also:
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