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Square number
A square number, sometimes also called a perfect square, is the result of an integer multiplied by itself. 1, 4, 9, 16 and 25 are the first five square numbers. In a formula, the square of a number n is denoted n^{2} (exponentiation), usually pronounced as "n squared". The name square number comes from the name of the shape; see below.
Square numbers are nonnegative. Another way of saying that a (nonnegative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.
Contents
Examples
The squares (sequence A000290 in OEIS) smaller than 60^{2} are:
 0^{2} = 0
 1^{2} = 1
 2^{2} = 4
 3^{2} = 9
 4^{2} = 16
 5^{2} = 25
 6^{2} = 36
 7^{2} = 49
 8^{2} = 64
 9^{2} = 81
 10^{2} = 100
 11^{2} = 121
 12^{2} = 144
 13^{2} = 169
 14^{2} = 196
 15^{2} = 225
 16^{2} = 256
 17^{2} = 289
 18^{2} = 324
 19^{2} = 361
 20^{2} = 400
 21^{2} = 441
 22^{2} = 484
 23^{2} = 529
 24^{2} = 576
 25^{2} = 625
 26^{2} = 676
 27^{2} = 729
 28^{2} = 784
 29^{2} = 841
 30^{2} = 900
 31^{2} = 961
 32^{2} = 1024
 33^{2} = 1089
 34^{2} = 1156
 35^{2} = 1225
 36^{2} = 1296
 37^{2} = 1369
 38^{2} = 1444
 39^{2} = 1521
 40^{2} = 1600
 41^{2} = 1681
 42^{2} = 1764
 43^{2} = 1849
 44^{2} = 1936
 45^{2} = 2025
 46^{2} = 2116
 47^{2} = 2209
 48^{2} = 2304
 49^{2} = 2401
 50^{2} = 2500
 51^{2} = 2601
 52^{2} = 2704
 53^{2} = 2809
 54^{2} = 2916
 55^{2} = 3025
 56^{2} = 3136
 57^{2} = 3249
 58^{2} = 3364
 59^{2} = 3481
There are infinitely many square numbers, as there are infinitely many natural numbers.
Properties
The number m is a square number if and only if one can compose a square of m equal (lesser) squares:
m = 1^{2} = 1  
m = 2^{2} = 4  
m = 3^{2} = 9  
m = 4^{2} = 16  
m = 5^{2} = 25  
Note: White gaps between squares serve only to improve visual perception. There must be no gaps between actual squares. 
A square with side length n has area n^{2}.
The expression for the nth square number is n^{2}. 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 (shown in magenta). The formula follows:
 [math]n^2 = \sum_{k=1}^n(2k1).[/math]
So for example, 5^{2} = 25 = 1 + 3 + 5 + 7 + 9.
A square number can end only with digits 0, 1, 4, 6, 9, or 25 in base 10, as follows:
 If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the digits preceding the ending 0s must also form a square.
 If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four.
 If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even.
 If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four.
 If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd.
 If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.
A square number cannot be a perfect number.
All fourth powers, sixth powers, eighth powers and so on are perfect squares.
Special cases
 If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m × (m + 1) and represents digits before 25. For example the square of 65 can be calculated by n = 6 × (6 + 1) = 42 which makes the square equal to 4225.
 If the number is of the form m0 where m represents the preceding digits, its square is n00 where n = m^{2}. For example the square of 70 is 4900.
 If the number has two digits and is of the form 5m where m represents the units digit, its square is AABB where AA = 25 + m and BB = m^{2}. Example: To calculate the square of 57, 25 + 7 = 32 and 7^{2} = 49, which means 57^{2} = 3249.
Odd and even square numbers
Squares of even numbers are even (and in fact divisible by 4), since (2n)^{2} = 4n^{2}.
Squares of odd numbers are odd, since (2n + 1)^{2} = 4(n^{2} + n) + 1.
It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.
As all even square numbers are divisible by 4, the even numbers of the form 4n + 2 are not square numbers.
As all odd square numbers are of the form 4n + 1, the odd numbers of the form 4n + 3 are not square numbers.
Squares of odd numbers are of the form 8n + 1, since (2n + 1)^{2} = 4n(n + 1) + 1 and n(n + 1) is an even number.
Notes
References
 Weisstein, Eric W., "Square Number" from MathWorld.
Further reading
 Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, pp. 30–32, 1996. ISBN 038797993X
Other websites
 Learn Square Numbers. Practice square numbers up to 144 with this children's multiplication game
 Dario Alpern, Sum of squares. A Java applet to decompose a natural number into a sum of up to four squares.
 Fibonacci and Square Numbers at Convergence
 The first 1,000,000 perfect squares Includes a program for generating perfect squares up to 10^{15}.

