Square number: Difference between revisions
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{{more footnotes|date=February 2012}} |
{{more footnotes|date=February 2012}} |
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In [[mathematics]], a '''square number''', sometimes also called a '''perfect square''', is an [[integer]] that is the |
In [[mathematics]], a '''square number''', sometimes also called a '''perfect square''', is an [[integer]] that is the square of an integer;<ref>Some authors also call squares of rational numbers perfect squares.</ref> in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. |
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The usual notation for the formula for the square of a number ''n'' is not the product |
The usual notation for the formula for the square of a number ''n'' is not the product ''n'' × ''n'', but the equivalent [[exponentiation]] ''n''<sup>2</sup>, usually pronounced as "''n'' squared". The name square number comes from the name of the shape. This is because a square with side length ''n'' has [[area]] ''n''<sup>2</sup>. |
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Square numbers are [[non-negative]]. Another way of saying that a (non-negative) number is a square number, is that its [[square root]] is again an integer. For example, |
Square numbers are [[non-negative]]. Another way of saying that a (non-negative) number is a square number, is that its [[square root]] is again an integer. For example, √9 = 3, so 9 is a square number. |
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A positive integer that has no perfect square [[divisor]]s except 1 is called [[Square-free integer|square-free]]. |
A positive integer that has no perfect square [[divisor]]s except 1 is called [[Square-free integer|square-free]]. |
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For a non-negative integer |
For a non-negative integer ''n'', the ''n''th square number is ''n''<sup>2</sup>, with 0<sup>2</sup> = 0 being the [[zeroth]] square. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square (e.g., 4/9 = (2/3)<sup>2</sup>). |
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Starting with 1, there are <math>\lfloor \sqrt{m} \rfloor</math> square numbers up to and including |
Starting with 1, there are <math>\lfloor \sqrt{m} \rfloor</math> square numbers up to and including ''m'', where the expression <math>\lfloor x \rfloor</math> represents the [[floor function | floor]] of the number ''x''. |
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==Examples== |
==Examples== |
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==Properties== |
==Properties== |
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The number |
The number ''m'' is a square number if and only if one can arrange ''m'' points in a square: |
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{| cellpadding="8" |
{| cellpadding="8" |
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|m = 1<sup>2</sup> = 1 |
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|[[Image:Square number 1.png]] |
|[[Image:Square number 1.png]] |
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|- |
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|m = 2<sup>2</sup> = 4 |
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|[[Image:Square number 4.png]] |
|[[Image:Square number 4.png]] |
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|- |
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|m = 3<sup>2</sup> = 9 |
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|[[Image:Square number 9.png]] |
|[[Image:Square number 9.png]] |
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|- |
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|m = 4<sup>2</sup> = 16 |
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|[[Image:Square number 16.png]] |
|[[Image:Square number 16.png]] |
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|- |
|- |
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|m = 5<sup>2</sup> = 25 |
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|[[Image:Square number 25.png]] |
|[[Image:Square number 25.png]] |
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|} |
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The expression for the |
The expression for the ''n''th square number is ''n''<sup>2</sup>. This is also equal to the sum of the first ''n'' [[odd number]]s 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: |
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:<math>n^2 = \sum_{k=1}^n(2k-1).</math> |
:<math>n^2 = \sum_{k=1}^n(2k-1).</math> |
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So for example, 5<sup>2</sup> = 25 = 1 + 3 + 5 + 7 + 9. |
So for example, 5<sup>2</sup> = 25 = 1 + 3 + 5 + 7 + 9. |
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There are several [[recursion#Recursion in mathematics|recursive]] methods for computing square numbers. For example, the |
There are several [[recursion#Recursion in mathematics|recursive]] methods for computing square numbers. For example, the ''n''th square number can be computed from the previous square by <math>n^2 = (n-1)^2 + (n-1) + n = (n-1)^2 + (2n-1)</math>. Alternatively, the ''n''th square number can be calculated from the previous two by doubling the (''n'' − 1)-th square, subtracting the (''n'' − 2)-th square number, and adding 2, because ''n''<sup>2</sup> = 2(''n'' − 1)<sup>2</sup> − (''n'' − 2)<sup>2</sup> + 2. For example, 2 × 5<sup>2</sup> − 4<sup>2</sup> + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 6<sup>2</sup>. |
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:{{math|1= 2 × 5<sup>2</sup> − 4<sup>2</sup> + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 6<sup>2</sup>}}. |
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A square number is also the sum of two consecutive [[triangular number]]s. The sum of two consecutive square numbers is a [[centered square number]]. Every odd square is also a [[centered octagonal number]]. |
A square number is also the sum of two consecutive [[triangular number]]s. The sum of two consecutive square numbers is a [[centered square number]]. Every odd square is also a [[centered octagonal number]]. |
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Another property of a square number is that it has an odd number of divisors, while other numbers have an even number of divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs. |
Another property of a square number is that it has an odd number of divisors, while other numbers have an even number of divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs. |
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[[Lagrange's four-square theorem]] states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form |
[[Lagrange's four-square theorem]] states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4<sup>''k''</sup>(8''m'' + 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 4''k'' + 3. This is generalized by [[Waring's problem]]. |
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A square number can end only with digits 0, |
A square number can end only with digits 0,1,4,6,9, or 25 in base 10, as follows: |
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#If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the [[Numerical digit|digits]] preceding the ending 0s must also form a square. |
#If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the [[Numerical digit|digits]] preceding the ending 0s must also form a square. |
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#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. |
#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. |
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In base 16, a square number can end only with 0, |
In base 16, a square number can end only with 0,1,4 or 9 and<br> |
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- in case 0, only 0,1,4,9 can precede it,<br> |
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- in case 4, only even numbers can precede it.<br> |
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In general, if a [[prime]] |
In general, if a [[prime]] ''p'' divides a square number ''m'' then the square of ''p'' must also divide ''m''; if ''p'' fails to divide <math>m/p</math>, then ''m'' is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number ''m'' is a square number if and only if, in its [[canonical representation of a positive integer|canonical representation]], all exponents are even. |
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Squarity testing can be used as alternative way in [[factorization]] of large numbers. Instead of testing for divisibility, test for squarity: for given |
Squarity testing can be used as alternative way in [[factorization]] of large numbers. Instead of testing for divisibility, test for squarity: for given ''m'' and some number ''k'', if ''k² − m'' is the square of an integer ''n'' then ''k − n'' divides ''m''. (This is an application of the factorization of a [[difference of two squares]].) For example, 100² − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from ''k − n'' to ''k + n'' where ''k'' covers some range of natural numbers ''k'' ≥ √''m''. |
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A square number cannot be a [[perfect number]]. |
A square number cannot be a [[perfect number]]. |
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==Special cases== |
==Special cases== |
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* If the number is of the form ''m''5 where ''m'' represents the preceding digits, its square is ''n''25 where |
* If the number is of the form ''m''5 where ''m'' represents the preceding digits, its square is ''n''25 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. |
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* If the number is of the form ''m''0 where ''m'' represents the preceding digits, its square is ''n''00 where |
* If the number is of the form ''m''0 where ''m'' represents the preceding digits, its square is ''n''00 where ''n'' = ''m''<sup>2</sup>. For example the square of 70 is 4900. |
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* If the number has two digits and is of the form 5''m'' where ''m'' represents the units digit, its square is ''AABB'' where |
* If the number has two digits and is of the form 5''m'' where ''m'' represents the units digit, its square is ''AABB'' where ''AA'' = 25 + ''m'' and ''BB'' = ''m''<sup>2</sup>. Example: To calculate the square of 57, 25 + 7 = 32 and 7<sup>2</sup> = 49, which means 57<sup>2</sup> = 3249. |
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==Odd and even square numbers==<!-- This section is linked from [[Irrational number]] --> |
==Odd and even square numbers==<!-- This section is linked from [[Irrational number]] --> |
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Squares of even numbers are even (and in fact divisible by 4), since |
Squares of even numbers are even (and in fact divisible by 4), since (2''n'')<sup>2</sup> = 4''n''<sup>2</sup>. |
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Squares of odd numbers are odd, since |
Squares of odd numbers are odd, since (2''n'' + 1)<sup>2</sup> = 4(''n''<sup>2</sup> + ''n'') + 1. |
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It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd. |
It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd. |
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==Uses== |
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Since the product of [[Real number|real]] [[negative number]]s is positive, and the product of two real [[positive number]]s is also positive, it follows that no square number is negative. It follows that no [[square root]] can be taken of a negative number within the system of [[real number]]s. This leaves a gap in the real number system that mathematicians fill by postulating [[complex number]]s, beginning with the [[imaginary unit]] ''i'', which by convention is one of the square roots of −1. |
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Squaring is used in [[statistics]] in determining the [[standard deviation]] of a set of values. The deviation of each value <math>x_i\,</math> from the [[mean]] <math>\overline{x} \,</math> of the set is defined as the difference <math>x_i - \overline{x} \,</math>. These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the [[variance]], and its square root is the standard deviation. In [[finance]], the [[volatility (finance)|volatility]] of a financial instrument is the standard deviation of its values. |
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==See also== |
==See also== |
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* [[Square root]] |
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* [[Methods of computing square roots]] |
* [[Methods of computing square roots]] |
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* [[Quadratic residue]] |
* [[Quadratic residue]] |
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* [[Polygonal number]] |
* [[Polygonal number]] |
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* [[Euler's four-square identity]] |
* [[Euler's four-square identity]] |
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* [[Cube (algebra)]] |
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* [[Fermat's theorem on sums of two squares]] |
* [[Fermat's theorem on sums of two squares]] |
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* [[Pythagorean theorem]] |
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* [[Parallelogram law]] |
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* [[Brahmagupta–Fibonacci identity]] |
* [[Brahmagupta–Fibonacci identity]] |
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* ''[[The Book of Squares]]'' |
* ''[[The Book of Squares]]'' |
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== Further reading == |
== Further reading == |
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*[[J.H. Conway|Conway, J. H.]] and [[R.K. Guy|Guy, R. K.]] ''The Book of Numbers''. New York: Springer-Verlag, pp. |
*[[J.H. Conway|Conway, J. H.]] and [[R.K. Guy|Guy, R. K.]] ''The Book of Numbers''. New York: Springer-Verlag, pp. 30-32, 1996. ISBN 0-387-97993-X |
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==External links== |
==External links== |
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* Dario Alpern, [http://www.alpertron.com.ar/FSQUARES.HTM Sum of squares]. A Java applet to decompose a natural number into a sum of up to four squares. |
* Dario Alpern, [http://www.alpertron.com.ar/FSQUARES.HTM Sum of squares]. A Java applet to decompose a natural number into a sum of up to four squares. |
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*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1296&bodyId=1433 Fibonacci and Square Numbers] at [http://mathdl.maa.org/convergence/1/ Convergence] |
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1296&bodyId=1433 Fibonacci and Square Numbers] at [http://mathdl.maa.org/convergence/1/ Convergence] |
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* [http://www.naturalnumbers.org/psquares.html The first 1,000,000 perfect squares] Includes a program for generating perfect squares up to 10 |
* [http://www.naturalnumbers.org/psquares.html The first 1,000,000 perfect squares] Includes a program for generating perfect squares up to 10^15. |
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[[Category:Figurate numbers]] |
[[Category:Figurate numbers]] |
Revision as of 18:09, 3 September 2012
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2012) |
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer;[1] in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3.
The usual notation for the formula for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "n squared". The name square number comes from the name of the shape. This is because a square with side length n has area n2.
Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number, is that its square root is again an integer. For example, √9 = 3, so 9 is a square number.
A positive integer that has no perfect square divisors except 1 is called square-free.
For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zeroth square. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square (e.g., 4/9 = (2/3)2).
Starting with 1, there are square numbers up to and including m, where the expression represents the floor of the number x.
Examples
The squares (sequence A000290 in the OEIS) smaller than 602 are:
- 02 = 0
- 12 = 1
- 22 = 4
- 32 = 9
- 42 = 16
- 52 = 25
- 62 = 36
- 72 = 49
- 82 = 64
- 92 = 81
- 102 = 100
- 112 = 121
- 122 = 144
- 132 = 169
- 142 = 196
- 152 = 225
- 162 = 256
- 172 = 289
- 182 = 324
- 192 = 361
- 202 = 400
- 212 = 441
- 222 = 484
- 232 = 529
- 242 = 576
- 252 = 625
- 262 = 676
- 272 = 729
- 282 = 784
- 292 = 841
- 302 = 900
- 312 = 961
- 322 = 1024
- 332 = 1089
- 342 = 1156
- 352 = 1225
- 362 = 1296
- 372 = 1369
- 382 = 1444
- 392 = 1521
- 402 = 1600
- 412 = 1681
- 422 = 1764
- 432 = 1849
- 442 = 1936
- 452 = 2025
- 462 = 2116
- 472 = 2209
- 482 = 2304
- 492 = 2401
- 502 = 2500
- 512 = 2601
- 522 = 2704
- 532 = 2809
- 542 = 2916
- 552 = 3025
- 562 = 3136
- 572 = 3249
- 582 = 3364
- 592 = 3481
The difference between any perfect square and its predecessor is given by the identity . Equivalently, it is possible to count up square numbers by adding together the last square, the last square's root, and the current root, that is, .
Properties
The number m is a square number if and only if one can arrange m points in a square:
m = 12 = 1 | |
m = 22 = 4 | |
m = 32 = 9 | |
m = 42 = 16 | |
m = 52 = 25 |
The expression 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 (shown in magenta). The formula follows:
So for example, 52 = 25 = 1 + 3 + 5 + 7 + 9.
There are several recursive methods for computing square numbers. For example, the nth square number can be computed from the previous square by . Alternatively, the nth square number can be calculated from the previous two by doubling the (n − 1)-th square, subtracting the (n − 2)-th square number, and adding 2, because n2 = 2(n − 1)2 − (n − 2)2 + 2. For example, 2 × 52 − 42 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 62.
A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.
Another property of a square number is that it has an odd number of divisors, while other numbers have an even number of divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.
Lagrange's four-square theorem states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4k(8m + 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 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.
In base 16, a square number can end only with 0,1,4 or 9 and
- in case 0, only 0,1,4,9 can precede it,
- in case 4, only even numbers can precede it.
In general, if a prime p divides a square number m then the square of p must also divide m; if p fails to divide , then m is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number m is a square number if and only if, in its canonical representation, all exponents are even.
Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k, if k² − m is the square of an integer n then k − n divides m. (This is an application of the factorization of a difference of two squares.) For example, 100² − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from k − n to k + n where k covers some range of natural numbers k ≥ √m.
A square number cannot be a perfect number.
The sum of the series of power numbers
can also be represented by the formula
The first terms of this series (the square pyramidal numbers) are:
0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201... (sequence A000330 in the OEIS).
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 = m2. 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 = m2. Example: To calculate the square of 57, 25 + 7 = 32 and 72 = 49, which means 572 = 3249.
Odd and even square numbers
Squares of even numbers are even (and in fact divisible by 4), since (2n)2 = 4n2.
Squares of odd numbers are odd, since (2n + 1)2 = 4(n2 + n) + 1.
It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.
Uses
Since the product of real negative numbers is positive, and the product of two real positive numbers is also positive, it follows that no square number is negative. It follows 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 complex numbers, beginning with the imaginary unit i, which by convention is one of the square roots of −1.
Squaring is used in statistics in determining the standard deviation of a set of values. The deviation of each value from the mean of the set is defined as the difference . These deviations are squared, then a mean is taken of the new set of numbers (each of which is positive). This mean is the variance, and its square root is the standard deviation. In finance, the volatility of a financial instrument is the standard deviation of its values.
See also
- Square root
- Methods of computing square roots
- Quadratic residue
- Square-free polynomial
- Polygonal number
- Euler's four-square identity
- Cube (algebra)
- Fermat's theorem on sums of two squares
- Pythagorean theorem
- Parallelogram law
- Brahmagupta–Fibonacci identity
- The Book of Squares
- Integer square root
- Square triangular number
- Automorphic number
- Exponentiation
- Power of two
Notes
- ^ Some authors also call squares of rational numbers perfect squares.
References
Further reading
- Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30-32, 1996. ISBN 0-387-97993-X
External links
- 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.