Find the relation between odd numbers and perfect square numbers.
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In mathematics, a square number or 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. For example, 9 is a square number, since it can be written as 3 × 3.
The usual notation for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "nsquared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with side length n has area n2.
Square numbers are non-negative. Another way of saying that a (non-negative) integer 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 = 0being the zeroth one. 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., {\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}}.
Starting with 1, there are ⌊√m⌋ square numbers up to and including m, where the expression ⌊x⌋ represents the floor of the number x.
ExamplesEdit
The squares (sequence A000290 in the OEIS) smaller than 602 = 3600 are:
02 = 012 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81
102 = 100112 = 121122 = 144132 = 169142 = 196152 = 225162 = 256172 = 289182 = 324192 = 361
202 = 400212 = 441222 = 484232 = 529242 = 576252 = 625262 = 676272 = 729282 = 784292 = 841
302 = 900312 = 961322 = 1
The usual notation for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "nsquared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (1 × 1). Hence, a square with side length n has area n2.
Square numbers are non-negative. Another way of saying that a (non-negative) integer 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 = 0being the zeroth one. 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., {\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}}.
Starting with 1, there are ⌊√m⌋ square numbers up to and including m, where the expression ⌊x⌋ represents the floor of the number x.
ExamplesEdit
The squares (sequence A000290 in the OEIS) smaller than 602 = 3600 are:
02 = 012 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81
102 = 100112 = 121122 = 144132 = 169142 = 196152 = 225162 = 256172 = 289182 = 324192 = 361
202 = 400212 = 441222 = 484232 = 529242 = 576252 = 625262 = 676272 = 729282 = 784292 = 841
302 = 900312 = 961322 = 1
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