Properties of Square Numbers
Square numbers:
Numbers that can be expressed as the square of a number are called square numbers or perfect squares. If a and b are two natural numbers such that a = b2, then a is called a square number or a perfect square of b.

Every natural number has a square, but every natural number is not a perfect square itself. Numbers that cannot be expressed as the square of another number are not perfect squares. Since there are infinite natural numbers, there are infinite numbers of perfect squares.

The numbers that have 0, 1, 4, 5, 6 or 9 in their units place maybe perfect squares where as the numbers that have 2, 3, 7 or 8 in their units place are never perfect squares. 
  1. All numbers ending with 1 or 9 have 1 in the units place of their squares. All squares having 1 in their units place are squares of the numbers ending with either 1 or 9.
  2. All numbers ending with 2 or 8 have 4 in the units place of their squares. All squares having 4 in their units place are squares of the numbers ending with either 2 or 8.
  3. All numbers ending with 3 or 7 have 9 in the units place of their squares. All squares having 9 in their units place are squares of the numbers ending with either 3 or 7.
  4. All numbers ending with 4 or 6 have 6 in the units place of their squares. All squares having 6 in their units place are squares of the numbers ending with either 4 or 6.
  5. All numbers ending with 5 have 5 in the units place of their squares. All squares having 5 in their units place are squares of the numbers ending with 5.
  6. All numbers ending with 0 have 0 in the units place of their squares. All squares having 0 in their units place are squares of the numbers ending with 0.
  7. The square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.
  8. The square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1.
For any natural number n, (n + 1)2 - n2 = (n + 1) + n.

Square of a number ending with zero(s) contains double the number of zeroes than the number. All square numbers contain an even number of zeroes.

Odd square numbers are squares of numbers ending with 1, 3, 5, 7 or 9. Even square numbers are squares of numbers ending with 0, 2, 4, 6 or 8.

The numbers whose dot patterns can be arranged as triangles are called triangular numbers. The sum of any two consecutive triangular numbers is a square number.

triangular numbers, triangle, dot patterns

For two consecutive numbers n and n+1, there are 2n non square numbers between n2 and (n+1)2.

Sum of first n odd natural numbers is n2. The square of any odd number can be written as the sum of two consecutive positive integers.

132 = 169
1032 = 10609
10032 = 1006009
100032 = 100060009
1000032 = 10000600009.

Summary

Square numbers:
Numbers that can be expressed as the square of a number are called square numbers or perfect squares. If a and b are two natural numbers such that a = b2, then a is called a square number or a perfect square of b.

Every natural number has a square, but every natural number is not a perfect square itself. Numbers that cannot be expressed as the square of another number are not perfect squares. Since there are infinite natural numbers, there are infinite numbers of perfect squares.

The numbers that have 0, 1, 4, 5, 6 or 9 in their units place maybe perfect squares where as the numbers that have 2, 3, 7 or 8 in their units place are never perfect squares. 
  1. All numbers ending with 1 or 9 have 1 in the units place of their squares. All squares having 1 in their units place are squares of the numbers ending with either 1 or 9.
  2. All numbers ending with 2 or 8 have 4 in the units place of their squares. All squares having 4 in their units place are squares of the numbers ending with either 2 or 8.
  3. All numbers ending with 3 or 7 have 9 in the units place of their squares. All squares having 9 in their units place are squares of the numbers ending with either 3 or 7.
  4. All numbers ending with 4 or 6 have 6 in the units place of their squares. All squares having 6 in their units place are squares of the numbers ending with either 4 or 6.
  5. All numbers ending with 5 have 5 in the units place of their squares. All squares having 5 in their units place are squares of the numbers ending with 5.
  6. All numbers ending with 0 have 0 in the units place of their squares. All squares having 0 in their units place are squares of the numbers ending with 0.
  7. The square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.
  8. The square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1.
For any natural number n, (n + 1)2 - n2 = (n + 1) + n.

Square of a number ending with zero(s) contains double the number of zeroes than the number. All square numbers contain an even number of zeroes.

Odd square numbers are squares of numbers ending with 1, 3, 5, 7 or 9. Even square numbers are squares of numbers ending with 0, 2, 4, 6 or 8.

The numbers whose dot patterns can be arranged as triangles are called triangular numbers. The sum of any two consecutive triangular numbers is a square number.

triangular numbers, triangle, dot patterns

For two consecutive numbers n and n+1, there are 2n non square numbers between n2 and (n+1)2.

Sum of first n odd natural numbers is n2. The square of any odd number can be written as the sum of two consecutive positive integers.

132 = 169
1032 = 10609
10032 = 1006009
100032 = 100060009
1000032 = 10000600009.

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