• The unit digit of a perfect square can be only 0, 1, 4, 5, 6 or 9.
• The square of a number having:
1 or 9 at the units place ends in1.
2 or 8 at the units place ends in 4.
3 or 7 at the units place ends in 9.
4 or 6 at the units place ends in 6.
5 at the units place ends in 5.
• There are 2n natural numbers between the squares of numbers n
• A number ending in odd numbers of zeroes is not a perfect square.
• The sum of first n odd natural numbers is given by n2.
• Three natural numbers a, b, c are said to form a pythagorean
triplet if a2 + b2 = c2.
• For every natural number m > 1, 2m, m2–1 and m2 + 1 form a
• The square root of a number x is the number whose square is x.
Positive square root of a number x is denoted by
• The cube root of a number x is the number whose cube is x. It is
• Square root and cube root are the inverse operations of squares
and cubes respectively.
• If a perfect square is of n digits, then its square root will have
⎛ n +1⎞
digit if n is even or ⎜
⎟ digit if n is odd.
⎝ 2 ⎠
• Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9
end with digits 0, 1, 4, 5, 6 and 9 respectively.
Describe what is meant by a perfect square. Give an example.
Explain how many square roots a positive number can have. How are
these square roots different?