1. Perfect Square:
A natural number x is a perfect square if there exists a natural number y such that x=y2. In other words, a natural number x is a perfect square, if it is equal to the product of a number with itself.
2. Properties of Squares Numbers:
(i) A number ending in 2, 3, 7, or 8 is never a perfect square.
(ii) The number of zeroes in the end of a perfect square is never odd. So, a number ending in an odd number of zeroes is never a perfect square.
(iii) Squares of even numbers are always even.
(iv) Squares of odd numbers are always odd.
3. General Properties of Perfect Squares:
(i) For any natural number n, we have n2= (Sum of first n odd natural numbers)
(ii) The square of a natural number, other than 1, is either a multiple of 3 or exceeds a multiple of 3 by 1 .
(iii) The square of a natural number, other than 1, is either a multiple of 4 or exceeds a multiple of 4 by 1.
(iv) There are no natural numbers p and q such that p2=2q2
4. Pythagorean Triplets:
For any natural number n greater than 1, (2n, n2−1, n2+1), is a Pythagorean triplet.
5. Square roots:
The square root of a given natural number n is that natural number which when multiplied by itself gives n as the product and we denote the square root of n by n. Thus, n=m⇔n=m2.
6. Finding Square Roots:
(i) In order to find the square root of a perfect square, resolve it into prime factors; make pairs of similar factors and take the product of prime factors, choosing one out of every pair.
(ii) For finding the square root of a decimal fraction, make the even number of decimal places by affixing a zero, if necessary; mark off periods and extract the square root; putting the decimal point in the square root as soon as the integral part is exhausted.
7. Properties of Square Roots:
For positive numbers a and b, we have
(i) ab=a×b
(ii) ab =ab