Squares and Square Roots
If n = m2, then n is known as the square root of m and it can be written as m = square root of n.
When a number is multiplied by itself, then the product is the square of the number.
If a natural number ‘m’ can be expressed as n2, where n is also a natural number, then m is a square number.
The square of an odd number is odd.
The square of an even number is even.
Square root of a number is denoted by the symbol √.
A perfect square has only 0, 1, 4, 5, 6, or 9 digits in its unit place.
If the digits at the unit place of a number are 2, 3, 7 or 8, then it is not a perfect square.
If a number has 1 or 9 in its unit place, then its perfect square has 1 in its unit place.
If a number has 2 or 8 in its unit place, then its square has 4 in its unit place.
If a number has 3 or 7 in its unit place, then its square has 9 in its unit place.
There are 2n non-square numbers between square of any two consecutive numbers.
The product of two consecutive even or odd natural numbers can be represented as one subtracted from square of their middle number.
A perfect square number can be expressed as a sum of successive odd natural numbers starting with one.
For any natural number m > 1, we have
(2m)2 + (m2 – 1)2 = (m2 + 1)2. So, 2m, m2 – 1 and m2 + 1 form a Pythagorean triplet.
Square root is the inverse operation of square.
Square root of a given perfect square number can be calculated with the help of prime factorisation.
When the numbers are large, the method of finding square root by prime factorisation becomes lengthy and difficult. To overcome this problem, we use long division method.