Points to Remember: Squares & Square Roots | Mathematics (Maths) Class 8 PDF Download

What’s a Perfect Square?

A natural number n is a perfect square if n = m², where m is a natural number. So, when a number is the square of another number, it’s called a perfect square!
Example:
9 = 3² (The square of 3 is 9)
25 = 5² (The square of 5 is 25)

Digits of a Perfect Square:

• If a number ends in 2, 3, 7, or 8, it can never be a perfect square.
• The squares of even numbers are even, and the squares of odd numbers are odd.
• A number ending in an odd number of zeros cannot be a perfect square.

Between Two Squares:

• Between n² and (n+1)², there are exactly 2n non-perfect square numbers!

The Pythagorean Triplet Connection:

• For any natural number n greater than 1, the numbers 2n, (n² - 1), and (n² + 1) form a Pythagorean triplet. 

Square Roots

• The square root is the opposite of squaring a number!
  Example: If you square 4, you get 16. If you take the square root of 16, you get 4.

Number of Digits in Square Numbers

If a number n has n digits, then the number of digits in its square root is:

• n/2 if n is even.
• (n+1)/2 if n is odd.

Square of a Number: 

• When you multiply a number by itself, it's called squaring that number.
  Example:
  3 × 3 = 9 (This means 9 is the square of 3!)
  5 × 5 = 25 (This means 25 is the square of 5!)

Perfect Square: 

• A number like 16, which can be expressed as 4², is called a perfect square! But remember, not every number is a perfect square. For example, 32 is not a square number. So, always check if a number is the square of another number.

Remember

All-natural numbers are not perfect squares or square numbers, 32 is not a square number. In general, if a natural number ‘m’ can be expressed as n², where n is also a natural number, then ‘m’ is the perfect square. The numbers like 1, 4, 9, 16, 25, and 36 are called square numbers. 

Table: Square of numbers from 1 and 10. 

Properties of Square Number

From the table, we conclude that:

Property 1: “The ending digits (the digits in the one’s place) of a square number is 0, 1, 4, 5, 6 or 9 only.”

Some Interesting Patterns

• Triangular numbers are: 1, 3, 6, 10, 15, 21, etc. If we combine two consecutive triangular numbers, we get a square number.
  1 + 3 = 4, ‘4’ is a square number
  3 + 6 = 9, ‘9’ is a square number
  6 + 10 = 16, ‘16’ is a square number
  and so on.

• 1² =1
  11² = 121
  111² = 12321
  1111² = 1234321
• 7² = 49
  67² = 4489
  667² = 444889
  6667² = 44448889 and so on.

Solved Examples

Sol. The unit’s digit in the square of the following is:
1. 12487 is 9 (as 7² = 49, 9 in the unit’s place).
2. 1324 is 6 (as 4² = 16, 6 in the unit’s place).
3. 91478 is 4 (as 8² = 64, 4 in the unit’s place).
4. 1251 is 1 (as 1² = 1, 1 in the unit’s place).

Sol. The square of an even number is always an even number and the square of an odd number is always an odd number. The square of an even number will always have 4, 6, or even zeros in its unit’s place. And the square of an odd number will always have 1, 5, or 9 in its unit’s place.