Revision Notes on Squares and Square Roots

Square Number

Any natural number ‘p’ which can be represented as y², where y is a natural number, then ‘p’ is called a Square Number.

Example

4 = 2²

9 = 3²

16 = 4²

Where 2, 3, 4 are the natural numbers and 4, 9, 16 are the respective square numbers.

Such types of numbers are also known as Perfect Squares.

Some of the Square Numbers

Properties of Square Numbers

• We can see that the square numbers are ending with 0, 1, 4, 5, 6 or 9 only. None of the square number is ending with 2, 3, 7 or 8.

• Any number having 1 or 9 in its one’s place will always have a square ending with 1.

• Any number which has 4 or 6 in its unit’s place, its square will always end with 6.

• Any number which has 0 in its unit’s place, its square will always have an even number of zeros at the end.

Some More Interesting Patterns

1. Adding Triangular Numbers

If we could arrange the dotted pattern of the numbers in a triangular form then these numbers are called Triangular Numbers. If we add two consecutive triangular numbers then we can get the square number.

2. Numbers between Square Numbers

If we take two consecutive numbers n and n + 1, then there will be (2n) non-perfect square numbers between their squares numbers.

Example

Let’s take n = 5 and 5² = 25

n + 1 = 5 + 1 = 6 and 6² = 36

2n = 2(5) = 10

There must be 10 numbers between 25 and 36.

The numbers are 26, 27, 28, 29, 30, 31, 32, 33, 34, 35.

3. Adding Odd Numbers

Sum of first n natural odd numbers is n².

Any square number must be the sum of consecutive odd numbers starting from 1.

And if any natural number which is not a sum of successive odd natural numbers starting with 1, then it will not be a perfect square.

4. A Sum of Consecutive Natural Numbers

Every square number is the summation of two consecutive positive natural numbers.

If we are finding the square of n the to find the two consecutive natural numbers we can use the formula

Example

5² = 25

12 + 13 = 25

Likewise, you can check for other numbers like

11² = 121 = 60 + 61

5. The Product of Two Consecutive Even or Odd Natural Numbers

If we have two consecutive odd or even numbers (a + 1) and (a -1) then their product will be (a²- 1)

Example

Let take two consecutive odd numbers 21 and 23.

21 × 23 = (20 - 1) × (20 + 1) = 20² - 1

6. Some More Interesting Patterns about Square Numbers

Finding the Square of a Number

To find the square of any number we needed to divide the number into two parts then we can solve it easily.

If number is ‘x’ then x = (p + q) and x² = (p + q)²

You can also use the formula (p + q)² = p² + 2pq + q²

Example

Find the square of 53.

Solution:

Divide the number in two parts.

53 = 50 + 3

53² = (50 + 3)²

= (50 + 3) (50 + 3)

= 50(50 + 3) +3(50 + 3)

= 2500 + 150 + 150 + 9

= 2809

1. Other pattern for the number ending with 5

For numbers ending with 5 we can use the pattern

(a5)² = a × (a + 1)100 + 25

Example

25² = 625 = (2 × 3) 100 + 25

45² = 2025 = (4 × 5) 100 + 25

95² = 9025 = (9 × 10) 100 + 25

125² = 15625 = (12 × 13) 100 + 25

2. Pythagorean Triplets

If the sum of two square numbers is also a square number, then these three numbers form a Pythagorean triplet.

For any natural number p >1, we have (2p) ² + (p² -1)² = (p² + 1)². So, 2p, p²-1 and p²+1 forms a Pythagorean triplet.

Example

Write a Pythagorean triplet having 22 as one its member.

Solution:

Let 2p = 6

P = 3

p² + 1 = 10

p² - 1 = 8.

Thus, the Pythagorean triplet is 6, 8 and 10.

6² + 8² = 10²

36 + 64 = 100

Square Roots

The square root is the inverse operation of squaring. To find the number with the given square is called the Square Root.

 2² = 4, so the square root of 4 is 2

10² = 100, therefore square root of 100 is 10

There are two square roots of any number. One is positive and other is negative.

The square root of 100 could be 10 or -10.

Symbol of Positive Square Root

Finding Square Root

1. Through Repeated Subtraction

As we know that every square number is the sum of consecutive odd natural numbers starting from 1, so we can find the square root by doing opposite because root is the inverse of the square.

We need to subtract the odd natural numbers starting from 1 from the given square number until the remainder is zero to get its square root.

The number of steps will be the square root of that square number.

Example

Calculate the square root of 64 by repeated addition.

Solution: