Finding the square of a number

Finding the square of a number is a straightforward procedure. To find the square number, we must multiply the supplied integer by itself. A integer raised to the power of two is always used to represent the square term. The square of 7 is 7 multiplied by 7, for example, 7 × 7 = 7² = 49

Square of the single digit number:-

We may calculate the square of single-digit values by multiplying them by themselves. We may also easily get the square of a number by learning the tables from 1 to 10 ,i.e., as follows;

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

Finding square by splitting the number:-

We’re going to stretch the square to discover the square of a natural number. Let’s start with some examples:

Find the square of 26 .

We would expand 26 as (20 + 6) or (30 – 4) then would easily find the square using the formula

     (p + q)² = p² + 2.p.q + q²

     (p – q)² = p² – 2.p.q + q²

So,

    26² = (20 + 6)²

           = (20² + 2.20.6 + 6²)

           = (400 + 240 + 36)

           = 576

Or,

   26² = (30 – 4)²

          = (30² – 2.30.4 + 4²)

          = (900 – 240 + 16)

          = (916 – 240)

          = 576

Finding the Square Using Patterns:-

We may notice some patterns while squaring the numbers that will aid us in remembering the squares. Let’s look at various patterns:

25² = 625 = 600 + 25 = 6 x 100 + 25 = (2 × 3) hundreds + 25

35² = 1225 = 1200 + 25 = 12 x 100 + 25 = (3 × 4) hundreds + 25

75² = 5625 = 5600 + 25 = 56 x 100 + 25 = (7 × 8) hundreds + 25

125² = 15625 = 15600 + 25 = 156 x 100 + 25 = (12 × 13) hundreds + 25

We can observe from the pattern above that all the squared integers have a 5 at their unit’s location. Let’s say m5 is a squared number. Thus, we may write the generalized expression by looking at the above patterns.

(m5)² = (10m + 5)²

= 10m(10m + 5) + 5(10m + 5)

= 100m² + 50m + 50m + 25

= 100m(m + 1) + 25

= m(m + 1) hundreds + 25

As a result, the shortcut for finding the square of numbers with 5 as their unit is:

(m5)² = m(m + 1) hundreds + 25

Illustration:

Calculate the square of 85.

Here the unit’s place is 5.

And, m = 8

As a result, we can find the square of a number with 5 at unit place using the aforementioned pattern:

(m5)² = m(m + 1) hundreds + 25

m = 9 in this case.

(85)² = 8(8 + 1) × 100 + 25

          = 72 × 100 + 25

          = 7200 + 25

          = 7225

As a result, the square of 85 equals 7225.

Finding square of number using Pythagorean Triplets form:

While studying right triangles, we learned that the Pythagoras theorem allows us to find the length of any side of a triangle given the length of the other two sides.

The hypotenuse, perpendicular, and base are the three sides of a right triangle. According to Pythagoras’ theorem,

Perpendicular² + Base² = Hypotenuse²

Assume the sides’ lengths are as follows:

Perpendicular = 3

Base = 4

Hypotenuse = 5

If we do the math, the square of the hypotenuse equals the sum of the squares of the perpendicular and the base.

                   5² = 3² + 4²

             Or,25 = 9 + 16

             Or,25 = 25

As a result, we can deduce that the numbers 3, 4, and 5 are Pythagorean triplets.

One of the other examples of Pythagorean Triplet is 6 , 8 and 10.

In general, we can define the Pythagorean Triplet as follows;

Suppose ‘n’ is a natural number.

So the Pythagorean Triplet can be described as;

(2n)² + (n² – 1)² = (n² + 1)²

That means any three numbers in the above form is a Pythagorean Triplet.

Conclusion:-

A perfect square, also known as a square number, is an integer that is the square of another integer; in other words, it is the product of two integers. 16 is a square number, for example, because it equals 3² and may be expressed as  × 4.