# Squaring a 3 digit no mentally BY MR. VEERARAGAVAN EX. RBI OFFICER

**Squaring a 3 Digit Number mentally**

Can you work out the square of a 3 digit no mentally?

Yes if you can add two 3 digit numbers mentally and multiply the result by a number from 1 to 5 and add the product mentally to a 5 or 6 digit number.

The following steps will show how?

**First, let us find the square of a 2 digit number.**

Consider the no 73

73^{2 } = 70^{2} + 3 (70 + 73)

= 4900 + 429

= 5329

**The same can also be worked by the formula**

73^{2 } = 80^{2} – 7 (73 +80)

= 6400 – 171

= 6329

But we would prefer to use the first form as the multiplicand is 3 for the second terms while it is 7 in the second calculation.

In both formulae, for a given no N we choose a convenient no which is a nearest multiple of 10, say m.

Then ** N ^{2 } = m^{2} + (N –m ) (N + m)**

By taking m close to N so that N-m is from 1 to 5, the square can be worked out mentally and easily.

**Note:**

**When m is smaller than N, (N-m) is positive.**

**When m is greater than N, (N–m) is negative.**

**We select m so that the numerical value of N-m is one of the numbers from 1 to 5.**

It is easy to find the square of m as its unit digit is 0 and 10s digit will be from 1 to 9.

**Examples:**

49^{2} = 50^{2} – 1X (50 +49) = 2401

It is quite easy to do this mentally because 5^{2} is 25. By adding two 0’s and subtracting 99 mentally we get the answer

86^{2 } = 90^{2 }– 4 X (90 +86) = 8100 – 704 = 7396

Let us try a 3 digit no between 100 and 200, say 163^{2}

Applying the above formula we get

163^{2} = 160^{2} + 3 (160 + 163) = 25600 + 969 = 26569

We can easily remember squares of numbers upto 30. By practice we can mentally add two 3 digit numbers and multiply the sum by numbers upto 5 and then add the product mentally to a 5 digit or 6 digit number.

**For numbers above 200 we can use the following approach**

439^{2 } = 440^{2} – 1 × (440 + 439) = (44^{2}) × 100 + 879

= (40^{2} + 4 × 84) × 100 + 879

= 193600 – 879

= 192721

993^{2} = 990^{2} + 3 × 1983

** = **980100 + 5949

= 986049