Vedic Math - Squaring of numbers near '50'

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In this article, we pick another special case of squaring i.e. squaring numbers which are near 50. It can have two cases, which are:
  • Case 1: Numbers greater than 50.
  • Case 2: Numbers lesser than 50
In both the cases, we need to take 50 as the base value. First let us take 'Case 1' i.e. 'Numbers greater than 50'.

Case 1 -
Take an example: Say, 542=2916 
Consider this answer in two parts: 29 (first part) and 16(second part). Now let us study, how Vedic Math can help us to achieve this answers or both of these parts.

As we are taking '50' as base, so the number presentation will be like 50 + 4. So the first part is 50, and second part is 4.

For the first part of the answer:
  1. Pick the first part i.e. 50.
  2. Pick the first digit i.e. 5
  3. Square this digit i.e. 52 > 25
  4. Add 4(second part) to it
  5. And we get our first part of the answer i.e. 29 (25 + 4).
For the second part of answer, following are the steps:
  1. Pick second part i.e. 4
  2. Square this digit i.e. 42 > 16
  3. And we get our second part of the answer i.e. 16
So the answer is 2916

Let us understand it with another example to make it more clear. Say, 612=3721
As we are taking '50' as base, so the number presentation will be like 50 + 11. So the first part is 50, and second part is 11.

For the first part of the answer:
  1. Pick the first part i.e. 50.
  2. Pick the first digit i.e. 5
  3. Square this digit i.e. 52 > 25
  4. Add 11(second part) to it
  5. And we get our first part i.e. 36 (25 + 11).
For the second part of answer, following are the steps:
  1. Pick second part i.e. 11
  2. Square this digit i.e. 112 > 121
  3. Now the result is of three digit. So the first digit (1) will be added to the first part i.e. 1 + 36 = 37
  4. So first part becomes 37 now
  5. And second part of answer will be 21 
So the answer is 3721. Refer to image below for visual representation.



Now, let us study 'Case 2' i.e. Numbers lesser than 50.

Case 2 -
Take an example: Say, 482=2304
As we are taking '50' as base, so the number presentation will be like 50 - 2. So the first part is 50, and second part is (-2). For numbers below 50, we take the deficiency from 50 (2 in this case), to get the number (48 in this case); and use the square of the deficiency (22= 4 in this case) for calculation.

For the first part of the answer:
  1. Pick the first part i.e. 50.
  2. Pick the first digit i.e. 5
  3. Square this digit i.e. 52 > 25
  4. Add (-2) (i.e. second part) to'25'
  5. And we get our first part i.e. 23 (25 + (-2)).
For the second part of answer, following are the steps:
  1. Pick second part i.e. (-2)
  2. Square this digit i.e. (-2)2  =  4
  3. And second part of answer will be 04
So the answer is 2304. Refer to image below for visual representation.


In the second case, we explore the sub-sutra "Whatever the deficiency lessen by that amount and set up the square of the deficiency"

The Algebra behind this method is:
  • (50 + a)2 = 100 (25 + a) + a2   (if number is above 50)
  • (50 - a)2 = 100 (25 - a) + a2    (if number is below 50)

Here are few exercises for your practice:

432 = ?

492 = ?

562 = ?

522 = ?

622 = ?


Try it. I hope by now you would have understood the method. Even then if you have any difficulty, post your doubts here. Enjoy!!

Please do share your views that would be having great value for us and will encourage us.

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1 comments:

Rohtash Singh said...

When you are getting the square of any number (2 digit number only), here is an easy formula:

suppose you are getting the square of 54:
Steps: -
1.) Multiple the last number with itself. i.e 4 X 4 = 16
2.) Pick the last digit of result. i.e. 6 and remeber carray i.e. 1.
3.) Now multiply each digits with next digit and then multiply every by 2. i.e. 5 X 4 X 2 = 40.
4.) Add last carry in the result. i.e 40 + 1 = 41.
5.) Now repeat step 2.
6.) Now multiple the first digits with itself. i.e. 5 X 5 = 25 and add carry in it.i.e 25 + 4 = 29.
7.) Get together all last digits and at the last the full number.
i.e. 2916.

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