How To Mentally Square Any Number Ending In 5?

Let us say that you are required to mentally square any number ending in 5. It could be that such a request comes up in a job interview. Or it could be that the quick mental calculation would benefit you and your meeting panel whilst assessing the viability of a decision. Such situations come up in life now and then. So, just maybe, such a skill could come in handy.

I admit that this is a tough sell. You don’t need this in your life. But you might. It is a ‘nice-to-have’ skill. If I’m being honest with you, I find the mathematical aspect of such a task fascinating. That has been my motivation behind exploring it. So, if this essay doesn’t end up enriching you with a handy new skill, I hope that it at least instills curiosity about an intriguing ancient Indian approach to mathematics.

In this essay, I will first demonstrate the mental algorithm behind squaring any number ending in 5. Following this, I will explore the reason why this works using an algebraic proof of sorts. So, let’s get right into it.

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The Mental Algorithm to Square Any Number Ending in 5

To execute this mental algorithm, you need just two pre-requisite skills:

1. The knowledge that 5² = 25, and

2. The ability to mentally multiply two numbers.

The Indian Vedic ‘Sutra’ gives the following algorithm to compute the square of any number ending in 5:

1. Decompose the number (ending in 5) into the last digit (5) and the digits preceding the last digit.

2. Consider the digits preceding the last digit as a number — n. Now compute n*(n+1).

3. The final answer consists of 2 parts: the left-hand side (LHS) and the right-hand side (RHS). The RHS will always be 25 (considering 5² = 25). The LHS is the result of n*(n+1).

4. Write the result as LHSRHS to arrive at the final answer.

If you find the above verbal breakdown of the algorithm confusing, it is perhaps easier to look at examples.

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Worked Out Examples

Let us say that you are required to compute 65². You can mentally apply the algorithm as follows (with the visual illustration below the steps):

1. For 65, 5 is the last digit and 6 is the number from the digit(s) preceding 5 (the last digit).

2. Let us consider that n = 6. We can now compute n*(n+1) as 6*7 = 42.

3. LHS = 42. We know that RHS = 5² = 25.

4. The final result is given by LHSRHS = 4225.

Let us work out one more example. What is 205²? (the question from the title image)

1. For 205, 5 is the last digit and 20 is the number from the digits preceding 5 (the last digit).

2. Let us consider that n = 20. We can now compute n*(n+1) as 20*21 = 420.

3. LHS = 420. We know that RHS = 5² = 25.

4. The final result is given by LHSRHS = 42025.

Now that we have covered a couple of illustrated examples, let us turn our attention to a more interesting question: Why does this algorithm work?

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The Mathematics Behind the Vedic Algorithm to Square Any Number Ending in 5

We can understand the mechanics of the Vedic algorithm using algebra.

Consider the following algebraic result:

In the case of numbers ending in 5, b is necessarily 5 and x is necessarily 10. Consider 95 as an example. It can be expressed as follows:

95 = 90 + 5 = 9*(10) + 5 (where a = 9, x = 10, and b = 5)

Consequently, our original algebraic result can be resolved as follows:

Thus, using algebra, we clearly see why the Vedic algorithm works. It just takes advantage of some convenient decimal properties of numbers ending in 5.

Final Thoughts

The algorithm that I have demonstrated here is quite simple. The ability to mentally square any number ending in 5 is not a life-saving skill either. However, what I truly find fascinating is the ancient Indian (Vedic) approach to mathematics.

I am linking a reference document to Vedic mathematics and methods at the end of this article. If you are interested, do check it out! It features a vast array of intriguing mathematical methods and tricks just like the one that I covered in this essay.

If I stumble upon any other interesting and insightful Vedic math method or concept, I will be sure to cover it in a future essay.

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Reference: Vedic Mathematics.

Credit: Presh Talwalkar.

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Further reading that might interest you: How To Multiply Any Number By 11 In The Head? and Why Do You See Mirrors Flipping Words?