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A Brief Introduction.

Introducing Vedic Maths: Tricks & Mental Maths, Myths & Facts

Vedic Mathematics is often dubbed as the 'Yoga' of mathematical thoughts. This is because, its unique techniques encourage mental calculations, and provides mechanisms that aid in enhancing interest and understanding of numbers and mathematical concepts.
The most common apprehension in using the 'un-conventional tricks' of Vedic Mathematics, is that the solutions look too magical to believe that they would actually work (in all cases). More importantly, they seem to be devoid of 'mathematical logic' - whatever that means!
As a simple illustration to introduce Vedic Mathematics, let us consider squaring 99:
Since, 99 is close to (slightly lower than) a power of 10 (100, in this case), let us use Upasūtra:
यावदूनं तावदूनीकृत्य वर्गं च योजयेत्
yāvadūnam tāvadūnīkṛtya vargañca yojayet
(Lessen by the deficiency, and set up square of the deficiency) - for 992, in simple steps:
  Steps 992
1. Consider the nearest power of 10 as the Base In this case, the Base, closest to 99, is 100.
2. Find the deficiency In this case, the Deficiency =
100 - 99 = 1
3. Subtract the deficiency from that number In this case, 99 - 1 = 98
4. Set-up as many zeroes, as the digits of the number In this case, 99 being a 2-digit number, we get 9800
5. Square the Deficiency, and add to get the answer 1 being the Deficiency, and 12 = 1
9800 + 1 = 9801, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like 9962:
  Clearly, the deficiency is 4, and 42 = 16
Also, 996 - 4 = 992
And, 992,000 + 16 = 992,016
Thus, 9962 = 992,016
Of course, it is so amazingly simple that it can be done mentally, with some practice - within a few seconds. Taking it ahead, for something like 999882:
  Clearly, the deficiency is 12, and 122 = 144
Also, 99988 - 12 = 99976
And, 99976,00000 + 144 = 99976,00144
Thus, 999882 = 9,997,600,144
Now, that should be magical for somebody who is newly introduced to Vedic Mathematics.
The irony is: Most of us, knowingly or unknowingly, have been learning and using the 'tricks' of Vedic Mathematics through preparations of various competitive examinations - albeit, at a later stage, and during a race against time.
Speed does matter. Especially in today's life, when the results can be obtained faster than typing the digits in a calculator.
But, not just speed - the real power of Vedic Mathematics is in calculating the seemingly impossible calculation - that too, with ease. Let us consider the above example for a 20-digit number, say 999999999999999999982:
  All we have to do is stay calm, and have faith in Vedic Mathematics.
Clearly, the Deficiency is 2, and 22 = 4
Also, 99999999999999999998 - 2 = 99999999999999999996
(The operation is actually on the last digit)
And, 99999999999999999996, 00000000000000000000 + 4
= 99999999999999999996, 00000000000000000004
(The operation is again on the last digit, of 20 trailing zeroes)
Thus, 999999999999999999982 =
9,999,999,999,999,999,999, 600,000,000,000,000,000,004
Don't bother to check with a calculator, we don't expect it to work.
This is the reason that Vedic Maths, more often than not, is promoted as an alternate mathematical methodology which is 'superior' to the conventional methods of calculations, or tricks, and even mental maths for 'fastest' mathematical calculations that brings out the 'human calculator' within oneself!
It is a fact that steps of conventional methods are too tedious to be performed mentally, which is why Vedic Mathematics is the source of thousands of 'Mental Maths' books/tutorials/videos doing the rounds - not to mention the innumerable 'tricks', 'short-cuts' and 'secrets' that educational 'gurus' demonstrate to provide their students an edge in competitive examinations, all over the world.
And considering the fact, that the above example is based (only) on an Upasūtra - not even a Sūtra - should provide a fair idea of what Vedic Mathematics is about.
But, why does it work? For the above Upasūtra, let us consider the following:
  Assuming N is a number close (and less) to a power of 10, then N = a - b
'a' being the power of 10, and 'b' being the Deficiency
As in examples above, 99=100-1, 996=1000-4, and 99988=100000-12
Now, N2 = (a - b)2
= a2 - 2ab + b2
= a (a - 2b) + b2
But, N = a - b. So, substituting a = N + b
N2 = a ([N + b] - 2b) + b2
= a (N - b) + b2
This is exactly what this Upasūtra makes us do:
992 = [100 × (99 - 1)] + 12 = 9800 + 1 = 9801
9962 = [1000 × (996 - 4)] + 42 = 992000 + 16 = 992,016
999882 = [100000 × (99988 - 12)] + 122 = 99976,00000 + 144 = 997,600,144
Needless to state that the principles of Vedic Mathematics work because they are mathematical.
The above Upasūtra, being mathematical, would work even for numbers not close to the Base, but that will not be very smart - especially when there are other methods (like Upasūtra: dvaṅdvayoga - discussed here ») to do the same. For example, using this Upasūtra for 772 would require us to find 232 (23 being the Deficiency). That would be foolishness, and not failure of Vedic Mathematics.
Obviously, further explanations and details (including the number being close to, but greater than the Base) are presented in the respective Upasūtra: yāvadūnam tāvadūnīkṛtya vargañca yojayet (discussed here »), but the point is: Vedic Maths provide no magical trick, and is certainly not devoid of 'mathematical logic'. Fact is, it is practical step ahead of the 'Algebraic Rules' - that we are comfortable with.
The standpoint typical to Vedic Scholars is that the Vedic Civilization was aware of the Algebraic Rules, and generalized them for practical usage with easy-to-remember phrases (read Sūtras). But, we don't want to get into that, and just stick to the mathematical structure.
Having stated the above, irrespective of everything, Vedic Maths can indeed lead to amazingly fast calculations, with some practice, for the mathematically inclined. There is no denying that the principles of Vedic Maths provide methods to convert cumbersome and complicated calculations into simple steps, which can be mentally executed to reach the solution at ease. However, Most of our above standpoints are opposed to hardcore 'Vedic Mathematics' (and even Hindutva) fanatics. Nevertheless, this web endeavour should speak volumes of our love, appreciation and respect for Vedic Mathematics and its amazing principles.
All said and done, there is no denying that Vedic Mathematics is far more than 'wise shortcuts'. It provides techniques that encourage us to execute the computations mentally - thereby stretching and excercising our capability to deal with mathematical thoughts.
The techniques improve our overall mathematical capability, by clarifying the positional notation (as in śūddha, discussed here »), by making us visualize the numbers to mentally execute calculations (as in ūrdhva tiryagbhyāṃ, discussed here »), by helping us to think closely with the number system (as in ānurūpyeṇa, discussed here »), and by forcing us to anticipate consecutive steps (as in dhvajāṅka, discussed here ») - to list a few positive influences.
Vedic Mathematics is truly the 'Yoga' of mathematical thoughts, and its use can help overcome 'Maths Phobia' by making one 'play' with numbers and mathematical concepts.
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