It is important to understand that (theoritically) the Vedic Mathematical Sūtras and Upasūtras were developed to mentally execute cumbrous calculations - for practical usage. Thus, they mostly deal with special cases (like a number ending with 5, or a number close to a power of 10) based on the demand of usage in the time-period that they were developed, and for practical need for measurements to aid architectural constructions, and trading.
This is also a reason that some of the Upasūtras are far more powerful and generic - as mathematical formulae, than some of the Sūtras. Nevertheless, the generic methods were overshadowed as corollaries, by specific rules - which are case dependent, but in tune with the need of the time-period that they were developed in.
This section explains the application of Vedic Mathematics in the context of current mathematical expectations. Accordingly, all rules within the Vedic Mathematical structure is considered under a common umbrella of 'Principles of Vedic Mathematics' - irrespective of them being a Sūtra or an Upasūtra.
Our unbiased observations in applying the principles Vedic Mathematics are as follows:
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The principles that overshadow and outshine conventional methods, and even the rest of Vedic Mathematics, are:
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śūddha (Purification)
(discussed here »)
It is an Upasūtra for addition (x + y) and subtraction (x - y) of two numbers. Although, it does not provide a marked advantage over conventional methods, it instils a good practice that reduces mistakes. However, in combination with Rekhanks, it provides a powerful method to simultaneously execute addition and subtraction of more than two numbers.
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ūrdhva tiryagbhyāṃ (Vertically and crosswise)
(discussed here »)
It is a Sūtra for multiplication (x × y) of two numbers, that enables mental execution of multiplication of any (n-digit) number with another (m-digit) number. The unique technique that that this Sūtra implements, helps is clarifying the very concept of the number system.
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dhvajāṅka (On the flag)
(discussed here »)
It is an Upasūtra that is considered the 'crown jewel' of division (x ÷ y) of two numbers, and often dubbed as the 'ūrdhva tiryagbhyāṃ' of divisions. The technique implemented by this Upasūtra drastically reduces the complexity of a division, and can work to generate decimal figures in a single operation.
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dvaṅdvayoga (Sum of pairs)
(discussed here »)
It is an Upasūtra that provides different techniques for square (x2), cube (x3), square root (√x) and cube root (∛x) of a number. These techniques are so effective, that the 'Duplex Methods' have earned mentions in formal mathematics.
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The above principles are truly generic, as they are not case-dependent. Most practitioners of Vedic Mathematics rely heavily on the afore-mentioned principles, especially because they serve as fail-safe techniques when any other case-dependent technique get cumbersome.
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The principles that dazzle with numbers close to a power of ten (10n) are:
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nikhilaṃ navataścaramaṃ daśataḥ (All from 9, and the last from 10)
(discussed here »)
It is a Sūtra for subtracting (x - y) a number from any power of ten (10n). Not only is this a very effective method in practical mathematics, this Sūtra lends its value for many other methods (including Vedic Mathematical methods). Furthermore, it provides a technique for multiplication (x × y) that is uniquely effective for numbers close to a power of ten (10n).
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yāvadūnāṃ (By the deficiency)
(discussed here »)
It is a Sūtra for multiplication (x × y) of two numbers, both close to a power of ten (10n). However, more often than not, it is referred as the 'Yāvadūnāṃ Trio', along with its two powerful Upasūtras (corollaries):
yāvadūnaṃ tāvadūnaṃ (discussed here »)
yāvadūnam tāvadūnīkṛtya vargañca yojayet (discussed here »)
The Upasūtras provide techniques for cube (x3) and square (x2) of a number close to a power of ten (10n), respectively.
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Real-life mathematics deal with umpteen situations which are close to a power of 10. This is the reason that, despite the fact that these principles are case-dependent, they are popularly used by practitioners of Vedic Mathematics - especially because of their capability to compute figures easier than the generic principles.
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The principles that amaze with mathematical concepts, and work wonders when used in conjunction with other principles are:
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ānurūpyeṇa (Proportionately)
(discussed here »)
It is an Upasūtra for multiplication (x × y) of two numbers, that implements the multiplication technique of the Sūtra: nikhilaṃ navataścaramaṃ daśataḥ, for bases other than 10n. However, its very interesting conceptual method finds application in numerous methods (conventional and Vedic).
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parāvartya yojayet (Transpose and adjust)
(discussed here »)
It is a Sūtra for division (x ÷ y) of two numbers, that implement an interesting technique by using a 'Transposition' of the Divisor. This technique drastically reduces the complexity of a division, and reduces it to a few multiplications and additions.
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vilokanaṃ (By mere observation)
(discussed here »)
It is an Upasūtra that is perhaps the most misunderstood concept amongst all the principles. More often than not, we tend to mechanically execute methods (whether conventional or vedic) when an investigation may help us finding a solution to a given mathematical problem. It does not promote guesswork, but encourages oneself to take an neutral look at the problem to fit patterns - before commencing with the relevant methods. It is a very powerful concept when actively practised.
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The above-mentioned principles open vast opportunities to deal with numbers, equations, and other mathematical concepts - that every practitioner of Vedic Mathematics tend to exploit to the fullest. Apart from the advantages, these principles encourage mathematical thought-process amongst practitioners and learners.
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In order to maximize the benefit of using Vedic Mathematics, one must have the knowledge of 'when' and 'how' to use the principles of Vedic Mathematics - which forms the basis of confusion, amongst most practitioners of Vedic Maths - during the early phase.
We intend to keep building on this list because the principles of Vedic Mathematics are so generic that they can be applied in many spheres of mathematics, and not limited to basic algebraic operations. We request our readers to keep an eye on this page, as we hope to include videos for more helpful tutorials - obviously, for free!
