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Vedic Mathematics is based on 16 sutras and 13 sutras.Many people doubt how we can apply these sutras to all branches of Mathematics.
The same sutra can be applied in various areas and suits in different branches of Mathematics. And most importantly these sutras are related to philosophy also.The Vedic maths sutras to be considered in a holistic way
Let us take the case of the sutra : व्यष्टि समष्टि Vyasti-Samasti Individuality and Totality Or Specific and General.
In Chapter 19 of Vedic Mathematics by Swami Bharthi Krishna Theertha ji uses the sutra to solve a special case of Biquadratic equations.
in which two binomials on the left are both raised to the 4th power and there stands a single number on the right. The average of the two binomials is used to break the equation down into a simpler one. (x + 7) ^4 + (x + 5)^ 4 = 706
Let x + 6 = a ; The average of the two binomials
(a +1)^ 4 + (a −1) ^4 = 706 owing to the cancellation of Odd powers X^3 and x
2a^ 4 +12a ^2 + 2 = 706
From which, a = ±4 or ± √22
x = -2 or -10 Or ± √22 - 6
The sutra can be applied in calculus concerning the derivatives of combined functions, Conventionally this is taught and practised using the chain rule, which is an application of the Transpose and Adjust sutra. When students have understood and practised the chain rule they usually repeatedly use the pedantic method as if proving it every time they want to use it. But the sutra can be used to obtain the answer in one step.
Another application of the sutra occurs when a single entity gets referred to the whole, in order to solve a problem.
Here is an example. Suppose Wimbledon Men’s Singles tennis competition has 128 players. It is a knockout competition. How many matches are there in the whole competition from the first round through to the final? One method is to figure that there are 64 matches in the first round, 32 in the second round and so on until one match in the final. The sum of each round will give the answer. But the sutra can be employed to provide a more intuitive answer. The argument goes like this: each match has a loser. The winner of the competition is the only one who has not lost and so there are 127 losers and hence 127 matches.
The concept within Vyashti Samashti lies at the heart of the whole realm of probability and statistics. Much of statistical theory is based on the issue of providing a simple representation of a large amount of data. That representation can take the form of a mean, a regression line, a range, a distribution, and so on. It never gives a complete picture but shows some aspect of the whole. The normal distribution curve is intriguing because it seems to connect each individual to the whole.
We can easily refer the sutra to Vedanta philosophy of Advaitha: Individuality and Totality also.
Ref:Vyashti Samashti – A sutra from Shankaracarya Bharati Krishna Tirtha/ James Glover