When one talks about the History of Mathematics,one must also remember to talk about Numbers and their origin. Numbers, in general ,have been classified into three types.

Worshippers of Lord Siva recite Rudram with 11 sections followed by Chamakam with 11 sections as a routine prayer every day. This is called the daily nyasam or mode of worship. In the Rudram part, the devotee pays repeated obeisance to Lord Siva and prays for his blessings for human well being. But on special occasions, the number of times the recitation is done is increased.

In Rudra Ekadasi, Rudram is recited 11 times and Chamakam is recited once. After Rudram is recited once, one section or anuvaka ofChamakam is recited in order.

In Atirudram, 11 Maharudrams are recited; that is, Rudram is recited 11⁴ = 14641 times and Chamakam is recited 11³ = 1331 times.

In the Chamakam, in anuvakas or sections 1 to10, the devotee prays for almost everything needed for human happiness and specifies each item.  But in the 11th anuvaka or 11^th section of Chamakam, the devotee prays for the desired things not specifically but in terms of numbers, first in terms of odd numbers from 1 to 33 and later in multiples of 4 from 4 to 48, as follows:

“Let these be granted to me. One, three, five, seven, nine, eleven, thirteen, seventeen, nineteen, twenty one, twenty three, twenty five, twenty seven, twenty nine, thirty one and thirty three as also four, eight, twelve, sixteen, twenty, twenty four, twenty eight, thirty two, thirty six, forty, forty four and forty eight”.

The Vedas are not mathematical texts; they are merely hymns to the Vedic gods. However, the word Veda means “knowledge”, and when analyzed closely it actually contains many mathematical references, especially in the section on Jyotisa, or “the constellations”. Unfortunately, almost all of these references are implied, so much of the interpretation is largely guesswork. Another reason that the Vedas are hard to interpret is that because it was an oral document, there are no symbols for numbers or operations — only words. It is highly likely, however, that they did use symbols, because without them math becomes very tedious. For example, consider doing a multiplication problem using “four thousand six hundred and thirty-seven times two hundred and eighty-eight.” You would most likely convert the words to symbols, do the math on a piece of paper, and then probably only take the time to convert the answer back into words…..And yet another oddity of the Sanskrit language involves what happens with compound numbers, numbers with more than one “digit” (like “thirty-four”). In normal Sanskrit, compound words (like “servant of the king”) came from left to right in order of prevalence (so our example would be “king-servant”; “servant-king” would mean a servant who was treated well). However, compound numbers are written the opposite way, with the higher digits on the right. (Our number 529 would be written “nine-two-five”.) It is always like this, and there is even a rule included: ankanam vamato gatih, which literally means “the understanding of the numbers in the reverse way…

A good example of a story from which we can extract mathematics is one about a man named Manu.³ Manu had ten wives, who had one, two, three, four, etc. sons each (the first wife had one son, the second wife two, etc.). The one son allied with the nine sons, and the two sons allied with the eight, and so on until the five sons were left by themselves. They asked Manu for help, and so he gave them each a samidh or “oblation-stick”. The five sons then used these sticks to defeat all of the other sons.

On the surface, this is just a silly fable, but it shows several things about Vedic mathematics. Because the ten sons did not ally with anyone, and the nine did with the one, eight with two, and so on, the mathematicians must have been thinking that nine plus one, and eight plus two equal ten. This obviously shows that they practiced addition, and it also implies that they used a base 10, or decimal, system. For the second part of the story, the authors probably added the tens up to find that there were 50 allied sons. When the five remaining sons asked their father for help, it is likely that he gave them just enough mathematical power to defeat the others. This would mean that each stick equaled the strength of 10 men, for a total of 50. With the five sons added to that, they were able to defeat the 50. (Or maybe the father gave them 50 men worth of sticks, thinking it would be an equal battle, but not realizing that the five sons would throw off the balance.) But this 50 business implies both multiplication and division as well, because there were five groups of ten sons allied, or five times ten. Then, when the father went to decide the power of the sticks, he would have had to divide that 50 by five, to split the power equally among the five sons…even further, the story can be shown to symbolize the idea of positional notation — the idea of place values in numerals. (For an example of positional notation, 218 is the same as 200 + 10 + 8, or 2 x 102 plus 1 x 101 plus 8 x 100. In summary, the order of numerals tells how big the numbers are.) The “oblation-sticks” are obviously thought of as very powerful, just as 10 might be thought of as more “powerful” than a lowly 1. So when the 5 “lowly” sons were “added” to the 5 “powerful” sticks, this could have symbolized the 50 and 5 making 55, which is a bigger number (and therefore more powerful) than 50. This view of things also gives further evidence to the fact that they used base 10.

So this simple story shows examples of addition, multiplication, division, base 10, and even positional notation. The Vedas are full of these stories, and many more examples are given throughout of all these concepts, along with subtraction, fractions, and squares.5 There are even instances of arithmetic and geometric sequences, which are series of numbers that increase by adding or multiplying a certain number (arithmetic sequence: 2 (+3 =) 5, 8, 11, 14, … ; geometric sequence: 2 ( x 3 =) 6, 18, 54, … Source. Zimmerman, Francis. “Lilavati, Gracious Lady of Arithmetic.” UNESCO Courier. Nov, 1989, p 20. [bibliography entry]
2 Pandit, M. D. Mathematics as Known to the Vedic Samhitas. Dehli, India: Sri Satguru Publications, 1993, p 153. [bibliography entry]

3 From Maitrayani Samhita (part of the Vedas), section 1.5.8, trans. M. D. Pandit. [bibliography entry]

4 Most of the ideas in this paragraph come from Pandit, p 102.

5 See Pandit for more details

A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The most commonly used system of numerals is decimal. Indian mathematicians are credited with developing the integer version, the Hindu–Arabic numeral system.The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the number seven would be represented by ///////….The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + − − /// without any need for zero. This is called sign-value notation. More elegant is a positional system, also known as place-value notation. Again working in base 10, ten different digits 0, …, 9 are used and the position of a digit is used to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1 or more precisely 3×102 + 0×101 + 4×100. Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to “skip” a power. The Hindu–Arabic numeral system, which originated in India and is now used throughout the world, is a positional base 10 system.In computers, the main numeral systems are based on the positional system in base 2 (binary numeral system), with two binary digits, 0 and 1. Source.https://en.m.wikipedia.org/wiki/Numeral_system