Like the crest on the peacock’s head,
Like the gem in the cobra’s hood,So stands mathematics 
at the head of all the sciences. 

-       Vedanga Jyotisa (Sanskrit text , 4^th century BC)

I will begin my article series on the ancient Indian excellence with their knowledge in mathematics.

No work on mathematics can begin without talking about Aryabhatta and his contributions. 

Now without much hush bush , who is this Aryabhatta?

Aryabhata was a great indain mathematician who was born around 476CE. He is the author of many great works in the field of maths and astronomy. Lets have a look into few of his excellent contributions. But before we can understand his contribution , we need to understand aryabhatiya number system.

What is Aryabhatiya number system?

Ariyabhatiya number system is the system of numerals based on the Sanskrit phonemes.

It was introduced by Aryabhata in the first chapter called “Gitika Padam” in his classic work “Aryabhtiya”.

Āryabhaṭa’s code is a mapping of numbers to words that employs first a division of the Sanskrit alphabet into two groups of consonants and vowels .

The first 25 consonants called the varga letters  of the Sanskrit alphabet are:

 k=1 kh=2 g=3 gh=4 ṅ =5 c=6 ch=7 j=8 jh=9 ñ=10 ṭ=11 ṭh=12 ḍ=13 ḍh=14 ṇ=15 t=16 th=17 d=18 dh=19 n=20 p=21 ph=22 b=23 bh=24 m=25

The remaining 8 consonants are the avarga  letters: y=30 r=40 l=50 v=60 ś=70 ṣ =80 s=90 h=100 The varga letters (k through m), V, stand for squares such as 1, 100, 10000, and so on.

The avarga letters (y through h), A, stand for non-squares, such as 10, 1000, and so on. The following vowels are the place holders as shown: a=1 i=10² u=10⁴ ṛ =10⁶ ḷ =10⁸ e=10¹⁰ ai=10¹² o=10¹⁴ au=10¹⁶

So, ka to ma 1 to 25 Number, and ya is 30, ra is 40, la is 50 and upto sha 80. So, ka is 1, if you are putting ikaaram, ki x 100, if you are putting ukaaram, ku x 10000, if you are putting rikaaram, kr x 1000000

Aryabhata used this number system for representing both small and large numbers in his mathematical and astronomical calculations. This system can even be used to represent fractions and mixed fractions. 

For example nga is 1/5, nja is 1/10 and Jhardam (jha=9; its half) = 4½. 

Why should i understand all this number system?

Aryabhatta has used this substitution methodology to code all his legendary works as sanskrit poems / texts. and to understand them , we first have to understand this number system.

Let me give an example for how he has coded values in sanskrit words. Aryabhata has said in one of his sutras "Neela Bhuvyasa" which means "The diameter of earth is 1050 Yojanas".

Now we all may wonder what on earth is this "Neela" and "Yojana".

Lets break it down. According to Aryabhata's number system explained above , "Na" gets the value 10 and if any consonant is added to by "Ikara" i.e "EE" it gets multiplied by 100. so Nee (NA+EE) becomes (10*100=1000) 1000 and "La" has the value 50. so the diameter of earth has become 1050 Yojanas.

Now lets breakup what is One Yojana.

One Yojana is said to be "Sachangulo Gahasthonara shi yojana", sa is 90, cha is 6, sachangulo is 96 angulam, one angula is average diameter of your finger, middle finger, 1.6 cm approx. x 96, i.e. your average height or gahastha tha 1, kha 2, ga 3, tha 4 gahastha from this tip to this tip x 4, that is your average height. Like that, nara shi yojana, sha 80, shi 8000. So 8000 the average height of man is one yojana is equal to 12.11 kms.

So how do you translate this in terms of the diameter of earth?

When he says "Neela Bhuvyasa" i.e the diameter of earth is 1050 Yojanas , it means the diameter of earth is 1050 * 12.11Kms approx = 12715.5Kms

Now what does the modern value say about the diameter of earth ?

A picture speaks 1000 words. So below is the snapshot from my best friend Mr.Google on the diameter of Earth.

Now how close is the modern value to the one perdicted by Aryabhatta ? Approximately 27Kms less . 

I enjoyed writing this article as much as you enjoyed reading this.

Look forward to my next article to know more on what Aryabhata says on the rotation of Earth.

Infinity refers to something that has no end. Something without any limit. In mathematic terms, we can express infinity as a numberless number. It does not increase by adding anything or decrease by subtracting anything from it.

Infinity

Can anything guess what is Infinity+1 called? Or What is Infinity-1 called? The answer is also Infinity.

So lets now have a sneak peak on ancient Indian knowledge on Infinity. What does our vedas and Upanishads talk about infinity. What does ancient Indian masters and scholars tell about infinity.

The concept of infinity was known in vedic times. The Opening lines of Shukla Yajur veda says:

Om Purnamadah Purnamidham Purnaath  Purnamudachyate

Purnasya murnamaataya Purnamevavasishyate

We can interpret this sloka in 2 ways. One metaphysical and the other is mathematical.

Roughly we can translate this as “Infinity is born from Infinity and when we take infinity out of infinity, only infinity remains”. 

Here, it is very interesting to note the relationship between zero and infinity .Anything divided by zero is infinity. The first purnam relates to the infinity and the later one to zero which is a root for infinity.

Asmin vikara khahare na raasaavapi praveshteshvapi ni: srutheshu bahushvapi syaallaya srushtikaale_nanthe_ chyuthe bhoothaganeshu yaddhath -  Bramhmasputah siddhantha.

Nothing happens to the (huge number) infinity, when any number enters (added) or leaves (subtracted) the infinity.

This sloka uses this analogy to compare with the Bramhan or the ultimate being. It says, During pralaya many things get dissolved in Mahavishnu and after pralaya, during srushti all those things get out of him.

This happens without affecting the lord himself. Like that, whatever number is added to infinity or whatever is subtracted from it, the infinity remains unchanged.

In Taittariya Updanishad, Brahman has been described as "satyam gynanam anantam brahma", which means Brahman is truth, knowledge and infinity.

There are also various other references towards infinity and other large numbers used by ancient Indians. Watch out for those in the next post.