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| Aryabhata's ghana citi for 2025 |
Most of us studied not only numbers, but also series of numbers and sums of series in school in mathematics. I am sure most people remember, that the sum of the series from 1 to any number N is given by the formula N*(N+1)/2.
In other words, 1+2+3+4…..+N = (N * (N+1))
/2
In some school mathematic text books, the name of Carl Friedrich Gauss, the
great German genius, is mentioned in association with this series. It seems a German
teacher asked his class of nine or ten year olds, what is the sum of the first
hundred natural numbers. And Gauss, who was in this class quickly responded, 5050.
When asked how we calculated it so quickly, Gauss responded, that he added the smallest
and largest numbers 1 and 100, which came to 101; then he added 2 and 99 the
second smallest and second largest numbers, which also came to 101; next 3 and
98, then 4 and 97, also each adding to 101. He realized that there were 50 such
pairs, each adding to 101, so the sum is 50 times 101 which is 5050. Gauss went
on to do amazing things in mathematics, and became one of the greatest
mathematical geniuses the world ever saw.
Nice story. Every student and teacher can relate to it. Why don’t
we have such stories about Indian mathematicians, except for the famous taxi
number 1729 of Ramanujan?
While some formulae in mathematics have names attributed to their
inventors or discoverers, there are several mathematical formulae that remain
anonymous. All the formula that have names in either physics or mathematics
have European or American scientist or mathematician’s names. So, we have Pythagoras
theorem, Newton’s formula, Einstein’s formula, Euler identity, and so on. Have you ever wondered why? Why is this Gauss
story told without mentioning that Aryabhata gave us this formula for sum of series?
We also learnt that Indians invented zero – we are wrongly
told that Aryabhata invented zero. No, Aryabhata did not invent zero. Zero was at
least a few hundred years old before Aryabhata was even born. Besides Aryabhata
at least Bhaskara is famous as a great mathematician in India. Why do we never
learn about some Aryabhata theorem or Bhaskara formula.
Also, even if Aryabhata discovered or invented zero, he must
have invented something else also?
Let us discuss one set of things Aryabhata presented, which are
given in school textbooks throughout India and the world without mentioning his
name. Aryabhata gave not just the formula
for the sum of series of numbers, he gave formula for the sum of series of
squares and the sum of series of cubes.
In Sanskrit books, the word citi is used for series. Citi (Sanskrit चिति
Tamil சிதி) is
literally the word for series of bricks with which a yagna or fire altar for
Vedic rituals is made. Aryabhata uses these terms for these formulae
citi: for sum of series of numbers (1+2+3+4… +N) = N*(N+1)/2
varga citi: for sum of series of squares (1^2 + 2^2 + 3^2+ 4^2
…. + N^2) = N*(N+1)*(2N+1)/6
ghana citi: for sum of series of cubes (1^3 + 2^3 + 3^3+ 4^3
…. + N^3) = (N*(N+1)/2)^2
Varga (वर्ग
வர்க) and ghana (घन கன)are
the words used in most Indian languages for square and cube. Varga moola and ghana
moola are the words used for square root and cube root – incidentally Aryabhata
also gave us algorithms to calculate varga moola and ghana moola, but that is a
topic for another day.
We learn these formulae in school with Greek notations,
invented by European mathematicians in the 18th and 19th
century like sigma for sum.
Interestingly the sum of the series of the cubes upto 9, that is, 1^3
+ 2^3 + 3^3+ 4^3+…9^3 is equal to 2025, which is the Christian year that comes
up shortly. I am sure social media will be full of posters and jpegs and gifs
and short videos telling you this interesting fact, and perhaps bated breath
narrations of Gauss. And zero mention of Aryabhata. So, here is Aryabhata
wishing you a happy 2025.
Ironically Aryabhata knew nothing about this Christian calendar
adopted in Constantinople and the Roman empire, a few decades before he was
born. He used the Kali Yuga notation in his book on astronomy, giving his own
year of birth as 23 years before the 3600th year of the Kali
calendar. As 499 AD is Kali year 3600, historians of mathematics believe he was
born in 473 AD. In the Kali yuga calendar 2025 is the year 5126 – I am sure
enterprising mathematicians will come up with interesting ways to compute this
number using Aryabhata’s various formulae.
My essay in Swarajya magazine about Aryabhata
Aryabhata - CSIR NiScPR Posters
