## Tools to aid thinking

An article by Sonali Ranade on Rediff would improve tremendously if tools developed by ancient Indian greats were used:

Mumbai and its suburbs have a population of 20.50 million. Maharashtra’s total population is 112 million. Nearly every fifth Maharashtrian lives in Mumbai. That is a huge proportion.

This is a trivial truth if you count everyone living in Maharashtra as a Maharashtrian. However, that does not seem to be the whole intent of her article. So Ms Ranade must have a more restricted definition in her mind: where some residents of Maharashtra are not counted as Maharashtrian.

If that is so, then it is time to roll out that old tool developed by ancient Indian mathematical genuises: algebra. Suppose a percentage p of Mumbai’s population is not Maharashtrian, then the number of Maharashtrians in Mumbai is (1-p/100) X 20.50 million. What this means is that if by Ms Ranade’s definition it turned out that 50% of Mumbaikars were not Maharashtrian then the number of Maharashtrians in Mumbai would be 10.25 million. On the other hand, if Ms Ranade gave a different meaning to being Maharashtrian, so that non-Maharashtrians were only 10% of Mumbai’s population, then this number would be 18.45 million.

Whatever is Ms Ranade’s definition of a Maharastrian, we could take it beyond Mumbai. By the same definition, suppose that a percentage q of Maharashtra’s population did not count as Maharashtrian, then the number of Maharashtrians in the state would be (1-q/100) X 112 million.

What do we know about q? We certainly know that the number of non-Maharashtrians in Maharashtra cannot be smaller than the number of such people in Mumbai. So (q/100) X 112 ≥ (p/100) X 20.50. A little use of the same old Indian tool tells us that q ≥ p (20.50/112), and therefore q ≥ 0.18 p. Also, if Mumbai is more of a magnet for non-Maharashtrians than any other part of the state, then one should have q < p.

What this means is that is p were 50% then q is between 9% and 50%. If, on the other hand, p were 10% then q is between 1.8% and 10%.

So finally we are in a position to attempt to do the computation that Ms Ranade accomplishes with elan. The fraction of Maharashtrians (by whatever definition Ms Ranade hands us) who live in Maharashtra is (1-p/100) 20.50 / (1-q/100) 112 = 0.18 (1-p/100)/(1-q/100). This is 18% only if p=q, which would happen if Mumbai were no more attractive to non-Maharashtrians than the rest of Maharashtra. But when we take the other limiting case q=0.18 p, then we find that the fraction of Maharashtrians in Mumbai is not so easy to calculate, since it depends on p. If p were 50% then it would turn out that only 10% of Maharashtrians live in Mumbai. On the other hand, if it were as low as 10%, then 16.7% of Maharashtrians would live in Mumbai.

The important point about all this is that the fraction of Mumbai's population which is Maharashtrian is different from the fraction of Maharashtrians who live in Mumbai. The former is 1-p, the latter is 0.18 (1-p/100)/(1-q/100). Without knowing what Ms Ranade's definition of Maharashtrian is, it would be hard to find the values of p and q. You could take the position that all people residing in Maharashtra are Maharashtrians, so p=0 and q=0. You could have a friend who takes the extreme position that Mumbaikars are Mumbaikars and are distinct from the rest of Maharashtra, in other words that p=100 and q=18. In either case some parts of Ms Ranade's arguments would fail. The remainder are robust arguments, which must be correct, and therefore are worth quoting in full:

Maharashtra has 288 members in its legislative assembly of which Mumbai city contributes 9. Add to that number the seats from the greater metropolitan area, the number grows to 36 or roughly 12 per cent of the total strength of the legislative assembly. In contrast, 18 per cent of all Maharashtrians live in Mumbai.

Representative democracy means that every member of a legislature must represent an equal number of people. A legislator’s vote on the budget or a law should represent the vote of a fixed number of people. One may argue that this is not the perfect system for giving equal voice in government to each citizen of India, but this is the system which our constitution sets down. In order to make it work, the boundaries of constituencies must be redrawn every now and then so that the population in all constituencies remain equal. This has not been done even once after independence!

The ancient Indian mathematicians, living and working in Rajasthan (Brahmagupta, 6th century), Bihar (Aryabhata, 5th century), Kerala (Madhava, 14th century) and Maharashtra (Bhaskaracharya, 12th century) developed tools for aiding thought which remain as useful and practical today as they were in their lifetimes.

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