# Tamil Fractions

A friend asks for REASONS:

THAERTTHUGAL n. The smallest named fraction in the ancient Tamil language = 1/2323824530227200000000.

— The QI Elves (@qikipedia) April 7, 2014

A who to the what now? A twelve-letter word, a thaerrhugaL, representing a number somewhere in the region of twenty-three ninety-ninths of a sextillionth.

It’s hardly unreasonable to demand REASONS. It’s not an ‘obvious’ number - a power of anything, for example. Its reciprocal is (brilliantly) $\frac{9,800,000}{27}\times 18!$; dear readers, I shall be honest with you: that wasn’t the first thing I guessed.

The Wikipedia page on Tamil fractions sheds a little more light, though:

- 1/160= araikkaaNi
- 1/320= munthiri
- 1/102,400= keezh munthiri
- 1/2,150,400= immi
- 1/23,654,400= mummi
- 1/165,580,800= aNu
- 1/1,490,227,200= kuNam
- 1/7,451,136,000= pantham
- 1/44,706,816,000= sggtta
- 1/312,947,712,000= vintham
- 1/5,320,111,104,000= naagavintham
- 1/74,481,555,456,000= sinthai
- 1/1,489,631,109,120,000= kathirmunai
- 1/59,585,244,364,800,000= kuralvaLaippidi
- 1/3,575,114,661,888,000,000= veLLam
- 1/357,511,466,188,800,000,000= nuNNmaNl
- 1/2,323,824,530,227,200,000,000= thaertthugaL

Spot the pattern? No, me either. However, it turns out that each fraction is related to the one before:

$\frac{1}{160} = \frac{2}{320}$ $\frac{1}{320} = \frac{320}{102,400}$ - this could be a square measurement, perhaps? $\frac{1}{102400} = \frac{21}{2150400}$ $\frac{1}{2150400} = \frac{11}{23654400}$ $\frac{1}{23654400} = \frac{7}{165580800}$ $\frac{1}{165580800} = \frac{9}{1490227200}$ $\frac{1}{1490227200} = \frac{5}{7451136000}$ $\frac{1}{7451136000} = \frac{6}{44706816000}$ $\frac{1}{44706816000} = \frac{7}{312947712000}$ $\frac{1}{312947712000} = \frac{17}{5320111104000}$ $\frac{1}{5320111104000} = \frac{14}{74481555456000}$ $\frac{1}{74481555456000} = \frac{20}{1489631109120000}$ $\frac{1}{1489631109120000} = \frac{40}{59585244364800000}$ $\frac{1}{59585244364800000} = \frac{60}{3575114661888000000}$ $\frac{1}{3575114661888000000} = \frac{100}{357511466188800000000}$ $\frac{1}{357511466188800000000} = \frac{13}{2\times 2323824530227200000000}$

Oh, those wacky Tamils! What a ludicrous way of dealing with naming fractions! Where does that 17 come from, for example? Why suddenly $\frac{13}{2}$ as a divisor? There’s no way you’d catch us modern-day cosmopolitan Westerners dividing - let’s say - a mile into 4,561,920 equal parts and calling them each a point, eh? That would be completely preposterous and backwards.