A popular way to have fun with arithmetic is to find a way to express a number in terms of other numbers. For example, here’s 69 expressed with the digits 1,2,3,4,5 in order:
69 = 1·√2 [sup]3·4[/sup] + 5
I just watched a Numberphile YouTube that demonstrated an impressive equation. Express Euler’s Number (2.7182818…) using each of the digits 1,2,3,4,5,6,7,8,9 one time:
e = (1+9[sup]-4[sup]6·7) [sup] (3[sup]2[sup]85[/sup][/sup])[/sup][/sup][/sup]
Wait! Didn’t Charles Hermite prove that Euler’s Number is transcendental? This number, however ridiculous, isn’t transcendental. And, sure enough, the “equation” is only an approximation. … But it approximates eaccurately to 18 trillion trillion digits! Isn’t that close enough for you?
If it’s confusion with numbers you seek, learn Thai! พันพัน /phan phan/ (literally ‘thousand thousand’) means ‘several thousands’, and หมื่นหมื่น /muen muen/ (literally ‘ten-thousand ten-thousand’) means ‘several tens of thousands’, but, since there are no specific words for numbers bigger than a million, ล้านล้าน /la:n la:n/ (literally ‘million million’) can mean either ‘several millions’ or ‘trillion.’ I suppose ambiguity is often avoided since even the brothers who own CP Group (the vast Thai conglomerate which dominates pork and poultry here, owns the telecom giant True, has expanded into China and other countries, and owns the Thailand franchises for KFC, 7-Eleven and Heineken beer) are not quite baht trillionaires yet.
Even the Brits have quietly given up on their long trillion. First they lost the Empire where the Sun never set. Then they lost their super-sized billion and trillion. (The quaint notion that a penny represented a penny-weight of silver was lost long ago: in 1464 King Edward IV was minting almost 39 shillings in pennies from a troy pound of silver, in contrast to the 31-shilling pound of his predecessor.)
There must be a way to estimate the error. Using a calculator just now I see that (1 + 10^-100)^(10^100) agrees with e to a hundred decimal digits and (1 + 10^-200)^(10^200) equals e to 200 decimal digits. Apparently this extrapolates in the obvious way, because it appears 3^2^85 ~= 10^{18.4 trillion trillion}.
What makes the weird expression work, of course, is that 9^4^(6*7) = 3^2^85 exactly.
Yeah, they didn’t calculate both to 10^24 digits and compare them-- Even if we could calculate either that precisely, the comparison alone would take far too long. They calculated the error directly.
I used to have that problem; even Lorazepam didn’t put me to sleep.
But I found a way to solve the problem! I moved my laptop outside our air-con bedroom; the last two (hot) evenings I’ve lain down for a brief rest in the air-con room … and woken up eight hours later!
Nitpick: Since the digits of e are unknown beyond the first trillion or so, it is possible that, at the key point, e = 2.7…400000000000000000000000000…0001… where there are a septillion or more consecutive zeros, with the approximation in OP then having a septillion consecutive nines: 2.7…39999999999999999…
In such a low-probability scenario, the approximation would be correct to only 17 trillion trillion digits instead of 18 trillion trillion. [/nitpick]
