987654321 / 123456789
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987654321 / 123456789
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Re: 987654321 / 123456789
#2Feels like a Temu version of Ramanujan's constant [0].
Re: 987654321 / 123456789
#3In a similar vein, e^pi - pi = 19.9990999792, as referenced in this XKCD: https://xkcd.com/217/
Re: 987654321 / 123456789
#4I like to think of 0.987654... and 0.123456... as infinite series which simplify to 80/81 and 10/81, hence the ~8 ratio.
Re: 987654321 / 123456789
#5More general analytic proof: https://math.stackexchange.com/questions/2268833/why-is-frac...
Re: 987654321 / 123456789
#6I like to think of 0.987654... and 0.123456... as infinite series which simplify to 80/81 and 10/81, hence the ~8 ratio.
Care to elaborate? Why does 0.987654 simplify to 80/81 and 0.123456 to 10/81?
Re: 987654321 / 123456789
#7In a similar vein, e^pi - pi = 19.9990999792, as referenced in this XKCD: https://xkcd.com/217/
Not really in a similar vein, because there's actually a good reason for this to be very close to an integer whereas there is no such reason for e^pi - pi.
Re: 987654321 / 123456789
#8I like to think of 0.987654... and 0.123456... as infinite series which simplify to 80/81 and 10/81, hence the ~8 ratio.
I didn't get where this comes from until I saw the second answer from the StackOverflow question another commenter shared.
https://math.stackexchange.com/a/2268896
Apparently 1/9^2 is well known to be 0.12345679(012345679)...
EDIT: Yes it's missing the 8 (I wrote it wrong intially): https://math.stackexchange.com/questions/994203/why-do-we-mi...
Interesting how it works out but I don't think it is anywhere close to as intuitive as the parent comment implies. The way its phrased made me feel a bit dumb because I didn't get it right away, but in retrospect I don't think anyone would reasonably get it without context.
Re: 987654321 / 123456789
#9I like to think of 0.987654... and 0.123456... as infinite series which simplify to 80/81 and 10/81, hence the ~8 ratio.
Care to elaborate? Why does 0.987654 simplify to 80/81 and 0.123456 to 10/81?
If you set x = 0.123456..., then multiplying it by (10 - 1) gives 9x = 1.111111..., and multiplying it by (10 - 1) again gives 81x = 10, or x = 10/81. I’m not writing things formally here but that’s the rough idea, and you can do the same procedure with 0.987654... to get 80/81.
Re: 987654321 / 123456789
#10I like to think of 0.987654... and 0.123456... as infinite series which simplify to 80/81 and 10/81, hence the ~8 ratio.
Care to elaborate? Why does 0.987654 simplify to 80/81 and 0.123456 to 10/81?
.123456... = x + 2 x^2 + 3 x^3 + ... with x = 1/10.
Then you have (x + 2 x^2 + 3 x^3 + ...) = (x + x^2 + x^3 + x^4 + ...) + (x^2 + x^3 + x^4 + x^5 + ...) + (x^3 + x^4 + x^5 + x^6 + ...) (count the number of occurrences of each power of x^n on the right-hand side)
and from the sum of a geometric series the RHS is x/(1-x) + x^2/(1-x) + x^3/(1-x) + ..., which itself is a geometric series and works out to x/(1-x)^2. Then put in x = 1/10 to get 10/81.
Now 0.987654... = 1 - 0.012345... = 1 - (1/10) (10/81) = 1 - 1/81 = 80/81.