Maths: why does 987654321 / 123456789 = 8.000000073

[jessica1994]

987654321/123456789 = 8.0000000729000006633900060368490...

which is very nearly 8, except for the isolated 729, 66339, 6036849 ... Now, 729 = 9^3. This isn't a coincidence, as we will see.


Explanation:
Using our prime factor finder program or built-in function, we find that:


729 = 9^3 = 9^3 * 91^0
66339 = 9^3 * 91 = 9^3 * 91^1
6036849 = 9^3 * 91^2 = 9^3 * 91^2

so we can conjecture that:


987654321/123456789 = 8 + 9^3*1E-10*SUM[N=0,N=INFINITE,(91*1E-10)^N]

where the sum goes from N=0 to N=infinite. This can be proved very easily using the well-known summation formula for geometric progressions, as this happens to be one.