Understanding the Constant in Division of Large Numbers

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In summary, the conversation discusses the realization that when dividing 987654321 by 123456789, there is a constant when using the sums of powers of 10. The conversation also mentions a formula for determining the sum of powers of 10.
  • #1
androol
9
0
like many people have done with their calculators, I was dividing 987654321 by 123456789. and I realized...

sum(0 to N) 987654321 * 10^(N*j)
divided by
sum(0 to N) 123456789 * 10^(N*j)

where N can be any positive integer and j is in the real domain, there is a constant.


Am I weird and stupid?
 
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  • #2
sum(0 to N) 987654321 * 10^(N*j) = 987654321 * sum(0 to N) 10^(N*j)
sum(0 to N) 123456789 * 10^(N*j) = 123456789 * sum(0 to N) 10^(N*j)
so their quotient is indeed constant.

I ended up not needing this: For N a positive integer, sum (j=0,N,10^(N*j)) = (10^(N^2+N)-1)/(10^N - 1).
 
  • #3
omg omg I'm like so stup.

something times (thing/thing) = something.

how can I be so stupid...
 

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