Understanding the Constant in Division of Large Numbers

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SUMMARY

The discussion centers on the mathematical concept of a constant arising from the division of large numbers, specifically 987654321 by 123456789. The user demonstrates that the quotient remains constant when expressed as a series involving sums of powers of ten. The key formula derived is the quotient of two sums, which simplifies to a constant value, confirming the user's insight into the behavior of these large numbers under division.

PREREQUISITES
  • Understanding of basic algebra and division
  • Familiarity with summation notation
  • Knowledge of exponential functions
  • Concept of series convergence
NEXT STEPS
  • Explore the properties of geometric series
  • Study the concept of limits in calculus
  • Learn about the behavior of large numbers in mathematical analysis
  • Investigate the applications of constants in numerical methods
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Mathematicians, students studying algebra and calculus, and anyone interested in the properties of large numbers and their division.

androol
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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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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).
 
omg omg I'm like so stup.

something times (thing/thing) = something.

how can I be so stupid...
 

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