Epilogue

As you are very well aware of by now if you're reading this, the problem of computing Fibonacci numbers is far deeper than it might first appear. Although you probably aren't going to need the ten billionth Fibonacci number any time soon, there's a lot of useful takeaways that will serve you well in your future mathematical endeavors:

  • Memoization and dynamic programming are fantastic tools when simple recursive approaches waste effort.
  • Recursion and iteration are equivalent: it's possible to rewrite one in terms of the other. In most common programming languages, iteration is faster.
  • Computers don't store infinite precision, and rounding errors can be a serious concern when doing computations with large numbers.
  • Linear algebra has incredible utility in many applied math problems: phrasing questions in terms of linear algebra often lets you stand on the shoulders of giants, borrow optimized code to solve your problem, or simply leverage existing theory to gain new insights.
  • Once you have an abstract solution, there are often significant gains to be made by exploiting specific features of your exact problem, losing that generality for bespoke optimizations.
  • Mathematics permeates the world around us—profound structure lurks behind even the most innocuous sequences.

I hope you've found this enjoyable. Thanks for reading!

Epilogue

As you are very well aware of by now if you're reading this, the problem of computing Fibonacci numbers is far deeper than it might first appear. Although you probably aren't going to need the ten billionth Fibonacci number any time soon, there's a lot of useful takeaways that will serve you well in your future mathematical endeavors:

  • Memoization and dynamic programming are fantastic tools when simple recursive approaches waste effort.
  • Recursion and iteration are equivalent: it's possible to rewrite one in terms of the other. In most common programming languages, iteration is faster.
  • Computers don't store infinite precision, and rounding errors can be a serious concern when doing computations with large numbers.
  • Linear algebra has incredible utility in many applied math problems: phrasing questions in terms of linear algebra often lets you stand on the shoulders of giants, borrow optimized code to solve your problem, or simply leverage existing theory to gain new insights.
  • Once you have an abstract solution, there are often significant gains to be made by exploiting specific features of your exact problem, losing that generality for bespoke optimizations.
  • Mathematics permeates the world around us—profound structure lurks behind even the most innocuous sequences.

I hope you've found this enjoyable. Thanks for reading!