So we recently covered the Pythagorean Theorem, and I am betting you’ve got at least one concern. No, I don’t mean about whether or not Planet X has begun cutting a destructive swath across our solar system. I mean in regards to calculating the length of the side of a triangle if you know the other two. Don’t have a right triangle? You may think you’re SOL. I am here to tell you there is hope. Enter the Law of Cosines.
It may very well be the second most famous equation of all time, outshone only by that braggart Einstein’s mass–energy equivalence equation. But for those of us that aren’t theoretical physicists, the Pythagorean Theorem is likely to play a fundamental role in many of the calculations we do whether we realize it or not.
Everyone knows the equation, but before we get into it’s proof, let’s re-cap what it means. Consider the following right triangle:
This triangle has sides and hypotenuse . The Pythagorean Theorem simply tells us then that the square of the length of the hypotenuse () is equal to the square of the length of side plus the square of the length of side (). Well, duh. And after we learn it in middle school many of us take it as axiomatic, but it does have “theorem” in it’s name so I feel its worth revisiting it’s proof every once in a while to keep me on my toes.
(Note: I lied. This will be my first “neural dump.” I began writing about Euler’s Formula, but felt what follows was worthy of its own post and a better foundation for what will follow when I tackle Euler.)
Complex numbers arose from the fact that there is no solution for in the equation in , the set of real numbers.
Early mathematicians being the devil-may-care mavericks that they were, were all like, “Screw it. Let’s just invent a new number. We’ll just call this number and say that the solution is .” Or, in other words, this new number they imagined up (see what I did there) is equal to .