MTH 495 – Blog Post 5 – Algebraic Rigor!

In one of the books that I have read recently, either “The Calculus Gallery: Masterpieces from Newton to Labesque” by William Dunham or “Isaac Newton” by James Gleick, it was said that Isaac Newton was known for his rigor with algebra. Looking up the definition of rigor on www.oxforddictionaries.com/us/, I found “The quality of being extremely thorough, exhaustive, or accurate.”

This, in addition to several experiences in class recently has piqued my interest in just what it means to have algebraic rigor. With this in mind, or discussion in class on Tuesday was on Georg Riemann and how he worked to define distance as a function and came to the conclusion that any distance function must have 3 properties:

1. D(x,y) = D(y,x)
2. D(x,x) = 0
3. D(x,y) + D(y,z) >= D(x,z)

We were then asked what function is normally used to determine the distance between two points on the coordinate plane. As is:

For point A(x₁,y₁) and point B(x₂, y₂), the distance, D, is:

D(A, B) = Sqrt((x₂ – x₁)² + (y₂ – y₁)²)

Okay, well how hard could it really be to prove that the distance formula fulfills the third property? Turns out, this is one problem that requires some significant algebraic rigor! After hours of work, I still haven’t finished and had to set it aside to work on this post. I am not ready to throw in the towel, I just needed to take a break. I am going to attach my work below, so take a look and let me know if you see anything I am missing or if you see a way to reduce the number of terms I am dealing with. I did use substitution to allow me to continue to work through the problem without having to rewrite everything over and over again. However, it seems that I have now reached a point where nothing is going to cancel out unless I expand everything out. I wonder if there are tricks for this that I am missing.

Anyways, take a look and let me know what you think.

Thank you for reading,

Jerry