The basic framework for geometry (IV) | Arithmetic and Geometry Math Foundations 26 | N J Wildberger

Angles don't make sense in the rational number system. The proper notion of the separation of two lines is the `spread' between them, which is a purely algebraic quantity and can be calculated easily using rational arithmetic only. This video highlights some of the advantages in replacing `angle' with `spread'. It also gives an explicit formula for the `inverse cosine' function, which rarely appears in trigonometry texts, despite the universal reliance on this function via calculators.

This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics.

Video Contents
00:00 Introduction
00:45 Angle on the circle
02:22 Spread
03:16 Defining angles precisely without using pictures
04:46 How to calculate a spread between two lines

************************

Here are the Insights into Mathematics Playlists:

Here are the Wild Egg Maths Playlists (some available only to Members!)

************************

[robindebreuil]

Commenting on a random video, but the way you approach math feels so much better than the way we do things now, wow. I’ve worked in programming all my life, and it always bothered me how far removed math theory is from actual math. It always felt more like a bag of tricks rather than a coherent system (much like bad software feels like). I was led to your videos while making software that deals with magnitudes in a way that tries to model variation more as we experience it (for ML). I’ve spent a few years on it and am getting to a system that feels similar to a lot of what you say (and a lot of your videos feel like I’ve found the answer key at the end of the book!). Anyway, I really appreciate you putting these ideas out there, more people need to think about these things. We all need to think about these things in fact. We’re built in a way that the easy questions are the hardest to ask, but that’s where the gold is. A very large (but not infinite!) number of kudos to you sir.

[hellenakinyi]

Video Content
00:00 Introduction
00:45 Angle on the circle
02:22 Spread
03:16 Defining angles precisely without using pictures
04:46 How to calculate a spread between two lines