MathFoundations23: The basic framework for geometry (I)

This video begins to lay out proper foundations for planar Euclidean geometry, based on arithmetic. We follow Descartes and Fermat in working in a coordinate plane, but a novel feature is that we use only rational numbers.
Points and lines are the basic objects which need to be defined.

Category:

Science & Technology

Tags:

• Foundations
• mathematics
• Euclidean
• planar
• geometry
• points
• lines
• rational
• numbers
• Wildberger

[enfomy]

If a point is a pair of rational numbers, then would it be safe to say your "points" are actually "lines?" I can certainly see this form of geometry being more mathematical and precise, but not more a priori. Euclidean geometry can be explained to a child, but not this. I also wonder if a grid would be a necessary tool for this form of geometry. It seems almost impossible to use without one, tho a grid is not always at hand for one who is doing some basic geometry.

[njwildberger]

@enfomy Not sure why you suggest points are lines, they are not. A piece of graph paper is a great tool for studying geometry, these are widely available, both physically and online.

[simpeltree]

At 7:31 you say that "the proportion a to b is the proportion

- (y.two - y.one) to (x.two - x.one), so a:b = y.one - y.two : x.two - x.one".

My question is how do you get that. I am lost with this proportion calculation...

Any help. Thanks.

[njwildberger]

Hi simpeltree, It is not too hard---remember that a:b=c:d precisely when ad-bc=0. So for example 2:3=4:6 since 2x6-3x4=0. Or a more complicated example 12:32=15:40 since 12x40-32x15=0.

So a:b=(y1-y2):(x2-x1) precisely when a(x2-x1)-b(y1-y2)=0 which is the same as what I wrote, except for replacing -b(y1-y2) with +b(y2-y1).

[andreirocks1992]

rat squared :D

[enfomy]

Perhaps you need to mention that there can be no singular point, or phrase it in the plural and not singular tense. With Euclid's geometry any two points can be a line which makes the 2 points individual entities. But in this, points are indicated by pairs, whose coordinates are represented on a number line, therefore u need to know what a line is to figure our what a point is. It's like knowing lines before your know points. This foundation starts off too complex i think. Is that not a problem?

[enfomy]

@njwildberger Not suggesting points in themselves are lines, Im only saying that the mental images of your "points" in this foundation for geometry would indicate a "line" rather than a point. I understand precision is what your looking for in this view of geometry, but a number line being somewhat necessary, when their is no natural number lines in our minds, makes the whole concept seem subjective and arbitrary. Everyone in the world doesn't have access to number lines.

[truwoman01]

im confused n lost

[simpeltree]

Hi njwildberger !

Very good help. Now I understand. Thank you very much.

[VolkColopatrion]

Why do i find () brackets more common instead of the [] for a point on a Cartesian graph?