Points, Line segment, Line

Background

We can follow paths off of The Number 1 to talk about more and more complex operations involving numbers. This path is going to follow paths involving shape and form.

Question

1. Graph a point. 
2. Graph another point. 
3. Graph a line segment between the two points. 
4. Extend the line segment in one direction. What's that called?
5. Extend the line segment in both direction. What's that called?

Answer

See below for the steps. A ray is a line segment extended in one direction. A line is a line segment extended in both directions.

Analysis

Without worrying about identifying where the point is put, let's just put a point on a page:



And now let's just graph another point:



Perfect. So now let's connect the two dots (we'll use the shortest distance possible, so no twisty lines or anything like that):



The green connector between the two dots is called a line segment.

Perfect! Now let's extend the line segment up and right so that it extends forever (so that if we had a large enough sheet of paper, the line segment would extend for as far as the paper reaches, and then continue even more - the only thing holding back the extension here is the limits of the image):



When we have a line segment that extends off into forever in one direction, this is called a ray.

And now let's extend the line segment in both directions:



A line segment extended in both directions into infinity is called a line.

At this point we might say "so what?". Why go through all this? Well, we've just worked through some of the work done by ancient peoples (the Greeks worked through this and other cultures might have as well. Euclid, in 300 BCE, as you can see in the link below, formalized what we just worked through - we just did the first two of his five geometrical postulates):

1. It is possible to draw a straight line from any point to any point, and
2. It is possible to extend a line segment continuously in a straight line

Vocabulary used:

For more information check out these links (comment to add your favourite link):

https://www.storyofmathematics.com/hellenistic_euclid.html
A pdf work (requires a download) at intellectualmathematics.com that takes Euclid's book, Elements, adds illustrations, and leads you through geometry: http://intellectualmathematics.com/geometry/

Where might you have come from?

Fact-orials Index

Numbers:

• The Number 1

Where might we go?

Geometry:

• Intersecting, Parallel, Perpendicular Lines
• Defining a Circle and Right Angles
• Perimeter and Area