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Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: petelamana on 18/02/2018 20:29:23

Title: Can the answer to a/b be arrived at geometrically?
Post by: petelamana on 18/02/2018 20:29:23

Geometry is a deep interest of mine, both Euclidean and non-Euclidean.  I am looking for geometric solution to the division of two integers, a/b.

Title: Re: Can the answer to a/b be arrived at geometrically?
Post by: evan_au on 18/02/2018 21:29:32

Quote from: petelamana

geometric solution to the division of two integers, a/b.

By "Geometric", doe you mean the whole "Ancient Greek Mathematician" version of geometry, which only accepts solutions that can be done in a finite number of steps, with just a compass and straight-edge?

If so, I think I have one solution:
- Create a number line, marked in integers
- At right angles to this, create another number line, marked in integers, and passing through 0* (ie now a number plane)
- Mark a on one axis, and b on the other
- Draw a line between a and b. This represents the ratio a/b.

Now, how useful it is depends on what you want to do with this ratio.

A typical application might be to divide a third number "c" in the ratio a/b:
- Mark c on one axis
- Draw a line through c, parallel to the a-b line
- Where this new line crosses the other axis is the result of c divided in the ratio of a/b

Maybe you can be more specific about what applications you have in mind for the ratio of a/b?

*Ancient Greek Mathematicians didn't really believe in zero. But this construction only uses a compass and straightedge...

Title: Re: Can the answer to a/b be arrived at geometrically?
Post by: Bored chemist on 18/02/2018 22:00:31

This sort of thing?

Title: Re: Can the answer to a/b be arrived at geometrically?
Post by: petelamana on 18/02/2018 22:03:46

Thank you chemist.  I need to study it a bit.  I'm wondering if what you show only works for 3:2, or if it would work for any a:b?

Title: Re: Can the answer to a/b be arrived at geometrically?
Post by: Colin2B on 18/02/2018 22:17:31

Quote from: petelamana on 18/02/2018 22:03:46

I'm wondering if what you show only works for 3:2, or if it would work for any a:b?

I use it a lot to create custom resizing scales and in navigation. You can see from the construction it works for any ratio, doesnt need to be integer.
If you have a parallel ruler, such as used in navigation, you can simplify by just drawing the first line joining B and X and then running parallel lines from the marks on AX over to AB