# Demonstration the idea of a Euclidian proof

I'm going to give a presentation involving mathematical proofs to people who are not really into math (in a Critical Thinking class).

I would like to use something from Euclid's elements. I would like it to be a geometric proof - but what is the most "audience friendly?" I need to be able to do it in a very short amount of time. (We have less than 10 minutes for our entire presentation, and the proof cannot take up the bulk of it).

I'm thinking Proposition 13 "If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles."

Ideas? I can have fun with this one...

-Dave K

## Answers and Replies

Stephen Tashi
Science Advisor
In the liberal arts the mathematical style of Euclid is still regarded as epitomizing mathematical proof. Euclid's style is still copied in secondary school textbooks. However, it is not a good example of modern "critical thinking" in mathematics. I'm talking about Euclid's general approach being antiquated, not about his subject matter (geometry, number theory).

In Euclid's approach, certain things are regarded as "self evident". These are set down as assumptions so the author can conveniently refer to them in writing proofs.. They aren't treated as assumptions with idea of allowing the possibility that there might be different fields of mathematics where some of the assumptions are false. His style of definition attempts to define a thing as a something everyone will be familiar with and after we understand it, we are allowed to assume that all the properties of the familiar thing apply without having them explicitly listed as part of the definition.

The Euclidean approach is useful in mathematics education since it emphasizes intuition.

Deveno
Science Advisor
i've always liked bhaskara's first proof of the pythagorean theorem.

1) the pythagorean theorem is likely something everyone will think is true, and will have seen somewhere.

2) the proof is highly visual, and takes maybe a minute or two to run through. the assumptions are minimal, and require little mathematical sophistication.

3) the theorem, nevertheless, is extremely important, and forms the basis of many important ideas.

(for an example of what i mean, see here: http://jwilson.coe.uga.edu/emt668/emt668.student.folders/headangela/essay1/pythagorean.html )

Euclid's proof of the infinitude of primes is another good candidate, illustrating the principle of "proof by contradiction" (aka reductio ad absurdum, "reducing to an absurdity"). a beautiful discussion of this proof can be found in:

http://www.math.ualberta.ca/~mss/misc/A Mathematician's Apology.pdf

in any case, good luck with your presentation! :)

lavinia
Science Advisor
Gold Member
I'm going to give a presentation involving mathematical proofs to people who are not really into math (in a Critical Thinking class).

I would like to use something from Euclid's elements. I would like it to be a geometric proof - but what is the most "audience friendly?" I need to be able to do it in a very short amount of time. (We have less than 10 minutes for our entire presentation, and the proof cannot take up the bulk of it).

I'm thinking Proposition 13 "If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles."

Ideas? I can have fun with this one...

-Dave K

For critical thinking in Euclidean geometry I suggest the insight that parallel lines exist. This does not require the parallel postulate but only simple ideas about lines and planes.

As I was taught it, a line in a plane separates a plane into two "half planes" which are intuitively imagined to be identical in all of their properties.

It is also assumed that two lines can intersect in exactly one point.

Now suppose that a line lies in a plane and two other lines are drawn at right angles to it in the plane.

Do these lines have to be parallel?

The answer is yes.

The argument appeals to our intuition that the half planes must be identical. If the two lines intersect in one of the half plane and not in the other then even though there is nothing about the lines or their angles of intersection that are different in either half plane ( they enter each half plane at right angles), somehow the half planes would be different because the two lines would intersect in one and not in the other.

But two lines can not intersect in two points so these lines must actually be parallel.

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