Mathematics with Minimum Raw Material, Part 2

Wasiq:

``...the area is proportional to the square of a side of a triangle...``

This is not true in general.

Area of the triangle = 1/2 x Base x Height

What you are saying would only be true in the case of a right triangle with two equal sides. So your proof would work in the special case not in general.

Khan:

I think following also rightfully belong to the list of great mathematicians:

Galois, Riemann, Hilbert, Hardy, Ramanujan, Godel, Neumann

If you are including Newton then you shouldn`t forget Witten.

Also don`t forget Russians and Americans.

Gauss is indeed known as the ``Prince of Mathematics`` but I will hold my vote for now. There is so much to come.

Re: Khan

I don’t think there are any clear winners in either category but if I had to pick I’d choose Euler and Gauss amongst the mathematicians. In the second category I’d vote for the Greeks simply because their logical rigor provided the foundation on which other giants could build.

Re: Wasiq

Yes, Cantor and part III it should be.

Here is an appetizing application of Pythagoras theorem.

A large pizza costs the same as a medium and small combination. You have to decide whether to buy the large pizza or the medium & small combo. What should you do?

Answer: Cut each pizza in half (so that you have three semi-circles) and arrange them so that the diameters form a triangle. If it is an acute angled triangle then choose the medium & small combo, if one of the angles is obtuse then choose the large pizza and in the event that the triangle is right angled then either choice is equally good (or bad if you are into healthy food!).

Reason: Let us say the three diameters are dl, dm and ds with respective areas as Al, Am, and As

If the triangle is right angled, then

sq(dl)=sq(dm)+sq(ds)

or (pi/4) x sq(dl)= (pi/4) x sq(dm)+ (pi/4) x sq(ds)

and so Al=Am+As

If the triangle is acute angled, then

sq(dl) *is less than * sq(dm)+sq(ds)

etc.

Fozia

[The last post got ``shorted`` because I used special characters; so here goes again]

Re: On Pythagoras`s Proof

I think the fact that the two new triangles formed by dropping the perp. onto the hypotenuse are similar (all corresponding angles equal) to the large one needs to be stated. Otherwise the ``area of a triangle being proportional to the square of a side`` is an ambiguous statement [in general the area of a triangle is not of the form K.SQ(a), where ``a`` is any side and K, a constant].

If the original triangles area is X, the area of the smaller triangles (being similar to the original one) is:

(SQ{AB/AC}).X and (SQ{BC/AC}).X

These sum to the total area:

(SQ{AB/AC}).X + (SQ{BC/AC}).X = X

Which gives you SQ(AB) + SQ(BC) = SQ(AC)

Just curious who would you all think of as the greatest mathematician ever? Gauss, Euler, Archimedes, Euclid, Pythagoras? What about Newton, and the Frenchies (Laplace, Lagrange, etc.)? Which nation/race contributed the most to Math (is there a clear winner): Greeks, Germans, French, Italian, Arabs, Ancient Egyptian?

Re: On Pythagoras`s Proof

I think the fact that the two new triangles formed by dropping the perp. onto the hypotenuse are similar (all corresponding angles equal) to the large one needs to be stated. Otherwise the ``area of a triangle being proportional to the square of a side`` is an ambiguous statement [in general the area of a triangle is not of the form K.a

Re: Fozia

Yep, they are absolutely beautiful. Maybe we should present Cantor`s proof in the next article in this series.

Your suggestion about cutting a Mobius strip is very interesting, I am sure people will be surprised at what happens.

Talking of simple proofs, here`s one short elegant proof of the Pythagoras theorem.

Take a right angled triangle ABC. Drop a perpendicular to the longest side AC from B. Since the area of the two new triangles ADB and BDC must sum to the the area of ABC, and the area is proportional to the square of a side of a triangle, the Pythagoras theorem follows:

AB.AB + BC.BC = AC.AC

regards

Thanks for posting this. Euclid`s proof of the infinitude of primes and Cantor`s demonstration of the countability of rational numbers are two of the most elegant yet simple proofs in all of mathematics.

Interested folks may want to try the following...
.... cut a Mobius Strip lengthwise all the way around. You will be surprised with the result!

Fozia