WildTrig2: Quadrance via Pythagoras and Archimedes

Think algebraically in this case and not geometrically. If you have rectangle with sides a and b, then Q=a*b,
further Q1=a*a, and Q2=b*b.

Then Q^2=(a*b)^2=(a^2)*(b^2)=(a*a)* (b*b)=Q1*Q2

which proves the lemma.

nice proof of the pythagorean theorem! :)

It took me some time to figure it out too. Just think of it this way:
Q = (one side of Q2 * one side of Q1)
since Q2 and Q1 are squares, the length of one side of them would be the square root of their area, so:
Q = sqrt (Q1) * sqrt (Q2)
then just square both sides

Q^2 = [sqrt (Q1) * sqrt (Q2)]^2
or
Q^2= Q1*Q2

The ancients rule!!! 5 stars

Suppose that the rectangle is m units wide and n units long. Then its area is mxn, and so Q squared will be (mxn)^2 .

But this is the same as (m^2)x(n^2), which is Q1xQ2.

These proofs are wonderful - many thanks..

Would I be being too pedantic to suggest that this explanation of the lemma seems to be using the notion of distance (am I right that we're using quadrance to avoid any reference to distance?)..
I admit the lemma seems completely obvious but I can't prove it too myself without considering distances - is there any way of doing this?..

Thanks again!

Thanks for the nice comment.

Distance is not really being used in the lemma. You can think of the area as being composed of lots of little unit squares, and we are just counting those squares by multiplying the number along the base by the number along the height.

Many thanks for the kind reply!

Neat yes - I think I follow..

just to check, do we have to allow the little squares to be infinitesimal? (If one of the sides of the rectangle were an irrational multiple of the other...)

Does this matter? (if I'm not just wrong) Would proving this lemma need any notion of limit?..

A good question.

You are quite right, and touching on a historically delicate point. If we restrict our attention to the rational grid plane, then limits are not needed. Otherwise we need limits to define our notion of area.

In my MathFoundations series, you will find that I prefer to work purely in the rational number plane, for exactly this kind of reason.

Thanks for the info - I'll take a look at MathFoundations..

I must admit, the more I look into the foundations of maths the more bewilderingly subtle it appears (tried looking at Category theory and the difference between a class and a set - mystifying :)

Just pondering, even in the rational grid plane would you still have irrational distances all over the place and so tricky areas - like the quadrance on the side of the unit square...(?)

Anyhow - thanks for all the videos and explanations.