Why Does Area Change Quadratically w/ Size? A Simple Question That Requires the Concept of Infinity

[@Jabber_Wock]

Thanks, I am enjoying this new lecture series!

[@theoremus]

Good job! Is what you are describing here called dilation scaling? Norman Wildberger starts his Algebraic Calculus with five propositions: Additivity, Linearity, Translation, Dilation and Normalization. In his FMP10 videos he claims that it took him a long time to prove these propositions. I have doubts, but I am just a software engineer. To me, these five propositions form what I call the Invariant (invariant properties of area and slopes). This is the starting point, not the Limit, which lacks a well defined definition (due to its dependence on infinite processes).

[@nomadr1349]

I would argue, imaginary numbers are not an "invention", but rather a discovery.

[@MathTheBeautiful]

Those are good arguments to have!

[@japedr]

My idea was to put the circle inside of a square (circle touching the middles of the square sides) and zoom the whole figure 4x. The square clearly has doubled its sides (and therefore, its area) and the circle has doubled its radius. Now, for the area of the circle we still need to say that the ratio of the areas remain constant for all shapes, which I am not sure is something we are "allowed" to assume.

[@MathTheBeautiful]

I did not see that simple argument, but I totally buy it!

[@akashpremrajan9285]

This video is completely excellent and brilliant. But could you clarify what is unclear about limits? The limit approach is very precise and cleverly avoids infinity. It just says no matter how close you want the answer to be to 4, I can make it that close. So either the answer does not exist (which is absurd, it clearly does). Or it's 4. Is it not?

[@akashpremrajan9285]

You actually said it is even more unclear with limits. I don't understand why that should be at all.

[@MathTheBeautiful]

It's a good question and I should make a video about it

[@akashpremrajan9285]

@MathTheBeautiful I guess the answer could be 4+dx, 4-dx, 4+(dx)^2, or things of that sort? Is that the concern?

[@anonymous_4276]

Any reference to know about how Cauchy's definitions etc "fool" us until we dig deeper?

[@MathTheBeautiful]

Unfortunately, no. It's just my opinion.

[@rickperez8975]

This was super good

[@MathTheBeautiful]

Thank you for letting me know!

[@YumekuiNeru]

I can sortof see how the argument is different in that area is not quite like circumference/length - but the argument still reminds me of the (incorrect) proof that pi=4 by drawing a 4-perimeter square around a unit circle and repeatedly folding the corners of it inwards so the path of the folded square approaches the border of the circle

[@MathTheBeautiful]

You're right - there are inherent contradictions everywhere

[@erikjohansson4021]

The core reason is the proportion a shape takes up as an inscribed shape in a rectangle. That ration, the area of rectangle over the area of the shape stays the same.

[@MathTheBeautiful]

Yes, I love that idea! It also shows how our intuition for objects with straight edges extends to curved edges.

[@orsoncart802]

Cue Bishop Berkeley and his “ghosts of departed quantities”! Let the ‘debate’ be rancorous! 😁 But seriously: 👍👍👍😁

[@Pluralist]

[@therealbrewer]

I'm not convinced that the "simple" geometric argument of filling up each circle with progressively smaller squares is easier to accept than some limit approach. Really, they are the same. I understand that you are trying to limit your methods to constructable geometry (without really explaining what that is...), but how do we know that the process of filling up the circle with smaller and smaller squares doesn't run into a problem eventually? It seems like an entirely faith based argument. I think the Borwein integrals (kzbin.info/www/bejne/bmaUhmhrbM9pfqcsi=p0DluukLgFfuzMQd) are a perfect example of unexpected outcomes when a process is repeated. To me, it is far easier to think of the square outside the circle, and how the fraction of the square's area that is covered by the circle must be the same at any scale.

[@MathTheBeautiful]

I welcome (and share) all of your doubts!