Reunion - Lesson 4 - On Circles and Giving Up - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act II

4. On Circles and Giving Up

4.10. Reunion

In this chapter we failed a lot, and we learned a surprising amount from our failures:

1.We attempted to figure out how much of a square was taken up by a circle. We eventually gave up and decided to simply give a name to the unknown answer. We defined our give-up number # by the sentence

2.We ran into a similar problem with a different give-up number , defined by the sentence

3.Despite our ignorance of both these numbers, we managed to figure out that they have to be the same:

Knowing now that these two numbers were really just one number, we decided to call it .

4.The standard textbooks use the symbol π to refer to what we’re calling . We still don’t know what specific number is, so we decided to keep our ignorance visible, refraining from calling it π until we’ve figured out how to calculate it for ourselves.

5.We discussed another problem, the reverse shortcut distance dilemma, and failed to solve it as well. As before, we encountered the surprising fact that two problems, neither of whose answers we know in isolation, can be shown to have the same answer:

6.Motivated by the discovery above, we chose the simpler abbreviations

and saw that textbooks refer to these machines by two rather archaic names:

7.We saw that textbooks have an additional set of goofy names for various simple combinations of V and H. These names included “tangent,” “secant,” “cosecant,” and “cotangent.” We won’t be needing these terms, so you’re welcome to forget them. If we ever encounter the concepts they refer to, we’ll just write them in terms of V and H.

8.At this point our only way of describing V and H is by drawing pictures. However, we still managed to figure out their derivatives by drawing pictures. We discovered that

9.This chapter provided our first direct encounter with the fact that the “prerequisites” to calculus often require calculus itself in order to be fully understood. It won’t be our last.