I don’t trust “experts”. I like to work things out for myself. The” experts” might not be experts at all, or they might be wrong, or they might be lying. And anyway, how do you tell if someone is an expert?
Last night I started wondering whether everything I’d learned from “experts” was wrong. And that maybe a lot of the mathematics I use is just plain wrong. There’s a number called π (or Pi) which is the ratio of the circumference of a circle to its radius, or maybe its diameter. And there’s a formula for working out the circumference of a circle using π. And there’s another one for working out the area of a circle using π.
I wondered if I could work all these things out for myself, ignoring untrustworthy mathematics textbooks, written by so-called “experts”.
The first puzzle was how to figure out the value of π. But as I was wondering how to do that, my eyes fell on the cylindrical ashtray in front of me. It was given to me as a present about 40 years ago, and intended to hold water when I was painting watercolours. But, being made of black glazed china, it was no good for that. And so I used it as an ashtray instead. And it’s been my main ashtray for the past 40 years. Amazingly, there’s not a single chip out of it.
I wondered if I could use it to empirically figure out the value of π. It was quite easy to find the approximate diameter of the cylinder with a ruler: it was almost exactly 100 mm. How could I measure its circumference? I got hold of a piece of string, and wrapped it round the ashtray, and marked the points where they overlapped with a felt pen. And then I took the piece of string and stretched it along the ruler, and found the distance between the marked points was 322 mm. If π = circumference/diameter, then π was 322/100 or 3.22.
I then thought that, since the piece of string I was using was several millimetres thick, that I might get a more accurate measurement using a length of fine cotton, and a biro rather than a felt pen to mark the overlap. This time I measured the circumference of the ashtray as 319 mm. And I had a new value of π as 319/100 or 3.19.
So after about 10 minutes I had a value for π of 3.19. And I could probably get an even more accurate value.
That was step 1. Step 2 was how to figure out the circumference of circle using π. And in fact that is built into the definition of π. For if π = circumference/diameter, then circumference = π × diameter. Or circumference = 2 × π × radius.
Step 3 was to somehow figure out the area of a circle. This looked like a pretty daunting problem. The only things whose areas I know how to measure are rectangles. I know that the area of a room that’s 6 m wide, and 5 m long, is 6 × 5 square metres, or 30 squ m. But what use was that in figuring out the area of a circle?
I wondered if it could be done using slices of a circle. If there were exactly 12 slices that made up a circle, then if I could find the area of each slice, the area of the circle would be 12 times the slice area. But I couldn’t see how to work out the area of a slice with an arc at one end. After all, I only know how to work out the area of rectangles.
But then I thought that if I used 1000 very, very thin slices, the arc at the end of the slice would get nearer and nearer to a straight line. And would look like this:
So I’d have a triangle with one side of length r, and another side of length c. And this triangle would have exactly half the area of a rectangle of side length c and r. I’d managed to turn a circular problem into a rectangular problem.
The length of c is 1000th of the circumference of the circle, which is 2.π.r. So it must be 2.π.r/1000.
So the area or rectangle r-c is r × 2.π.r/1000
And the area of the triangular slice is half this, or r × π.r/1000
And the area of the whole circle is 1000 times the area of a slice, or r × π.r (or π.r²).
Like I say, I don’t trust experts. I like to work things out for myself. And today I worked out that the circumference of a circle is 2 × π × radius, and that the area of a circle is π × radius × radius, and that π is roughly equal to 3.19.
And if any “expert” tells me anything different, I won’t believe a word he says, about anything, ever. I’ll only start to respect him a bit if he agrees with me – at least about the circumference and area of circles, and the value of π.