Inspired by the fifth post (and others) of this thread, a non-mathematical explanation of pi. To really play along you’ll need some string (a shoelace will do), a ruler, and a few circular objects such as CDs, coins, a mug, etc.
Measure how big around one of those circular objects is by wrapping the string tightly around it. Now, keeping the string pinched at the appropriate point, use the ruler to measure how much string was needed. Write this number down, then use the ruler to measure the item straight across the middle and write that down too. Now divide the numbers (using a calculator if necessary) and write the answer down.
Repeat the whole procedure for circular items of different sizes. Notice anything? Whether you start with a tiny circle like a ring or a big circle like a table, the answer is always a little more than 3.
This is very useful information because, as you have just seen, it is a lot easier to measure across a circle than around it. So if you need to know how big a circle is all the way around, you can cheat by just measuring across it, then multiplying that by 3 and rounding up a bit.
In mathematics of course it’s nice to be a bit more exact than just saying “a little more than 3”. A long time ago somebody decided to name this number a letter of the Greek alphabet - pi. There’s nothing mysterious about pi being called a “constant” - think of that as a mathematical term for number. In fact, the number 2 is a constant, and so is one half, and so is one million, etc. Pi is just a number a bit bigger than 3 but a bit less than 3 1/2.
For those wanting exact terms: all the way around the circle is called the circumference. Straight across the circle (at its widest point) is called the diameter.
When I was in high school, Pi was explained graphically to me. It stuck.
The class was all ushered into the gym. There’s a large circle on the floor that is bisected by the free throw line. The teacher had students line up, shoulder to shoulder on the circle. Then, he had the same students line up shoulder to shoulder on the free throw line. And what do ya know? Only a 1/3 as many kids could fit. He than explained that another .14159 of a kid should still fit, but we weren’t going to cut up students.
I have no memory of being taught Pi in school, though I’m sure I was. My father taught it to me years earlier at home, using a drinking glass and a piece of string. He said it would be a great way to win bets at school. What he didn’t get was that if I was such a know-it-all at school, I’d get beaten up.
Some of my strongest memories of childhood were of my father sitting down with me to explain things like that. Those were some great bonding moments.
I’ve never understood it. It’s so confusing and subjective; the top crust (occasionally a lattice top), the bottom crust, the variety of fillings (apple, cherry, blueberry, pecan, key lime, peach), that weird little frill around the edges…Christ, who COULD understand it?!
You can also estimate the value of pi using a dartboard and a sufficiently bad darts player. Imagine you have a dartboard 2 feet across. The radius is half the diameter or 1 foot. The area of the dartboard is defined as pi*r^2 (pi times r squared) or 3.14 square feet. Now mount this dartboard on a square piece of wood that is 2 foot by 2 foot, or 4 square feet and call this combination the target. Now start throwing darts at this target at random. This is the important bit, each dart must have an equal chance of landing anyplace on the target. Count the number of darts that land inside the dartboard and divide this by the total number of darts that hit the overall target. Multiply this ratio by the total size of the target (4) and you get pi! You’re going to need lots of darts. I’ve simulated this in software and you need to throw around 100000 darts to get a value correct to 2 decimal places.
I strongly recommend Petr Beckman’s book A History of Pi.
Beckmann was more than a brilliant writer who could render the intricacies of irrational numbers intelligible. He was also a world-class crank you hated the Roman Empire the way he hated the USSR, who thought Aristotle was a worthless drag on World Progress, and who had his own newsletter (the pre-internet equivalent of a regularly updated webpage) pushing his ideas. A great read.
I just taught this concept to my 10 year old daughter using an old CD-R. I used a sharpie to draw an arrow pointing on the circumference of the CD. We then placed the CD perpendicular to some paper we spread out on the table, so the rim (with the arrow pointing down) was on the paper like a wheel. We marked the point on the paper, where the arrow on the CD was pointing, then proceeded to roll it along, until the CD made one complete revolution (until the arrow was pointing straight down again). We marked that point on the paper.
[-----------------------------------]
Now, I took a straight edge, and drew a line connecting the two points.
[-----------------------------------]
I then, took the sharpie, and drew a line, straight through the center of the CD, bisecting it. We placed the CD, flat on the paper, and registered the bisection of the CD across the line on the paper. Starting from one end of the line, we measured the diameter of the CD, and drew a mark on the line. We scooted the CD to that mark, and made another mark in the same fashion. We did it one more time, and drew one last mark. It came just shy of the end of the line.
[----------|----------|----------|–]
3.14…
There’s three diameters in there, plus that little extra. Voila! Pi.
She totally got it, and it was really cool seeing her grasp it.
I don’t understand pi. I mean, I understand the whole ratio thing. I don’t understand how the number can just not stop. The whole irrational number thing never made sense to me, and it’s been explained to me, by people who knew what they were talking about, so I’m willing to accept it’s true. But it just seems wrong to me…that it shouldn’t be that way.
For what it’s worth, I feel the same way about a lot of quantum physics, though, so…
Have you ever met someone who constantly babbles on and on about how Obama isn’t a US citizen, 9/11 was staged by Jews, and half an eye is no good? And you just can’t shut them up?
That person is irrational and goes on forever. Pi is like that. The little bastard.
Of course, the reason pi is so interesting is that it doesn’t just turn up in geometry. For instance, suppose you add up 1 + 1/4 + 1/9 + 1/16 + 1/25 + …, and just kept on going until the numbers you were adding were small enough not to matter. What you would end up with is a number equal to pi[sup]2[/sup]/6. It doesn’t seem like that series should have anything to do with circles, and yet somehow, buried deep in the mathematics, it does.
As long as we’re discussing tricky math concepts, can someone please explain to me why e^(Pi*i) = -1? I absolutely do not understand this. Pi is an irrational constant, and e is an irrational constant, and i does not exist. How can the three of them combine, like some freaky mathematical Voltron, to form such a nice, simple result?
What’s the decimal of 1/3? 1.3333333333333333333333333333333333…
Same sort of thing. It just doesn’t solve evenly.
What I can’t grasp is how they measure it out to a million decimal places. Get a really big cylinder and wrap a string around it?
Actually I guess you can do it mathmatically. You know the x[sup]2[/sup] + y[sup]2[/sup] = r[sup]2[/sup]. You can then approximate your circle as a bunch of (n) steps instead of a curved line and then measure the hypotenuse of all the steps. As the size of the steps get smaller, n gets larger. As the step size approaches o (and n approaches infinity) some calculus happens and you get your length of the circle’s path.
