Why 2pi is better than pi

1  My opinions


1.1  2π is better than π


I spent an enjoyable few years studying maths. And during that time I kept seeing "2π" in formulas. I came to the conclusion that it is wrong that π has a special name and a symbol, and that 2π should have those privileges instead.


I didn't take it much further than that. Occasionally I would tell people my feelings on this issue in the pub, cos in those days I got to discuss maths in the pub all the time. I certainly didn't go on a massive internet crusade to do anything about it. Then years later, it was pointed out to me that this is not a new idea. And there is indeed a whole lot of people who agree with me. Check it out.


Today is the day when people stop what they are doing and remember "2π". Because in some countries the date looks like "6 . 28", which is the start of the decimal representation of this interesting value. Of course, the day is all rubbish, because some people like to write "28 . 6 . 2011". And there is nothing special about the decimal system either. Whereas the value is some magical intangible object which is completely independent of how we represent it in physical form.

Two pies

1.2 Why is it better?


Well, think about this. The circumference of a circle is πD where D is the diameter. Who uses diameters anymore? (Apart from midwives) Whereas, the circumference is also 2πr where r is the radius. And everybody likes to use radii. The above website will give you lots of information on the topic, so I am not going to bother. But in short, it boils down to the fact whenever you see a a 2π you have gone all the way around a circle. And when you see π you have gone half way, or magicked a half into your equation by some other means. (For example integrating - see below!)


2  Some maths


2.1  Area of a circle


Let's start with the junior school example, the formula for the area of a circle.

How do you derive this? How do you work out any area? You integrate! We want to integrate the constant function 1 over the inside of a circle of radius r, a region henceforth known as B_r for ball. The natural choice is to use polar coordinates, and changing to polar coordinates a sneaky r appears in the integrand.

2.2  Gaussian Distribution

Everybody's favourite probability distribution: the Gaussian distribution. If we take the "standard" one, then it has this formula:

What's that I see? A 2π. How does that get there. Well, being a probability distribution, the Gaussian has the property that

Looking at this another way, you might spot that the area under the curve

(1)

is actually √{2π}. Without going into too much detail, this 2π appears in the same way as the previous example, by integrating around a circle. It turns out that y(x) is very difficult to integrate, but if you square it and change to polar coordinates again, you end up trying to integrate

which, bizarrely, is possible. But not very interesting.


2.3  Stirling's approximation


Stirling's approximation gives you an approximation to the factorial function for large values. Here it is:

So you started by multiplying a few integers together and you end up with a 2π appearing in your maths. What!?! I did a little digging to see how this happens, and it is actually the same as the previous example, you need to evaluate the same integral integral (1). Perhaps it is not so incredible then?


2.4  Bernoulli numbers and the Reimann zeta function


The Reimann zeta function is an interesting thing, and people have written whole books on it. I just want to look at one of its many interesting properties. We only need to worry about the value of the zeta function for positive even integers, and for those values you can express the zeta function using this formula.

When you think about the zeta function at s = 2, you find you are summing the reciprocals of the squares. It's quite fun to work out the value of this sum, but I will just tell you.

Look at that, it's got π in it. Why? No idea. Let's try the same thing with s = 4.

That has a π⁴ in it. So one naturally senses a pattern forming after just two of these equations. And indeed, there is a pattern, it is usually written like this.

(2)

Where B_n are the Bernoulli numbers. Here are the first few...

If this is the first time you have seen the Bernoulli numbers then you might want to take (2) as their definition. Then you could incorporate the factor of 2²ⁿ into B_2n and you are left with a formula involving π and not 2π. However that would be a daft thing to do. The Bernoulli numbers turn up all over mathematics and most of the time, they aren't next to any pis, so you can't use them to sneakily try to change from π to 2π or back again. The appearance of 2π in this formula is pretty strong evidence that we have given a name and symbol to the wrong value.


2.5  The reduced Planck's constant


A little bit of history of physics now. Way back at the turn of the 20th century, before people had "invented" quantum mechanics, Max Planck discovered a relationship between energy and the frequency of some light and invented a new constant h to put in his equation. This came to be known as Planck's constant. Over the years, as quantum mechanics developed, physicists decided that it was better to measure the frequency of light, not in Hertz (cycles per second), but in radians. With the effect that everything gets multiplied by 2π. They found that h / 2π appeared all over their work, so invented a new constant

so that things looked neater. Even physicists want their work to look pretty.