Forget pi, it's cos squared that's wrong!
For a while now, a debate has been raging about whether we should scrap using pi in all our equations and instead write everything in terms of tau (which is 2 pi). Most of the time I stand at a distance from this debate, thinking it rather tedious and preferring instead to fun things with pi like draw its digits in chalk on the footpath. But every so often I get involved.
The last time I got involved, I made a video satirising the whole thing and suggesting that both pi and tau are wrong and we should instead use eta (which is pi/2). You can see it here. Every so often I check on the video to see how it's going and I read with some amusement the comments people have posted there. In one of these comments I learned that Michael Hartl, the main advocate for Tau with his Tau Manifesto, has actually added a reference to me in the latest version of his Manifesto, saying that eta is not that bad a choice after all for certain applications! How ironic (in a gratifying sort of way).
Anyway, because of this I have been drawn back into the debate again and have found myself watching YouTube videos and reading blogs on the topic, and commenting on these videos and blogs.
The time has come to write a blog post of my own...
One of the main arguments that our tauists put forward for using tau is that it makes teaching trigonometry easier. They claim that using pi is one of the main reasons people don't understand radian measure when learning trigonometry, saying that it's silly for a full turn to be 2 of something.
And this is where I am compelled to make this blog post: all that may possibly be true, but if you're going to pick something to fix in the way we teach trigonometry, which constant you use to describe how far it is around a circle is not the thing to pick. The pi versus tau issue pales in comparison to this little gem, which is a fundamentally wrong thing to write and causes all sorts of confusion:
| cos2 x + sin2 x = 1 |
The first and most basic confusion is that students are forever typing "cos^2 (x)" into their computer-marked assignments, and then asking me to help them because the computer doesn't mark it as correct.
And the reason the computer isn't marking it as correct is of course that the computer does not recognise that as a legitimate thing to write, for the very simple reason is that it's NOT a legitimate thing to write! The symbols cos and sin are functions which means that the true meaning of cos²(x) ought to be cos(cos(x)), as it is in all other situations when we use powers on functions. If we told them this is what it meant, then they wouldn't think it was the answer to their assignment question. I also believe it might make it a little easier for them to understand why cos⁻¹(x) is in fact the functional inverse of cos and not 1/cos(x).
On the other side of the coin, we don't do this with any other functions do we? We don't say either of these do we:
| (exp(x))2 = exp2(x) or (√ x)2 = √ 2 x |
That's ridiculous. Why do it for trig functions?
And finally, are we really that lazy? I went looking on the internet for the reasons why people write the abomination above, and every one of them just says "for brevity". Really? For brevity!? Honestly, it may be breif to you but it uses up hours of your students' and my time – time that could have been saved with four strokes of your pen!
Ok, so I got more and more passionate and less and less cohesive in each paragraph there, but I think the point still stands. I think the teaching of trigonometry and a lot of other things too would be much easier if we all just wrote:
| (cos x)2 + (sin x)2 = 1 |
These comments were left on the original blog post:
Sam Cohen 6 April 2013:
I like what you say, David, but I just wanted to flag up that I regularly write X^2 to mean the square of X, even when X is a random variable (and so, in my language, a function of the outcome \omega). I would argue that we do, frequently, want to write f^2 for the function x->(f(x))^2, particularly when thinking of functions as key objects in their own right, rather than as secondary entities to numbers, etc… Just my two cents.
David Butler 7 April 2013:
Fair call with the random variable thing Sam, if you view a random variable as a function from Omega to the real numbers. And yes, with functions like polynomials, we usually consider a polynomial as an abstract object constructed by addition and multiplication of other basis polynomials. So if p is a polynomial it probably does make more sense for p^2 to mean (p(x))^2. *sigh* It’s never simple is it? Still, I wish people gave this sort of reasoning rather than just “brevity”. 😉
Tomas 23 November 2013:
I love debates about Pi ,or Tau …. Pi or Tau were , are and will be always wrong and not accurate numbers. You all know this omfg 🙂 Pi was made up first with physical measurement .. you are unable to find it with any method , and juts still using this crap. I simply don’t get it. Start to think outside the box and find the real constant and new equation how to count circumference and stop using this made up crap and endless debates about, which are not solving anything (no offense). Just one advice 10:3= 3,2 this is the right result, solve it ,and you will maybe open new dimension of thinking and understanding for yourself . There is a reason ,why we using our stupid decimal math system . Peace. 😉