I LOVE maths. I enjoy maths I can do, and I enjoy maths I can't do yet. I enjoy spotting patterns, puzzling over abstract ideas, playing about and reaching beautiful (but ultimately wrong) conclusions, and the gorgeous, exciting moment when everything suddenly falls into place. Terry Pratchett said writing was the most fun you could have on your own, but he's wrong.

There's a dark side to maths, though. It's supposed to be this hyper-logical, completely consistent1 language, so that if you were given all of the definitions, you'd be able to work out exactly what everything meant. It's no coincidence that most of the fathers of modern mathematics spoke German2. However, it's NOT. It suffers from the most dangerous kinds of inconsistencies, the kind you don't notice because you KNOW maths is consistent. Here are a few things that, when I am King of Maths, I will change.

I like fractions -- I got the hang of them quite early and didn't even realise they were meant to be hard until I was in my teens3. But the way you write mixed fractions: $3\frac{1}{2}$ simply doesn't make sense.

If you write things next to each other -- almost always456 -- you multiply them. $x \frac{y^2}{z}$ doesn't mean '$x$, add $y^2$ $z$ths', it means '$x$ times $y^2$ $z$ths'. It really should be written $3 + \frac{1}{2}$ to avoid the ambiguity.

Look at it, that smug little cross: $\times$. Ugly little bugger, isn't it? So easily mixed up with an $x$ or a $+$ if you have bad handwriting7, and synonymous with 'wrong'. It's a lousy symbol.

'What's the right thing to do, then, Colin?' I hear you wail. 'How would you fix it?' Thanks for asking. I would say that either the centred dot -- not to be confused with a decimal point, of course8 -- or, better yet, brackets, would be fine. $(7)(8) = 56$. Simple.

This is more technical than the ones before. If you've got as far as C2, you'll have seen the ugly $\sin^2(x)$ notation floating around. If you've got as far as C3, you'll have also seen the beautiful $f^2(x)$ notation. Those twos, horrifyingly, mean completely different things.9

$f^2(x)$ means 'find $f$ of $x$ and find $f$ of the answer.' Completely logical. $\sin^2(x)$ means 'find the sine of $x$ and square the answer.' The horror, the horror.

Even worse, $\sin^{-1}(x)$, if sine was treated logically, should mean the same things as $\cosec(x)$ -- but no, it means the inverse of sine, which is morally consistent with the $f$ notation.

When I rule maths, $sin^2(x)$ will be mean sine of sine of $x$. If you're going to square the sine of an angle, you will write $\left[\sin(x)]^2\right]$, like you would with $\ln$.

Degrees will be abolished in favour of the logical and circle-related radians. Anyone caught using degrees will have their maths card revoked.

This one causes me no end of trouble. 'Square root' has two, subtly distinct meanings that I'd never noticed until someone bamboozled me with it. Here's what happens when you talk loosely:

"$\sqrt{25} = 5$." True statement.

"If $x^2 = 25$ and you square root both sides, you get $x = \pm5$." Another true statement.

See the problem? $\sqrt{25}$ -- pronounced 'the square root of 25' -- is five and five only; but if you square root 25, you get a positive and a negative solution. The function 'square root' and the verb 'to square root' have slightly different meanings.

This may just be a problem of loose language -- probably there's a better term for 'undoing a square' than 'square-rooting' -- but it's a hard one to untangle. I don't know exactly what I'd do as Imperial Overlord Of Maths to fix this.

What are your biggest bugbears about mathematical notation? Where does the subject not do what it's supposed to do for you?

* Edited 15/12/2013 for formatting and to count footnotes.

- This is called 'trolling Gödel'. [↩]
- Gödel, for instance. And Liebniz, Hilbert, Gauss, Euler, Bernoulli (probably), and I'll chuck in Einstein, Schrödinger and Heisenberg just to make up the numbers. I'm barely scratching the surface of German-speaking mathematicians. [↩]
- One thing I'd like to install in my classroom is a QI-style buzzer that flashes and clangs whenever someone says "I don't like fractions." I have other triggers, too. It'd be a good investment. [↩]
- 'Almost always' is the most weaselly phrase in maths [↩]
- I appreciate the number of footnotes is getting silly [↩]
- This is what happens if you spend an afternoon with Tom. [↩]
- And most mathematicians do. [↩]
- The Europeans, sensibly, use a comma instead of a decimal point. If you use a space as a number separator instead of a comma, the dot is completely freed up for use as a multiplication symbol. [↩]
- I'm not the first to point this out. Babbage said something similar in the 1800s. But he didn't have nine footnotes in a blogpost. [↩]

## Cav

Another great read! I have all these bug bears!