Yeah, I always feel the urge to poke my eyes out when I see someone violating maths by making one of these following mistakes and illegal maths moves...

The list isn't exclusive; there's more, but I can't think of more at once without passing out (seriously, not kidding...).

Putting an Equal Sign Inappropriately:

If it doesn't equal, then it DOESN'T EQUAL, so don't write the equal sign there.

If you want to do any operation with a number of your choice, please, be my guest. But when you do another operation, we're moving somewhere else, and we're no longer within the reach of the original number... Come on, don't tell me it doesn't matter.

Let's say there's a word problem: You and your two friends go on a trip. Two nights in the hotel will cost each of you £100. The return train ticket for each is £20, and you booked a tour guide for one day for £50. How much will the trip cost you in total?

100 + 20 = 120 * 3 = 360 + 50 = 410

Okay, the answer is correct; I even get your way of thinking – it's excellent! But once again, 100 + 20 does not equal 410.

Because look what you've done: 100 + 20 = ... = 410. (Not even thinking about the chaos in the middle.) No way. It doesn't. Period.

How to fix it?

Use implication (that's the arrow, like this one =>). This means we use something on the left side to create something out of it on the right side, and we can use it straight without fear that Veronika (or any other maths nerd) will poke her eyes out and run away screaming...

You can also use separate equations for each step.

How should it look?

100 + 20 => 120 * 3 => 360 + 50 = 410 (Yes, the last equal sign is okay, because 360 + 50 really IS 410)

or

100 + 20 = 120

120 * 3 = 360

360 + 50 = 410

Tadaa!

Wrong Rounding:

Or, better to say, no rounding at all. If the decimals behind the decimal point are crazy and either are countless or worse, truly never-ending (yeah, that means irrational numbers – not talking about recurring numbers, we know how they look "at the end"), you cannot just cut the tail and walk away (that's called truncation – which is ok on some cases, but not when you're asked to round mathematically).

The rules for rounding decimals are here for a reason. The best way to understand is to think about this easy example:

I have results from some research about colour preferences: 45.56% for red, 12.89% for blue, and 41.55% for yellow. If I want to make those numbers "better looking," I decide to get rid of the decimals... Okay, so without thinking, I say 45% red, 12% blue, and 41% yellow.

Hmm... that means 45% + 12% + 41% = 98% – woohoo! Where are the missing 2%? Did they disappear? BAD ROUNDING, my dears, bad rounding...

How to fix it?

Just do your rounding! Decide how many significant figures you need/want and do it properly. Don't just cut it off like a gangrenous limb.

How should it look?

Well, in our case, it's actually 45.5%, 13% (or 13.0% to stick to the same amount of decimals), and 41.5%. Because rounding to a whole number when having three numbers and expecting them to fit into 100% is unrealistic... But even leaving just one decimal simply by cutting the rest without thinking would lead to an error of 99.8% in total, so you still need to do your rounding properly – you want to have your cake whole (100%), not with one bite missing (99.8%); it looks really bad.

A tiny note about rounding: To make the subject even more complicated, in terms of statistics, we have a bit of a different rule for rounding to avoid those problems (on the other side, we could actually go over 100% when adding up to one whole, that's even crazier, right? =) So, in statistics, we round a bit differently; every other "5 goes up" rule actually goes down to equal things out and allow for matching full 100%.

And also let's mention "natural rounding": When we do our maths and see we need to buy 4.25 bottles of water to prepare enough drinks for all our friends coming to the party, of course, we'll buy 5 bottles, even when pure maths says rounding to a whole number in this case gives us 4. But then you'd be missing a bit, so always think first; do not let your friends be thirsty in the end! =)

Saying "Nothing" When the Result is Zero:

Zero is a number, come on! Why do you think you have "nothing" when you have zero as a result? In maths, zero isn't "nothing": An empty set has nothing in it, but if there's a zero, then it's got something (so yes, it's got that zero), and we can't say it's an empty set anymore. When getting nothing as a result of an equation, that means we really have no roots at all. If you get zero, that's still a lovely root (and very often brings some special features with it, which causes additional fun... or despair, depending on how you take it).

Can you imagine the craziness of not having a zero? There were ancient cultures without this important number... Yay, I wouldn't want to study their maths, really. We would miss the origin of any plots in geometry – 2D, 3D, or more-D, like nD (yes, there's n-dimensional geometry; I touched that a bit while studying at university – trust me, be happy with your A-level maths)! We wouldn't have a neutral balance – it would always be either positive or negative.

And there wouldn't be any fun while trying to divide by zero (well, never do that anyway!) – all the hours spent with limits and L'Hôpital's rule and others. Just because someone needs to divide by "nothing." Err, I meant by zero.

Speaking of L'Hôpital's rule... Anyone having a chuckle? No? Is it just me?

Right... let's get back to the "nothingness of zero".

How to fix it?

Just don't say it. Say proudly, "The result is zero."

How should it look?

Well, you should look like you understand that zero is a valuable member of the number family. If you have zero in your account balance, then okay, you can say, "There's nothing..." But not when talking about maths. Please.

This was actually just the first round. For you – to have maybe fun, maybe learn something (either how to improve your maths or how to cause me a headache on purpose, if that's what you're after, because now you know what it could cause). Anyway, three cases behind me, more might be added later, depending on how quickly I recover from writing (and thinking) about these first ones...