Yeah, I feel always the urge to poke my eyes out when seeing someone violating maths by doing one of these following mistakes...
The list isn't exclusive, there's more, but I can't think of more in once without passing out (seriously, not kidding...)
Putting 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 more withing 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 to the 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
Ok, the answer is correct, I even get your way of thinking, it's excellent, but once again, 100 + 20 doesn't 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 =>), that 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 like?
100 + 20 => 120 * 3 => 360 + 50 = 410 (yes, the last equal sign is ok, 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 decimal point are crazy and either are countless or worse, really never ending (yeah, that means irrational numbers - not talking about recurring numbers, we know how they look like "at the end"), you cannot just cut the tail and walk away.
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 of some research about colour preferences 45.56% for red 12.89% for blue and 41.55% for yellow. If I want to make those number "better looking", I decide to get rid of the decimals... ok, 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 numbers you need/want and do it properly. Don't just cut it off like a gangrenous limb.
How should it look like?
Well, in our case, it's actually 45.5%, 13% (or 13.0% to stick to the same number of decimals) and 41.5%. Because rounding to a whole number when having three numbers and expect to fit in a 100% is unrealistic... But even when leaving even 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.
Tiny note about rounding: To make the subject even more complicated, in terms of statistics, we have a bit different rule for rounding, to avoid those problems (on the other side, we could even 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" actually goes down.
And also when thinking about "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 you friends be thirsty at the end =)
Saying "nothing" when got 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, depends on how do you take it).
Can you imagine the craziness when 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've touched that a bit while studying at the university - trust me, be happy with your A-level maths). We wouldn't have neutral balance - it would be always 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'Hospital's rule and others. Just because someone needs to divide by "nothing". Err, I meant by zero.
How to fix it?
Just don't say it. Say proudly "The result is zero."
How should it look like?
Well, I should look like you understand, that zero is a valuable member of the number family. If you have zero on your account balance, then ok, you can say "There's nothing..." But not when talking about maths. Please.
So, this was 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 now when you know what it could cause). Anyway, three cases behind me, more on their way...
