Exceptions in mathematics

Before you read -
There are some things which one may not understand in this post. Students can ask there mathematics teachers/parents/friends about what is written. The adults may consult anyone (or anything) they like...




"Exceptions are always there..."

If you study chemistry as a main subject in your 11th and 12th standard here in India, you are going to hear the above phrase more than any other student. Why? Because chemistry is full of exceptions (and this is the reason why I don't like it - in general). See, we have never seen an atom, have we? So, things are kind of 'assumed' on the basis of observations (the assumptions may be correct) and exceptions arise. 

Then you may look up physics. It you don't go to a higher level and talked about physics thought at 11th or 12th standard, you will see that major of it is all 'ideal physics'. 

There are situations which we can not realize in practice. But i am not saying its useless. This physics is the 'base level'. When you know how to deal with ideal conditions, then only you will be able to deal with 'real' conditions. 

Talking about exceptions, I don't know if physics have any exception here on 'earth'. All laws of physics are applicable on earth and objects near it. But what about the universe? I have heard that the laws of physics do not hold good at some parts of the universe. So, shall we say that even physics have exceptions? I don't know... We may consider them as exceptions, we may not.

Let us now com to the main point. Are there exceptions in mathematics?

Well, my answer is YES and NO.

confused?

See.. its how you treat those 'exceptions'. You may consider them exceptions, you may not. It's up to you...

Mathematics (in my view) have the most number of assumptions than any subject. As i see it, it is an universally accepted assumption that 2 comes after 1 and that 5 precedes 6. But that is 'the magic' of mathematics. If you want to multiply 39 with 11 , you don't need to add 39 eleven times but just simply multiply as follows - 

39 * 11 = (30 * 11) + (9 * 11) = 330 + 99 = 429..

There are many ways to do this of course...

What i want to say is, you don't need to break down the problem in really small bits, solve it and then compile them again. If you have an identity or formula, you simply substitute the values and you have the answer. Sign convention etc. are all takes care off automatically and you never need to bother about it. 

Take trigonometry for example - 

sin(pi) =  sin (pi + 0) or sin (pi - 0)

Now, some can get confused what to use. See.. sin(pi + x) = - sin x while sin (pi - x) = sin x.

i.e. above pi, sin is negative and below it sign is +ve. As we see, pi divides the two 'regions'. So, do we take +ve or -ve at pi itself? 

I does not matter. That is the thing. Either way, answer comes out to be 0. Which is neither -ve nor +ve. 

This is how everything just fits in. Or does it?

We just saw that everything is taken care of automatically. Since this is the case, there is no question to be raised about mathematics having exceptions.  But there are. 

FACT : many things that people think are exceptions in mathematics are actually not exceptions.

take 0! (0 factorial) for example. We 'take' it as 1. actually, its a calculated figure (read my post 'factorial of negative numbers' to know how).

So, this is not an exception. 

What I consider as exception are the assumptions we make. Like a straight line being a part of a circle which has a radius equal to infinity. A 'point' has no dimensions. A line has no breadth or height, its just a length.

Again, 'infinity' can not be defined but still comes in use at various places. Anything divided by 0 is taken as infinity ans anything divided by infinity is zero are some.
But then, why you will not call them exceptions because they don't really contradict any 'thing' or any 'trend'. 

These are just extra tools used in mathematics. 'Assumed Tools'.

This is the fun in mathematics, everything just works in a 'harmony'. If you have the power to analyze and control this 'harmony', there is nothing more fun than mathematics.... 

(ps: I dont have any such power but I don't know why I still enjoy it)