Hi there!!!

For some people, it is Mental Abuse to Humans. For others, it is the Most Amazing Thing in History. Truly, the word MATH is possibly the one which elicits the most varied response from students all over the world.

The entire student community is split into three, regarding this subject. The first type includes fanatics of the first degree, who would live, eat, sleep and breathe mathematics, away from other trappings of the material world (I count myself among them).

The second type are ambivalent towards the subject, as prepared to hate it as to love it.

The third type are the math haters (grrr), the people who would make all kinds of excuses, appointments and emergencies, just to avoid a date with the subject.

Like it or not, mathematics is an essential field in today's life. People may not like it consciously, but they would baulk if they were to find out how much maths they actually use subconsciously in real life. Like the proverbial octopus, mathematics has spread its hypothetical tentacles across fields and spheres of life. Without it, life as we know it...would not remain as we know it.

Nevertheless, in all these years of mathemania, we have unearthed some very interesting ideas, incongrous to what we know and so powerful, that if they were true, the fabric of mathematics would come undone.

For instance,

Let a = b = 1.

Then, a = b

or, ab = b²

or, ab – a² = b² – a²

or, a(b – a) = (b + a)(b – a)

or, a = b + a

or, 1 = 2

Ouch.

For those who did not understand how this is possible, a word of advice. This is NOT possible. To find the error in this seemingly sound argument, just substitute the values in the expression.

You will find that to reach the 6th line of the (ahem) proof, you are dividing the whole equation by (b – a), i.e. 0, which is not possible.

In another case, let's take a right triangle.

Now, we all know that the side c can be given as a² = b² + c², which is the Pythagoras Theorem.

Now, consider this case:

Now, the sum of all the small bits and pieces of the lines is equal to (a + b), agreed? Please understand this statement to comprehend what I shall state now. No matter how many small pieces I make, the sum of their lengths shall always be equal to (a + b).

So, if I make infinitely small pieces of those pieces, we can assume that it will form a right triangle just like this:

in which case, a = b + c. Which we know is wrong. So, where's the mistake?

If one argues that you just can't take an infinite no. of sides as a single side, then tell me, shouldn't the entire branch of calculus - that is based on the assumption that something can be divided into an infinite no. of parts which can be worked upon, and vice versa - be wrong?

Like these problems, there are numerous fallacies in mathematics which provide a lot of mind-scratching and brain-boggling to the casual reader. In this way, we do find abnormalities in mathematics. But then, we the students find abnormalities everywhere. 

Still, I would maintain that mathematics is the most correct and effective form of communication, because everyone understands the language of mathematics. Like one little known scientist once said before he gained renown, "Pure mathematics is, in its way, the poetry of logical ideas." That man was Albert Einstein.

I would be very interested if you can share any more interesting and strange problems you know of and would like to share.

Till next week, then.

Yours in absolute abstraction
A student