# What is the use of Mathematics

Well, you might be asking this question in high school. You might have found that Math is a lot of formulae and manipulations similar to black magic in Algebra and wild imaginations in Geometry — I mean the proofs. So Math means prove this and that. Right?

I agree to some extent. Initially, it is sort of drab or *mechanical*. But, there is a good analogy. Think how you learnt writing the English alphabet — keep on drawing a big A, retracing it 10 times daily and perhaps, your Mom would have whacked you if you did not want to practise it. But, after you know English, the whole world of opportunities opens up for you; your vistas have widened. Exactly same is the case with Mathematics of high school and junior college. But, of course, there are applications of high school math which you learn later when you pursue…

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# Mathematics: A Very Short Introduction

By Professor Tim Gowers, Fields Medallist. Highly recommended for my young readers and students especially. Perhaps, it might also be recommended for Indian parents who want to know a bit more about Math and related exams like IITJEE Advanced.

Regards,

Nalin Pithwa.

# A cute little math pearl and some tips for studying for IIT-JEE or math Olympiads

Find out and compare roots of unity and roots of negative unity.

You should learn to play with such little gems of math on your own. This is creates ripples in the pond of the intellect, gently giving training to the subconscious mind. In this way we trick the mind rather create a natural uninterrupted flow of thoughts. It should not be that we can think of a math Olympiad or competitive programming contest or IIT-JEE problem only with so many textbooks in front of us… even in the bank queue or while taking shower one should be able to talk math to oneself…Of course, one does require physical solitude and also be introverted ( implies away from social media etc…)

Try it…and check for yourself…

Regards

Nalin Pithwa

# Check your mathematical induction concepts

Discuss the following “proof” of the (false) theorem:

If n is any positive integer and S is a set containing exactly n real numbers, then all the numbers in S are equal:

PROOF BY INDUCTION:

Step 1:

If $n=1$, the result is evident.

Step 2: By the induction hypothesis the result is true when $n=k$; we must prove that it is correct when $n=k+1$. Let S be any set containing exactly $k+1$ real numbers and denote these real numbers by $a_{1}, a_{2}, a_{3}, \ldots, a_{k}, a_{k+1}$. If we omit $a_{k+1}$ from this list, we obtain exactly k numbers $a_{1}, a_{2}, \ldots, a_{k}$; by induction hypothesis these numbers are all equal:

$a_{1}=a_{2}= \ldots = a_{k}$.

If we omit $a_{1}$ from the list of numbers in S, we again obtain exactly k numbers $a_{2}, \ldots, a_{k}, a_{k+1}$; by the induction hypothesis these numbers are all equal:

$a_{2}=a_{3}=\ldots = a_{k}=a_{k+1}$.

It follows easily that all $k+1$ numbers in S are equal.

*************************************************************************************

Regards,

Nalin Pithwa

# Observations are important: Pre RMO and RMO : algebra

We know the following facts very well:

$(x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}$

$(x-y)^{3}=x^{3}-3x^{2}y+3xy^{2}-y^{3}=()()$

But, you can quickly verify that:

$x^{3}+2x^{2}y+2xy^{2}+y^{3}=(x+y)(x^{2}+xy+y^{2})$

$x^{3}-2x^{2}y+2xy^{2}-y^{3}=(x-y)(x^{2}-xy-y^{2})$

Whereas:

$x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})$

$x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})$

I call it — simply stunning beauty of elementary algebra of factorizations and expansions

More later,

Nalin Pithwa