Monthly Archives: January 2015

A mathematician, a physicist and a layman

What exactly is the difference between a mathematician, a physicist and a layman? Let us suppose that they all start measuring the angles of hundreds of triangles of various shapes, find the sum in each case and keep a record. Suppose the layman finds that with one or two exceptions, the sum in each case comes out to be 180 degrees. He will ignore  the exceptions and say “the sum of  the three angles in a triangle is 180 degrees.” A physicist will be more cautious in dealing with exceptional cases. He will examine them more carefully. if he finds that the sum in them is somewhere between 179 degrees to 181 degrees, say, then he will attribute the deviation to experimental errors. He will then state a law, “The sum of the three angles of any triangle is 180 degrees.” He will then watch happily as the rest of the world puts his law to test and finds that it holds good in thousands of different cases, until somebody comes up  with the a triangle in which the law fails miserably. The physicist now has to withdraw his law altogether or else to replace it by some other law which holds good in all the cases tried. Even this new law may have to be modified at a later date. And, this will continue without end.

A mathematician will be the fussiest of all. If there is even a single exception, he will refrain from saying anything Even when millions  of triangles are tried without a single exception, he will not state it as a theorem that the sum of the three angles in any triangle is 180 degrees. The reason is that there are  infinitely many different types of triangles. To generalize from a million to infinity is as baseless to a mathematician as to generalize one to  a million. He will at the most make a conjecture and say that there is a strong evidence suggesting that the conjecture is true. But, that is NOT the same thing as proving a theorem. The only proof acceptable to a mathematician is the one which follows from earlier theorems by sheer logical implications. For example, such a proof follows easily from the theorem that an external angle of a triangle is the sum of the other two internal angles (or, by a suitable construction and using the properties of parallel lines, which in turn can be proved from axioms of plane geometry).

The approach taken by the layman or the physicist is known as inductive approach, whereas the mathematician’s approach is called the deductive approach.

In the former, we make a few observations and generalize. In the latter, we deduce from something which is already proven. Of course, a question can be raised as to on what basis this supporting theorem is proved. The answer will be some other theorem. But, then the same question can be asked about the other theorems Eventually, a stage is reached where a certain statement cannot be proved from any other proved statements and must, therefore, be taken for granted as true. Such a statement is known as an axiom or a postulate. Each branch of mathematics has its own postulates or axioms. For example, one of the axioms of plane geometry is that through two distinct points, there passes exactly one line. The whole beautiful structure of  plane geometry is based on five or six axioms such as this one. Every theorem in geometry can be ultimately deduced from these axioms.

More later

Nalin Pithwa

Basic Algebra for IITJEE Main and RMO

More basic algebra for you guys who are thirsting for more…The following is a nice problem indicating some basic concepts or tricks in problems involving logarithms/powers.

Solve for x: $4^{x}-3^{x-(1/2)}=3^{x+(1/2)}-2^{2x-1}$ (IITJEE 1978)

Solution:

Writing $4^{x}$ as $2^{2x}$ and bringing powers of the same number on the same side we get, $2^{2x}+2^{2x-1}=3^{x+(1/2)}+3^{x-(1/2)}$

The first term on the LHS can be written as $2^{2x-1} \times 2$, and hence, a common factor of $2^{2x-1}$ comes out from the terms on the LHS. As for RHS, we can rewrite the first term as $3^{x-(1/2)} \times 3$ and then the factor $3^{x-(1/2)}$ comes out as common. So, we get $2^{2x-1}(2+1)=3^{x-(1/2)}(3+1)$, that is, $3 \times 2^{2x-1}=4 \times 3^{x-(1/2)}$

Bringing all powers of 2 to  the left and all powers of 3 to the right, we get $2^{2x-3}=3^{x-(3/2)}$

By inspection, $x=3/2$ is a solution. But, how do we arrive at it systematically? Also, how do we know that there is no other solution? It is tempting to try to do this by saying that a power of 2 can equal a power of 3 only when are both are equal to 1. (Such a reasoning is indeed useful in solving equations in Number Theory where we mostly deal with positive integers and their factorization into integers). But, here it is inapplicable because we do not  know that the exponents are integers. Instead, let us express both the sides as the power of the same number. One way to do this  is to write 3 as $2^{\log_{2}3}$ in the RHS. Then, we can get $2^{2x-3}=2^{(\log_{2}3)(x-(3/2)}$

As the bases are the same, the equality of powers implies that of the exponents. So, we have $2x-3=(\log_{2}3)(x-(3/2))$

This can be solved easily to give $x=\frac{3-(3/2)(\log_{2}3)}{2-\log_{2}3}$

which simply equals $3/2$.

Hence, $x=3/2$ is the only solution.

AM GM inequality for IITJEE Main/Advanced, RMO and INMO Mathematics Hothouse

Here’s our classic hotline Wikipedia on various aspects of the AM-GM inequality. There are about 50 known proofs of the AM-GM inequality.

You may go through one or two proofs at a time, understand/read it very well, and reproduce the proof in your notebook without seeing. This is a way to learn Math.

Suggestions, comments, questions are welcome and even encouraged 🙂

More later…

Nalin Pithwa

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AM GM inequality for IITJEE Main/Advanced, RMO and INMO

Here’s our classic hotline Wikipedia on various aspects of the AM-GM inequality. There are about 50 known proofs of the AM-GM inequality.

You may go through one or two proofs at a time, understand/read it very well, and reproduce the proof in your notebook without seeing. This is a way to learn Math.

Suggestions, comments, questions are welcome and even encouraged 🙂

More later…

Nalin Pithwa