Math: Problem of the day-2

Dec 18th

Posted by admin in Algebra

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This problem was asked in IMO 1970, Hungary.

Q: For what natural numbers n can the product of some of the numbers n,n+1,n+2,n+3,n+4,n+5 be equal to the product of the remaining ones?

Note that since there are 6 numbers under consideration, if a prime number p divides any of these numbers, it cannot be greater than 5.

In the set of numbers n+1,n+2,n+3,n+4, there can be no common prime divisor greater than 2 or 3. Also, since two of these four numbers must be odd, they must be powers of 3. But this isn’t possible since no two powers of 3 differ by 2.

Therefore, there is no such n

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Math: Problem of the day-1

Dec 17th

Posted by admin in Geometry

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This problem was asked in IMO 1961, Hungary.

Q: Let a,b,c be the lengths of a triangle ABC whose area is S. Prove that a^2+b^2+c^2 \geq 4S \sqrt{3}. In what case does equality hold?

Solution: We can express the area in terms of the sides and angles of the triangle:

S=\frac{1}{2}bc sin A

Also, the cosine rule says that

a^2=b^2+c^2-2bc cos A

Therefore,

a^2+b^2+c^2 \geq 4S \sqrt{3} \iff b^2+c^2 \geq bc( \sqrt3 sin A + cos A) \\ \iff (b-c)^2+2bc(cos (A-60^0) \geq 0

Note that equality will hold above if b=c and a=60^0, that is, when the triangle is equilateral.

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On infinity

Nov 12th

Posted by admin in Reflections

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Infinity, represented as ∞ is a concept in mathematics and philosophy that refers to a quantity without any bound. Infinity is a strange concept, and has many intriguing implications.

  • A subset of an infinite set can be put into one-to-one correspondence with the original set. For example, the set of natural numbers and the set of even numbers are in one-to-one correspondence, even though the set of even numbers is a subset of the set of natural numbers. This rejects the concept that the whole cannot be the same size as the part.
  • The paradoxical nature of infinity can be illustrated using the idea of a grand hotel, which has infinitely many rooms, and all of them are filled with guests. Yet, a new guest can be accommodated by moving each of the existing guests, one by one, to the other rooms.
  • There exist different types of infinities. The infinity of natural numbers is different from the infinity of real numbers, for example. The set of naturals is countably infinite, while the set of reals is uncountably infinite.
  • The real line is the same size as any of its non-zero segments.  For example, the set R can be put into one-to-one correspondence with the set (-1,1) by defining a function that maps R to (-1,1). One example of such a function could be
    • f(x)=\frac{2}{\pi}tan^{-1}x
  • If there are different kinds of infinities, one might think that there will exist a biggest infinity. However, that’s not true. It can be mathematically proven that whatever infinity one might create, there will exist a bigger infinity.
  • Infinities have very interesting roles in cosmology. One question that physicists face is whether physical infinities actually exist. Is the universe finite, something like the earth, so that if you start moving from one point you will eventually return to that point again, or is it unbounded, so that you can keep moving in some direction for infinite time? Does space go on forever is an important question in cosmology.
  • Another question facing physicists is whether time is infinite or not. Has time always existed – or was there some initial point of time, t=0. Our present universe might keep on expanding forever, like it is doing presently, in which case we have an infinite amount of time ahead of us, or it may reverse its expansion, start contracting and end in a big crunch, which would effectively end time.

Perhaps the finite human mind will never be able to grasp the concept of infinity completely. George Cantor, one of the world’s leading mathematicians who worked on infinity, once said: “I see it, but I don’t believe it”.

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Is God a chemist

Oct 6th

Posted by admin in Chemistry

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Experimental science is difficult to comprehend and Chemistry is one of the oldest experiment based science. There are innumerable examples of chemical reactions in our daily life. The chemical reaction between the LPG and the air in the kitchen keeps the flame burning. The flame gives heat energy for cooking and even the food we cook is the end product of several chemical reactions- hydrolysis, oxidation, calcination, disproportionation and decomposition. The air we breathe in, burns carbohydrates and produces energy.  The color of flowers and the taste of fruits is all through a complex chemical reaction. Hundreds of chemical reactions are taking place within our body and in our surroundings all the time. Chemical reactions touch all aspects of our life – food, shelter, clothes, medicines, education, transport and recreation.

What would have been the first chemical reaction that occurred on this earth? What we know from our recent knowledge confirms that life started with a chemical reaction, product of which combined with other elements to create new reactions. Thus initiating a chain of reactions and ultimately after a long span of hit and trial gave the most stable product, humans, which can think for its own stability ( read survival). First chemical reaction performed by humans must have been the creation of fire. So you see everything is related through chemical reactions, one reaction initiating the other and supporting yet another. This world is moving via chemical reactions. We are living in a dynamic world indeed a dynamic world to its very core. Be it atoms, molecules, compounds, bacteria, algae, fungi, plants and vegetation, animals, humans or even the inanimate objects such as stones, walls or buildings, everything is dynamic and is a part of the chemical reaction. Our whole universe can be derived to be a chemical reaction.  The universe started with a big bang and since then is moving in the forward direction.

Scientists have often wondered about the fate of the universe. What is the lifespan of our universe and how is it going to die. Since our universe is chemical, answering this question through chemistry might be helpful. When would a chemical reaction stop? Ask this to yourself and if you have the basic knowledge of thermodynamics, you will find the answer to be extremely easy. Same theory (obviously not in the simple format) can be applied to the universal reaction which we are part of.

When we consider the intricacies of the molecular basis of our lives, we will feel that God is not a physicist,biologist or a mathematician, but is basically a practising chemist who has mastered other physical sciences and used them to perfect His creations.

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Physics: Problem of the day-19

Jun 18th

Posted by admin in Heat and Thermodynamics

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Q. A gas takes part in two thermal processes in which it is heated from the same initial state to the same final temperature. The two processes are represented on a p – V diagram by straight lines 1 →2 and 1 → 3:

img42

Figure-1

In which process is the amount of heat supplied to the gas larger?

The amount of heat exchange, by the first law of thermodynamics, is given by

img42

Figure-2

Since the final temperature of the gas is the same in both the processes, Δ U must be the same in both cases. We then simply have to compare the work done in the two cases:

img42

Figure-3

Therefore, heat supplied to the gas is more in the process 1 → 3.

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