Monthly Archives: July 2014

Math Basics A Fallacy in Geometry

You might think that crazy things like 1=2 can be *proved* if we make basic blunders in Algebra. But, in geometry also you can *prove* wild things like a right angle is an obtuse angle — errr…hmm…provided you make flaws in some fundamental assumption/axiom/theorem/property. So, check how strong are your basics in Euclidean/plane or high school geometry — point out my mistake in the proof “a right angle is an obtuse angle!”.

Please download the JPEG images of the two page *proof* 🙂page1geometryfallacy page2geometryfallacy

More later,

Nalin

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 your undergraduate program in Physics, Chemistry or Engineering and even Finance/Economics.

You might be amazed to know that there are about one hundred uses of Quadratic Equations. But, you have to progress further much beyond high school to encounter them. There is a well known quote by an immortal math genius, Carl Friedrich Gauss: Mathematics is the queen of all sciences, but actually, it is the hand-maiden of all sciences.

For those of  you who are more curious, let me reproduce an article on the same topic in the editorial of The Hindu some time back. Read and think — do you want to be a mathematician?

I also strongly recommend to you to read the classic book “Men of Mathematics” by E. T. Bell. It’s an old text about mathematicians from 2500BC up to the 20th century, but still widely read and available in e-stores like Flipkart or Amazon India.

More later,

Editorial the importance of mathematics

The Hindu

Aug 22 2010.

The fields of work of the seven recipients of the four high global awards at the ongoing International Congress of Mathematicians (ICM) at Hyderabad are indicative of the fast-disappearing boundaries between pure and applied mathematics. The field of mathematics has evolved tremendously since the great G.H. Hardy proclaimed in A Mathematician’s Apology that it was the very fact that pure mathematics had no practical applications that made it beautiful and of permanent aesthetic value — as against applied mathematics, which was dull and trivial. It was for the same reason that mathematics did not enter Alfred Nobel’s mind when he established the Nobel Prize primarily to honour inventions and discoveries of great practical benefit to humanity. The Fields Medal, the most important of the mathematics awards and regarded as the equivalent of the Nobel Prize, is traditionally given for exemplary achievement like solving an outstanding problem of great significance in pure mathematics. In recent years, this too has begun to recognise mathematical achievements in problems arising in physics and other subjects. This is very much in evidence in this year’s awards.

The institution by the International Mathematical Union of awards other than the Fields Medal to recognise significant mathematical achievements in information theory and other technologically important areas is yet another indicator of the increasing relevance of mathematics to diverse fields. The Rolf Nevalninna Prize and the Gauss Prize this year have honoured improved error-correcting codes in communications, which have applications in high speed modems, and the mathematical theory of wavelets, which has resulted in efficient data compression in imaging technologies with applications in digital movies and space-based astronomy. Public-key encryption, which is widely used in data security, is rooted firmly in number theory, Hardy’s field of specialisation, which he said was beautiful but of little practical value. Recent developments in theoretical computer science, financial mathematics, and derivative pricing are examples of important emerging applications in mathematics. Indeed, some very important talks in the Hyderabad Congress are in areas of pure mathematics inspired by problems in applied mathematics. It is this wonderful duality of mathematics — the joy of pursuit of pure mathematics for its intrinsic aesthetic experience, and its increasing relevance to real-life problems — that must be projected in greater measure to school and college students. It must become an essential component of mathematics education to promote the idea that successful careers are possible through the pursuit of mathematics.

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

 

Math Basics Division by Zero

Let’s pause Geometry for a little time and start thinking of some basic rules of the game of Math. Have you ever asked “why is division by zero not allowed in Math?” Try to do 1/2 in a calculator and see what you get!!

This was also a question an immortal Indian math genius, Srinivasa Ramanujan had asked his school teacher when he was a tiny tot. Note the following two arguments against the dangers of division by zero:

(a) Suppose there are 4 apples and two persons want to divide them equally. So, it is 4/2 apples per person, that is, 2 apples per person. But, now consider a scenario in which there are 4 apples and 0 persons. So, how can you divide 4 apples amongst (or by) 0 persons? You can think of any crazy answer and keep on arguing endlessly about it!!!!

(b) The cancellation law ac = bc implies a = b does not work when c = 0. For instance, the identity 1 x 0 = 2 x 0 is true, but if you carelessly divide both sides of the equality by 0 then you will obtain 1 = 2, which is nonsense. In this case, it was obvious that you are dividing by zero; but, in other cases it can be more “hidden”.

Let me show you an example of where it can be “hidden”.

Now, multiply both sides of this equation by c. Then, you get ac=bc. Then, add ab - b^{2} to both the sides to get ac + ab - b^{2}=bc+ab-b^{2}. Hence, you get ab-b^{2}=bc+ab-b^{2}-ac. Factorizing, you obtain

 

b(a-b)= (b-c)(a-b)

 

Cancelling the factor (a-b) on both sides of above equation yields

b=b-c

This forces c=0 always but at the start itself we had assumed that c is an arbitrary quantity. Thus, the conclusion b=b-c must hold for any b and any c, zero or non-zero. Taking b=5, c=1, we get the absurd answer 5=4!! Again, we must have gone wrong somewhere. Where?

The nonsensical stuff can be detected in manipulating equation 1 to get equation 2. It is true that the factor

(a-b) is common to both sides. But, while striking it off, you apparently forgot to observe that it is zero! Recall that we started with the assumption a = b so that a - b = 0.

You failed to observe the eleventh Commandment of Moses: Thou shall not divide by zero!!

More later…

Nalin

From scratch — the 3 medians of a triangle are concurrent

page1medianspage2mediansT

Please download the attachments, pages 1 and 2 respectively of the proof that the 3 medians of a triangle are concurrent. That’s called the power of axioms!! This is the example which mesmerized Albert Einstein as a child to Mathematics and later to  Physics.

More later…

Nalin