The Magic of Mathematics

When my college-bound daughter recently informed us that she wanted to study mathematics as an undergraduate, I was delighted. I have been passionate about mathematics since Class X, when the mathematics teacher Mr. Shawkat Hussain (the subject of a previous column) ignited this passion. The best grades I earned in undergraduate and graduate engineering education were in mathematics courses. If I had a second chance to attend university today I would probably study mathematics.

At any rate, in order to prepare her for the demanding courses she will have to face, I started working with my daughter on college calculus. She is self-reliant and learns quickly without help. Nevertheless, I made myself useful explaining some things to her: the big picture, or why a certain theorem or approach is useful (having seen its uses that she is yet to see.)

It was a wonderful experience working with her. But it was painful, too, when I saw certain problems and confidently claimed that I could solve them in a heartbeat - only be foiled halfway by the cobwebs of age. Then all I could do was mumble: “I used to know how to do this.” Along with the flood of memories came the feelings – I recalled what a pain integral calculus was. My most joyful moments came when she solved problems that confounded me.

But the hours spent with her also gave me time for reflecting on mathematics – particularly the dichotomy of discrete and continuous.

The calculus belongs to the world of continuous mathematics, closely mirroring the real world. It was the culmination of centuries of thinking about changing or moving things. Newton's (and Leibniz's) genius to some extent lay in the invention of “dx” – an infinitesimally small change that allowed the development of the calculus.

But in today's world, interesting mathematical applications involve discrete quantities. Thus, for example, when you take a digital picture, you create a very fine two-dimensional grid of numbers. Each of these numbers, called a pixel, represents a brightness level.  Computers, cellphones, television signals, music videos – all require mathematical computations on discrete (or digital) numbers.

But we don't see, hear, taste or smell the world digitally. Our senses understand analog, continuous input.  In general, things in the physical world are mostly analog.

How to connect the two? Digital and analog are connected by Nyquist's Sampling Theorem which points the correct way to move roundtrip between digital and analog worlds. Thus, for example, it tells us how many points we need in that grid of pixels inside the digital camera so the digital photograph is indistinguishable from one taken with a film camera, or how many “bits” we have to have in an “mp3” of a song. 

The philosophical implications of Nyquist's theorem are mind-boggling. For example, using it, a passport size photograph can be represented by a grid of 1024x1024 pixels, each pixel taking on one of 64 brightness levels. It follows that the number of all possible passport photographs is 64(1024x1024). This (very large) set of photographs will contain, for example, the faces of all humans who ever walked on earth – in fact, many many photographs of each!

Ah, the joys of mathematics. I hope you will join me in wishing my daughter wondrous discoveries on her journey.

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