"What is 1 divided by 0?" my high school mathematics teacher asked us twenty five years back.
Sweaty naked kneels clashing, compressed were we, in small benches misaligned in the general direction of the black board, which was more white than black.
We chorused "zero".
Short, strong, white flanneled and bare footed -- our village Swami Vivekananda -- our teacher smiled; then asked, "what is 1 divided by 1?"; we answered correctly. "What is 1 divided by 2?" Of course, "it is 0.5," we piped. So "What would be 1 divided by 4?" "0.25."
And then he paused. By now even us last-benchers sitting inside a little ramshackle of a room with thatched roof and limestone stained walls, got ourselves enthused.
"Do you all see what is happening? As you increase the denominator, the answer progressively decrease. Now, what if we take the denominator to its 'limit', i.e. infinity?"
We were silent; "answer is very small?," we stammered.
"Correct; if denominator 'tends to' infinity, the answer 'tends to zero'" he proceeded; "There is something more; What is 2 divided by 2? What is 2 divided by 4...."
We proceeded with this until we discovered to our wonder, 2 divided by infinity is ALSO zero. And so on until we arrived at our 'discovery' of any finite number divided by infinity is zero. We discovered the power of infinity.
Then with gleaming eyes our teacher went one up on us. "Now tell me kids, what if you reverse this; what is one divided by zero, the question we started with?" We again gaped. He ventured to help, "OK, you know what is 1 divided by 1; it is 1. What is 1 divided by 0.5?" "Of course, it is 2 sir."
And so on. when he asked "As you decrease the denominator, what happens to the answer?" we didn't let him down; "one divided by zero is infinity!" we yelled. Our teacher added, "and not only that, as was the case before, you put any number in the numerator and divided by zero, you get....what?" Infinity! we chimed. Now we have discovered the power of zero.
Our teacher smiled impishly; then he blew us up. "Now, you dearies are bright, aren't you all; Tell me what is zero divided by zero."
"Of course, infinity... ah ah..." few of us bit our tongue; the rest beamed embarrassment.
And then our beloved village school teacher, who on other occasions would be vivid about The Three Ws, great cricketers of West Indies, proceeded to brief us on indeterminate fractions and the power of nothing on nothing. And introduced us to one who had unraveled it; Srinivasa Ramanujan.
As he completed recounting the life story of The Mathematician, our teacher became emotional; we dispersed in stupor.
*
[source]
India is one of the few countries that still calender a "Teachers' day". It is one of the few countries where teachers get derided by 'visionaries' for their "lack of sacrifice", and continue to pass on knowledge to a generation of kids who upon graduation always earn more than them. It is a country where this passionate minority, like the street dog that gets whimsically stoned and could only lick its wounds and is expected to lie elsewhere in the same street for its livelihood, has to look only to its students for support; who go on to become policy makers, software merchants, verdict promulgators, event facilitators, adventurers, movie stars, treasure seekers...anything but a teacher.
When thankless instances frustrate me in academia, I remind myself of my beloved teachers. I remind myself of the debt I owe this passionate minority -- a debt I could only aim to repay with my life long struggle for professing.
Happy (belated) Teachers' Day.
Sweaty naked kneels clashing, compressed were we, in small benches misaligned in the general direction of the black board, which was more white than black.
We chorused "zero".
Short, strong, white flanneled and bare footed -- our village Swami Vivekananda -- our teacher smiled; then asked, "what is 1 divided by 1?"; we answered correctly. "What is 1 divided by 2?" Of course, "it is 0.5," we piped. So "What would be 1 divided by 4?" "0.25."
And then he paused. By now even us last-benchers sitting inside a little ramshackle of a room with thatched roof and limestone stained walls, got ourselves enthused.
"Do you all see what is happening? As you increase the denominator, the answer progressively decrease. Now, what if we take the denominator to its 'limit', i.e. infinity?"
We were silent; "answer is very small?," we stammered.
"Correct; if denominator 'tends to' infinity, the answer 'tends to zero'" he proceeded; "There is something more; What is 2 divided by 2? What is 2 divided by 4...."
We proceeded with this until we discovered to our wonder, 2 divided by infinity is ALSO zero. And so on until we arrived at our 'discovery' of any finite number divided by infinity is zero. We discovered the power of infinity.
Then with gleaming eyes our teacher went one up on us. "Now tell me kids, what if you reverse this; what is one divided by zero, the question we started with?" We again gaped. He ventured to help, "OK, you know what is 1 divided by 1; it is 1. What is 1 divided by 0.5?" "Of course, it is 2 sir."
And so on. when he asked "As you decrease the denominator, what happens to the answer?" we didn't let him down; "one divided by zero is infinity!" we yelled. Our teacher added, "and not only that, as was the case before, you put any number in the numerator and divided by zero, you get....what?" Infinity! we chimed. Now we have discovered the power of zero.
Our teacher smiled impishly; then he blew us up. "Now, you dearies are bright, aren't you all; Tell me what is zero divided by zero."
"Of course, infinity... ah ah..." few of us bit our tongue; the rest beamed embarrassment.
And then our beloved village school teacher, who on other occasions would be vivid about The Three Ws, great cricketers of West Indies, proceeded to brief us on indeterminate fractions and the power of nothing on nothing. And introduced us to one who had unraveled it; Srinivasa Ramanujan.
As he completed recounting the life story of The Mathematician, our teacher became emotional; we dispersed in stupor.
*
India is one of the few countries that still calender a "Teachers' day". It is one of the few countries where teachers get derided by 'visionaries' for their "lack of sacrifice", and continue to pass on knowledge to a generation of kids who upon graduation always earn more than them. It is a country where this passionate minority, like the street dog that gets whimsically stoned and could only lick its wounds and is expected to lie elsewhere in the same street for its livelihood, has to look only to its students for support; who go on to become policy makers, software merchants, verdict promulgators, event facilitators, adventurers, movie stars, treasure seekers...anything but a teacher.
When thankless instances frustrate me in academia, I remind myself of my beloved teachers. I remind myself of the debt I owe this passionate minority -- a debt I could only aim to repay with my life long struggle for professing.
Happy (belated) Teachers' Day.
2 Comments:
If only Justice Katju had such teachers.
Great article.
Everything we are is because someone has taught us to be this way. Be it a teacher in the form of a parent, a teacher at a formal education center, a friend who decides to be preachy, a book that gives you viewpoints you hadn't thought of before.
These are facts and utmost respect for teachers of any kind.
Thank you!
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