Teaching Mathematics and the Concept of Infinity

Mathematics

It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.

I love Mark Twain for various reasons, and this quote is one of the first few writings of Twain, which made me read most of his works. I have come across several people who argue things based on the assumption that they know a certain thing for sure (while they don’t)

Justice Markandey Katju, in a recent column in The Hindu titled Professor, teach thyself,wrote: “When I was a judge of Allahabad High Court I had a case relating to a service matter of a mathematics lecturer in a university in Uttar Pradesh. Since the teacher was present in court I asked him how much one divided by zero is equal to. He replied, “Infinity.” I told him that his answer was incorrect, and it was evident that he was not even fit to be a teacher in an intermediate college. I wondered how had he become a university lecturer. (In mathematics it is impermissible to divide by zero. Hence anything divided by zero is known as an indeterminate number, not infinity).”

Many bloggers, and a few articles even in top media websites, criticized Katju for giving wrong information. They claimed that only 0/0 is indeterminate and a non zero number divided by zero is infinity.

The argument of the bloggers are flawed.  The logic behind their argument can be demonstrated by the following example

1. Let ”a” and ”b” be equal non-zero quantities
a = b
2. Multiply through by ”a”
a^2 = ab 
3. Subtract b^2
a^2 – b^2 = ab – b^2 
4. Factor both sides
(a – b)(a + b) = b(a – b) 
5. Divide out (a – b) [This is the step where we divide zero by zero, which should not be permited]
a + b = b 
6. Observing that a = b
b + b = b 
7. Combine like terms on the left
2b = b 
8. Divide by the non-zero ”b”
2 = 1

So anything can be proved right in maths if we allow the 0/0, so zero divided by zero is indeterminate. They however say that a non zero number divided by zero is infinity. That argument should not be taken at face value.

In arithmetic, the expression of a/0 has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. This is nothing new, and even ancient Indian mathematicians debated about this and concluded that a/0is undefined. However infinitesimal calculus and trigonometry uses the term infinity (or a value that tends to infinity). This is based on Baskara II (who first came up with the concept of infinity), the purpose was not to assign a value but to solve certain trigonometrical problems and coming up with solutions in calculus. He did that through the extended, or in other words in a fraction a/b, where b approaches zero (and not becomes zero).

Even in trigonometry where we say tan 90 is infinity, we are not correct, because there cannot be a tan 90 (we can only approach a value close to 90, but not 90), because the total internal angle of a triangle is 180, and since we consider a right angle triangle, there ia already one angle that is of 90 degree and hence the other two angles put together should be 90 degree. In case tan 90 we are talking about one more angle of 90 degree, which is impossible. I find only in India, people still continue to say tan90 as infinity everywhere else people correctly call it as undefined.

I don’t like Katju for various reasons, he says idiotic things with a lot of arrogance like “90% of the Indians are idiots” and so on.. However for once he is correct.