The headteacher came round.

‘Sit down if you got 35.’ Some children sat.

‘Well done’, she said. Sit down if you got 34.’

You get the idea.

When only a few of us remained standing on our chairs, say at 6 out of 35, the headteacher smacked us on the backs of our bare legs until we cried.

I still hate Thursdays.

The problem came for me when I thought I’d understood.

On the bus home from school when I was about 7, I had a conversation a bit like this:

Keith: When you times a number by nought, why does it equal nought?
Me: Because timesing two by nought is the same as timesing nought by two. Two nothings are nothing.
Keith: But if you times two by nothing you’re not timesing it by anything, so it’s still two.
Me: Look, if you’ve got one nothing, or two nothings, or a squidrillion nothings, you’ve still got nothing.
Keith: But if you times a number by nothing, you’re not timesing it at all, so it stays the same.
Me: You smell.
Keith: You smell of wee.
Me: You smell of poo.

I never really did get maths, and as I had consistently high marks in English, languages and music, I was able to avoid maths-based subjects.

I still don’t understand why -x * -y is positive. 2 minuses make a plus because (minus sign) plus (minus sign on its side) = + was all the explanation I had.

Your site gives me hope that I may eventually understand the subject.

I got better at arguing, and my vocabulary increased considerably over the next 53 years.

But Keith really did smell.

On division by zero. I very much favor Abraham Walker, his non-standard analysis.*

*(H. Jerome Keisler provided open access to his book, Elementary Calculus: An Infinitesimal Approach, under a Creative Commons by-nc-sa license.
http://www.math.wisc.edu/~keisler/calc.html
The book provides a highly approachable explanation of non-standard analysis.)

Non-standard analysis (will be a funny name, if at some point in the future, it becomes standard) defines a range of positive numbers that are greater than zero and less than any positive real number. The additive inverse gives a negative range. And the inverses give infinite results.

I prefer this method for dealing with infinity because, at least to me, it does not seem reasonable to think of 1 divided by zero as positive or negative or even non zero.

To me, defining division by zero as undefined is not a bug. I think the bug is believing that we know nothing well enough assume that dividing by it should be defined (of course, if we do some day come do know nothing, then we may feel free to divide by it).

So for all practical intents and purposes, I like to use the inverse of positive and negative infinitesimal numbers to represent infinity because I know the sign of their inverses. (I don’t like taking the integral between negative and positive infinity because I don’t think it’s reasonable to say that infinity has a sign or non zero value).

On the other hand, I do think it may be reasonable to define zero divided by zero as 1, provided that the “1″ thus generated is given a universe of sets not compared with other divisions of zero by zero.

(Note: I am not familiar with set theory, so please correct and forgive any errors in my use of terminology. Specifically, I’m the concept of Universe that I learned from Lewis Carrol [Dodgson], his Symbolic Logic).

That is 0/0 of set universe A = 1 of set universe B, but 1_a does not equal 1_b. And 0_b/0_b = 1_c which does not equal 1_a or 1_b, or rather that 1_a = 1_b is one possibility out of an infinite set of roughly equivalent possibilities and therefore infinitely improbable.