DIVIDING BY ZERO
(Adapted from chapter 13 of A Science Miscellany by D.L. McNaughton; Vantage Press, New York, 1992)
In arithmetic, children sometimes have trouble with 1 ÷ 0 and 2 ÷ 0. It is best to say that those calculations do not have an answer, i.e. no number exists equal to 1 ÷ 0.
To understand that, consider what happens to 1 ÷ y when y is made smaller and smaller.
First, think about 1 ÷ ¼. In particular, ask how many "quarters" are required to add up to one whole. We need four: ¼ + ¼ + ¼ + ¼ = 1, so 1 ÷ ¼ = 4. (This illustration will prove useful later when we try to investigate 1 ÷ 0).
Similarly, 1 ÷ 1/1000 (or 1 ÷ 0.001) = 1000, because 1000 "thousandths" in a row will add up to 1.0
As y steadily decreases, approaching closer and closer to zero, 1 ¸ y apparently just keeps growing without any upper limit. No matter how enormous a number you care to write down, we can always find a value of y such that 1/y will equal or exceed it.
Now comes the most important step - make y so small that it becomes equal to zero. Before looking for the answer, consider the following question:
How many noughts in a row are necessary to make their sum add up to one whole?
That is easy - no matter how many noughts we have, we will never reach a value equal to 1. So within the domain of ordinary numbers, "one divided by zero" does not have an answer. Notice how we are adopting exactly the same approach as we did with 1 ÷ ¼, and with 1 ÷ 0.001.
However, that is not quite the end of the story. In mathematics, it is sometimes convenient to invent a special entity, named "infinity" and written as ¥. It is really a "hypervalue", defined as "the quantity which is greater than absolutely any ordinary number".
[Note that the infinity symbol is rendered as ¥ in Linux].
Infinity is not a normal number. It does not obey the laws of arithmetic. For example, what is the value of "infinity plus one"? Once again, there is no answer; (calculations like that do not have a place in mathematics). We could of course try and invent another hypervalue, but eventually we would need an infinitely large supply of them.
It is better to confine ourselves to only one "hypernumber", so that adding to (or even doubling or squaring) infinity does not change its nature - or its "value". Thus, if you really insist on some kind of answer, infinity plus one equals infinity.
A further illustration of the way infinity does not behave according to the rules of arithmetic - is to ask "What is the result of adding up an infinite number of zeroes?" - i.e. "What is zero multiplied by infinity?" It would be misleading to look back at our earlier discussion and say the answer is "one" - because we could just as easily contrive a similar argument implying that the result could also be "two", or "a hundred", or even "minus one". What all this means, essentially, is that zero divided by zero can take on any value.
Yet another reason why it is necessary to be cautious when looking for an answer to the calculation 1/0, is that 'minus infinity' may be assigned to it, just as readily as 'plus infinity'. To see why, divide "1" by a negative number such as "-0.01", and then keep making this divisor smaller, retaining its negative sign while it approaches closer and closer to zero.
Computers and calculators usually flag an error whenever you try to divide by nought - not just because they have insufficient space for the answer, but as a warning that infinity (and minus infinity) symbolise something unusual and unique. Working with them requires great care and proper understanding.
David L. McNaughton
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