Chasing down infinity
By Burkard Polster and Marty Ross
The Age, 26 May 2008
One of our favourite quotes comes from the writer Oliver Goldsmith: “Wherever the doctrines of infinity enter into philosophy, knowledge ceases, and we talk at random”. And it seems to be true. Infinity inspires all manner of fear and confusion and nonsense. Can anything worthwhile be said at all?
Yes. Mathematicians are perfectly comfortable with and clearheaded about infinity. In fact, it is fair to say that almost all modern mathematics, Newton’s calculus and beyond, depends upon an understanding of infinity.
How do we run into infinity, and how can we think clearly about it? The obvious example from school mathematics is infinite decimals. These can be repeating, when expressing a fraction such as 140/99 = 1.4141…. Or, they can be non-repeating, for irrational numbers such as √2.
We are asked by our teachers to accept infinite decimals, but it takes very little to get confused. Consider, for example, the infinite repeating decimal .999…. Is this equal to 1? Many are uncomfortable with this claim, or flatly deny it.
One objection is simply that 1 and .999… look different, that the distinct decimal expressions must mean the numerical quantities are different. But this is really a peculiar argument, since we equate mathematical quantities all the time. For example the fractions 6/4 and 3/2 look different, but they represent the same numerical quantity.
Another argument is that .999… is somehow less than 1. However, when asked how much less, the answer is always something like “an infinitely small amount”. Oliver Goldsmith then leaps to mind.
Here is an easy argument that .999… = 1. We’ll return to school and let X stand for the quantity .999…. We can then multiply X by 10, simply by moving the decimal point. This gives 10X = 9.99…. Subtracting X from 10X, we then have 9X = 9.000…. That is, X =1. Quite Easily Done!
This is a very simple proof, but is it valid, or is it just a “professor’s trick”? Yes, the proof is valid, but there’s a catch.
Infinite decimals are much more difficult than school texts suggest. For example, what do the dots really mean? We think of them as something like “and so on”, but does this really make sense for a non-repeating decimal?
It was only at the end of the 19th Century that decimals were suitably settled. In the above example, we sloughed over the logical foundations, and just assumed we could manipulate decimals as we do whole numbers and fractions. So, what we really proved is that if .999… is indeed a number which obeys the normal rules of arithmetic, then in fact .999… = 1.
Now that you’ve had your lesson on infinity it is time for your final exam. First of all, can you prove the equality 140/99 = 1.41414… which we gave above? And, can you make sense of the square root monster below? Email us if you need hints or if you wish to submit your exam.
www.qedcat.com
Copyright 2004-∞ All rights reserved.
|