I want to do some more tricky examples of using this ε idea, where I show two numbers have to be the same because the difference between them is smaller than every positive number. Before I do, I want to put out a problem where we can show two numbers are not the same, since I think that makes it easier to see why the proof works where it does. It’s easy to get hypnotized by the form of an argument, and to not notice that the result doesn’t actually hold, particularly if all you see are repetitions of proofs where things work out and don’t see cases of the proof being invalid.
What Numbers Equal Zero?
I want to give some examples of showing numbers are equal by showing the difference between them is ε. It’s a fairly abstruse idea but when it works amazing things become possible.
The easy example, although one that produces strong resistance, is showing that the number 1 is equal to the number 0.9999…. But here I have to say what I mean by that second number. It’s obvious to me that I mean a number formed by putting a decimal point up, and then filling in a ‘9’ to every digit past the decimal, repeating forever and ever without end. That’s a description so easy to grasp it looks obvious. I can give a more precise, less intuitively obvious, description, though, which makes it easier to prove what I’m going to be claiming.
Introducing a Very Small Number
Last time I talked mathematics I introduced the idea of using some little tolerated difference between quantities. This tolerated difference has an immediately obvious and useful real-world interpretation: if we measure two things and they differ by less than that amount, we’d say they’re equal, or close enough to equal for whatever it is we’re doing. And it has great use in the nice exact proofs of some sophisticated mathematical concepts, most of which I think I can get to without introducing equations, which will make everyone happy. Readers like reading things that don’t have equations (folklore has it that every equation, other than E = mc2, cuts book sales in half, although I don’t remember seeing that long-established folklore before Stephen Hawking claimed it in A Brief History Of Time, which sold a hundred million billion trillion copies). Writers like not putting in equations because web standards have evolved so that there’s not only no good ways of putting in equations, but there aren’t even ways that rate as only lousy. But we can make do.
The tolerated difference is usually written as ε, the Greek lower-case e, at least if we are working on calculus or analysis at least, and it’s typically taken to mean some small number. The use seems to go back to Augustin-Louis Cauchy, who lived from 1789 to 1857, who paired it with the symbol δ to talk about small quantities. He seems to have meant δ the Greek lowercase d, to be a small number representing a difference, and ε as a small number representing an error, and the symbols have been with us ever since.
Cauchy’s an interesting person, although it seems sometimes that every mathematician who lived in France anytime around the Revolution and the era of Napoleon was interesting. He was certainly prolific: the MacTutor biography credits him with 789 published papers, and they covered a wide swath of mathematics: solid geometry, polygonal numbers, waves, inelastic shocks, astronomy, differential equations, matrices, and a powerful tool called the Fourier transform. This is why mathematics majors spend about two years running across all sorts of new things named after Cauchy — the Cauchy-Schwarz inequality, Cauchy sequences, Cauchy convergence, Cauchy-Reimann equations, Cauchy-Kovalevskaya existence, Cauchy integrals, and more — until they almost get interested enough to look up something about who he was. For a while Cauchy was tutor to the grandson of France’s King Charles X, but apparently Cauchy had a tendency to get annoyed and start screaming at the uninterested prince. He has two lunar features (a crater and an escarpment) named for him, indicating, I suppose, that Charles X wasn’t asked for a reference.
Little Enough Differences
It’s as far from my workplace to home as it is from my workplace to my sister-in-law’s home. That’s a fair coincidence, but nobody thinks it’s precisely true. I don’t think it’s exactly true myself, but let me try to make it a little interesting. I’d be surprised if it were the same number of miles from work to either home. I’d be shocked if it were the same number of miles down to the tenth of the mile. To be precisely the same distance, down to the n-th decimal point, would be just impossibly unlikely. But I’d still make the claim, and most people would accept it, and everyone knows what the claim is supposed to mean and why it’s true. What I mean, and what I imagine anyone hearing the claim takes me to mean, is that the difference between these two quantities, the distance from work to home and the distance from work to my sister-in-law’s home, is smaller than some tolerable margin for error.
That’s a good definition of equality between two things in the practical world. It applies mathematically as well. A good number of proofs, particularly the ones that go into proving calculus works, amount to showing that there is some number in which we are interested, and there is some number which we are actually able to calculate, and the difference between those two numbers is less than some tolerated difference. If we’re just looking for an approximate answer, that’s about where we stop. If we want to do prove something rigorously and exactly, then we use a slightly different trick.
Instead of proving that the difference is smaller than some tolerated error — say, that the distance to these two homes is the same plus or minus two miles, or that these two cups of soda have the same amount of drink plus or minus a half-ounce, or so — what we do is prove that we can pick some arbitrary small tolerated difference, and find that the number we want and the number we can calculate must be smaller than that tolerated difference. But that tolerated difference might be any positive number. We weren’t given it up front. If the difference is smaller than any positive number, then, we can, at least in imagination, make sure the difference is smaller than every positive number, however tiny. The conclusion, then, is that if the difference between what-we-want and what-we-have is smaller than every positive number, then the difference must be zero. The two quantities have to be equal.
That probably read fairly smoothly. It’s worth going over and thinking about closely because, at least in my experience, that’s one of the spots where calculus and analysis gets really confusing. It’s going to deserve some examples.