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.