Now I want to do a little more complicated problem of showing two numbers are equal because the difference between them is so tiny. It struck me that if I wanted to do that, I’d have to do some setup to even start. What I really meant to do was to show that some number was equal to the square root of five. I picked the square root of five because I had it burned into my memory from a children’s book that knowing the first few digits of an irrational number would be sufficient to immobilize the mind-controlled population of an all-powerful computer dictator, and I’ve kept it in mind just in case ever since. I’m also glad to know on double-checking that I remembered the first couple digits of the square root of five well (2.236). I’m shakier on the square root of seven (2.something) so if it’s a more advanced computer we’re up against I’m in trouble.
Still, most square roots would do. It’s a neat little property of the whole numbers that the square roots of them are either whole numbers themselves — the square root of 4 is 2, the square root of 169 is 13, the square root of 4,153,444 is not worth thinking about — or else they’re irrational numbers, going on without ending and without repetition. Most people who’d read a mathematics blog on purpose have heard about how the irrationality of the square root of 2 was proven in ancient days, and maybe heard the story of how the Pythagoreans murdered the person who let slip the horrifying secret that there were irrational numbers and they represented real things that might be of interest, and a few are even aware we don’t really know with certainty that the story’s actually true. (At this point, I suspect it’s too strong a claim to say we know anything about the Pythagoreans for certain, but I haven’t looked closely. Maybe matters are not quite that dismal.) Whether true or not the legend of the Pythagoreans turning to murder is a fine way to get an algebra class’s attention. I just fear that what the students take away from it is, “if you learn any of this math stuff a cabal of mathematicians will murder you” and they stay oblivious for reasonable self-protection.
But anyone who’s understood a proof that the square root of two is irrational is perfectly able to show that the square root of three is irrational as well, or the square root of five, or any other such desired number. The proof that way runs just about the same route, but takes longer to get there.
Similarly, if you have a rational number that comes to an end, such as 0.49, then the square root either is a rational number that comes to an end, in this case 0.7; or else it never comes to an and and never repeats. That’s easy to prove, if you have that idea about the square roots of whole numbers. The square root of 4.9, for example, is not a rational number, although I can’t promise anything for its ability to halt world-spanning computers.