Chapter 2: Alien Bases Getting Your Arms around Binary and Hexadecimal

The Return of the New Math Monster

The year 1966. Perhaps you were there. New Math burst upon the grade school curricula of the nation, and homework became a turmoil of number lines, sets, and alternate bases. Middle-class parents scratched their heads with their children over questions like, "What is 17 in Base Five?" and "Which sets does the Null Set belong to?" In very short order (I recall a period of about two months), the whole thing was tossed in the trash as quickly as it had been concocted by addle-brained educrats with too little to do.

This was a pity, actually. What nobody seemed to realize at the time was that, granted, we were learning New Math—except that Old Math had never been taught at the grade school level either. We kept wondering of what possible use it was to know what the intersection of the set of squirrels and the set of mammals was. The truth, of course, was that it was no use at all. Mathematics in America has always been taught as applied mathematics—arithmetic—heavy on the word problems. If it won't help you balance your checkbook or proportion a recipe, it ain't real math, man. Little or nothing of the logic of mathematics has ever made it into the elementary classroom, in part because elementary school in America has historically been a sort of trade school for everyday life. Getting the little beasts fundamentally literate is difficult enough. Trying to get them to appreciate the beauty of alternate number systems simply went over the line for practical middle-class America.

I was one of the few who enjoyed fussing with math in the New-Age style back in 1966, but I gladly laid it aside when the whole thing blew over. I didn't have to pick it up again until 1976, when, after working like a maniac with a wire-wrap gun for several weeks, I fed power to my COSMAC ELF computer and was greeted by an LED display of a pair of numbers in base 16!

Mon dieu, New Math redux…

This chapter exists because at the assembly language level, your computer does not understand numbers in our familiar base 10. Computers, in a slightly schizoid fashion, work in base 2 and base 16—all at the same time. If you're willing to confine yourself to higher-level languages such as Basic or Pascal, you can ignore these alien bases altogether, or perhaps treat them as an advanced topic once you get the rest of the language down pat. Not here. Everything in assembly language depends on your thorough understanding of these two number bases. So before we do anything else, we're going to learn how to count all over again—in Martian.