1. Zero is a number. 2. It is the number that counts none or nothing. 3. It is marked 0 in our usual numeral writing. 4. The ancient Romans, however, had no numeral for it. Apparently, they did not think of zero as a number. 5. 0 is what you get when you take away 17 from 17, or when you subtract any number from itself. 6. If you add 0 to 23, you get 23; and if you add 0 to any number, you get that number unchanged. 7. If you subtract 0 from 48, you get 48; and if you subtract 0 from any number, you get that number unchanged. 8. If you multiply 0 by 71, you get 0; and if you multiply together 0 and any number, you get 0. 9. Usually you are not allowed to divide by 0: that is against the rules of arithmetic. 10. But if you do, and if you divide 12 by 0, for example—and there are times when this is not wrong—the result is called _infinity_ and is marked ∞, a sign that is like an 8 on its side. This is not all the story of zero; it is one of the most important of numbers. But, if you know these statements about zero, and have had some practice in applying them, you have a good _understanding_ of zero. Incidentally, a mechanical brain knows all these statements about zero and a few more; they must be built into it. For us to understand any idea, then, we pursue three aims: 1. We find out what it is called. 2. We collect true statements about it. 3. We apply those statements—we use them in situations. We can do this about any idea. Therefore, we can understand any idea, and the degree of our understanding increases as the number of true statements mastered increases. Perhaps this seems to be a rash claim. Of course, it may take a good deal of time to collect true statements about many ideas. In fact, a scientist may spend thirty years of his life trying to find out from experiment the truth or falsehood of one statement, though, when he has succeeded, the fact can be swiftly told to others. Also, we all vary in the speed, perseverance, skill, etc., with which we can collect true statements and apply them. Besides, some of us have not been taught well and have little faith in our ability to carry out this process: this is the greatest obstacle of all. But, there is in reality no idea in the field of existing science and knowledge which you or I cannot understand. The road to understanding lies clear before us. Supplement 2 MATHEMATICS In the course of our discussion of machines that think, we have had to refer without much explanation to a number of mathematical ideas. The purpose of this supplement is to explain a few of these ideas a little more carefully than seemed easy to do in the text and, at the end of the supplement, to put down briefly some additional notes for reference. DEVICES FOR MULTIPLICATION