• digital
• binary
• electronic

Modern computers are digital in that they represent each number in terms of its digits rather than as a single quantity.

Accordingly, to represent decimal digits, we would have to build a machine that had a mechanism in each digit that could represent 0 through 9—e.g., one of the gears in this photo. Such a gear could be called a ten-state device, because it has ten different positions it can be in, one for each possible number that can appear in the digit: 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.

Suppose the red gear represents the tens position and the white gear represents the ones position. Remember how we would write the number 58 in first grade? "Put a 5 in the tens place, and an 8 in the ones place," right? Similarly, for the above two gears to represent the number 58, the red gear would have to rotate into the position so that the 5 appears inside the little blue square, and the white gear would have to rotate into the position so that the 8 appears in the little blue square. Presto! You have a machine that can represent numbers!

Now, for the machine to "do" arithmetic, you would need to build the machine to mimic (model) the rules of arithmetic. Suppose the machine had an ADD button. Then, when you pressed the ADD button once (to add 1), the white gear would rotate one position so that 9 appeared in the blue window, and the red gear would stay at 5. Wow! Our machine just mimicked 58 + 1 = 59.


Of course, if we pressed the ADD button again (to add 1), the machine would have to built so the white gear rotates one position, so that 0 would be in the blue window. But our machine would also have to be built such that whenever the white gear goes from 9 to 0, the red gear gets advanced one position too —in this case, the red gear has to move from 5 to 6, mimicking the idea of carrying the one to the next column.

Still, as amazing as our little machine is—it can count by one from 0 to 99— it is very clear that our machine doesn't really do arithmetic. It is simply built to mimic (model) the rules of arithmetic. Our machine can't think. It is simply built so that, every case, the mechanisms representing the digits "flip" to the positions that represent the correct result.

And the same thing can be said of modern electronic computers: they are not intelligent machines. They are just built so that they mimic the rules of arithmetic.

As we will see, however, what is amazing about modern computers is

1. How fast they switch states in mimicking arithmetic, and
2. How many things in life can be represented as numbers and as arithmetic.