Modern Computers Are Binary (continued)

Why use Binary?
Now, why on earth would we use binary? Why would we write the number twenty-nine as 11101 instead of as good old 29?

Here's why:
Notice that the only numerals we have seen in any binary digit are a 0 or a 1. Why is that?

Consider this: when we write numbers in decimal form, we can count from 0 through 9 in any digit, and then we carry to the next digit. We never get to ten, because ten is the base of the number system; each digit represents another power of ten.

Likewise, if we count in binary representation:

(Remember: )

we never get to "2" in any digit, because we will always carry to the next digit. This is because two is the base; each digit is another power of two. Similarly, in base-10 counting, there are ten digits (0 through 9), but there is no digit for "ten" itself; that always involves carrying to the next digit, switching its state.

This is the key. Building a computer means building a machine with mechanisms that can represent the digits of numbers and that switch states according to the rules of arithmetic.

We saw with our gear example that in order to build a decimal (base-10) computer, for each digit we required a mechanism with ten possible states, one to represent each of the numerals 0 through 9 . Now, consider instead building a computer built upon the idea of binary digits instead of decimal digits. It's certainly true that this means working with odd-looking, less-familiar numbers. However, here's the trade-off: instead of having to build a machine based on mechanisms with ten possible states to represent 0 through 9, a computer based upon binary digits only needs to allow for the possibility of a 0 or a 1 appearing in any digit. Thus, each digit can be represented with a mechanism that has only two states. This makes for a much simpler machine!

A simple on/off switch can represent a binary digit ("bit"): ON = 1, and OFF = 0 . That is, if we need to represent a 1 in a binary digit; we set the switch to ON; if we need the digit to have a 0 in it, we set the switch to off.

This is a much simpler mechanism for representing a digit than a gear that has to rotate between ten different positions!