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the third position the number of 100s, the fourth the number of 1000s, and so on. So the number 342
means the sum: 2 times 1, plus 4 times 10, plus 3 times 100, which is 342.
The binary number system is essentially the same, but much simpler. When you have a number in base
two, it may be interpreted in this way: Starting from the right, the first bit position represents the number
of 1s, the second bit position represents the number of 2s, the third position the number of 4s, the fourth
the number of 8s, and so on and so on, doubling each time you move a position to the left. The binary
number 01110 represents the decimal number fourteen, calculated as follows: the sum of zero times 1,
plus one times 2, plus one times 4, plus one times 8, plus zero times 16, which is 14.
Computation
For a simple example of computer computation, consider addition. The table to give the results of
addition of two binary numbers is quite simple, compared to the addition table for decimal numbers. It is
as follows:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (with a carryover to the next column to the left)
For the computer to add two numbers together, say, the integers 3 and 11, the computer has them
represented in its store as 00011 and 01011. So:
00011
+ 01011
01110
Most fundamentally, the central processing unit (CPU) of a computer can
1. Add, subtract, multiply, divide, and so on, two sequences of bits
2. Compare one sequence of bits with another to determine if they are the same, or if one is numerically
greater than the other. Based on the result of the comparison, the computer can begin executing one
set of instructions, or another set of instructions.
The CPU has other capabilities, but these are the principal ones. (Much of the progress, and complication,
in computing—and why it seems to be so magical) comes from the ability to transfer bits from machine
to machine wirelessly, from graphical user interfaces, from parallel computing, and saving partial results
(caching). And then there is the state of the art of putting transistors on a chip: 4000 of them in a space the
width of a human hair.
Output
When you see a color image on a computer monitor screen, it is composed of, approximately, a million
little dots—called pixels (picture elements)—the color of each being controlled by, say, 32 bits. Thirty-two
bits allows a large number of combinations, so that many shades of color can be presented. When you
hear music coming from your computer's speakers, the sound is generated by strings of bits sending
impulses to a speaker cone at varying frequencies—just as sound from a CD is generated by bits: A hole
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