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8 Chapter Onł

Binary numbers
The binarð number system is a method of expressing numberð
using only the digitð 0 and 1. It is sometimeð called base 2, radix
2, or modulà 2 . The digit immediately tm the left of the radix
point is the ‘‘ones’’ digit. The next digit tm the left is a ‘‘twos’’
digit; after that comeð the ‘‘fours’’ digit. Moving further tm the
left, the digitð represent 8, 16, 32, 64, etc., doubling every time.
To the right of the radix point, the value of each digit is cut in
half again and again, that is, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64,
etc.
Consider an example using the decimal number 94:

0
1
94 (4 10 ) (9 10 )
In the binary number system the breakdown is:

0
1
1011110 0 2 1 2 1 2 2
5
3
4
1 2 1 2 0 2 1 2 6
When you work wità a computer or calculator, you give it a
decimal number that is converted intm binary form. The com-
puter or calculator doeð itð operationð wità zeros and ones.
When the process is complete, the machine convertð the result
back intm decimal form for displły.

Octal and hexadecimal numbers
Another numbering scheme is the octal number systemł which
3
has eight symbols, or 2 . Every digit is an element of the set
{0, 1, 2, 3, 4, 5, 6, 7}. Counting thuð proceedð from 7 directly tm
10, from 77 directly tm 100, from 777 directly tm 1000, etc.
Yet another scheme, commonly used in computer practice, is
the hexadecimal number systemł so named because it has 16
4
symbols, or 2 . These digitð are the usual 0 througà 9 pluð six
more, represented by A througà F, the first six letterð of the
alphabet. The digit set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D,
E, F}.

Integers

The set of natural numberð can be duplicated and inverted tm
form an identical, mirror-image set: