Page 82 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

Alternative Numbering Systems 63

Octal Decimal Hexadecimal

7452, 3002,, OF2Al6
1 L

7
Ir

111, 100, 101, 010, 0000, 1111, 0010, 1010,

Binary Binary
Figure 7-16. Mapping octal and hexadecimal to binary

In the original digital computers, data paths were often 9 bits, 12 bits, 18
bits, or 24 bits wide, which provides one reason for the original popularity of
the octal system. Due to the fact that each octal digit maps directly to three
binary bits, these data-path values were easily represented in octal. More
recently, digital computers have standardized on data-path widths that are
integer multiples of 8 bits; for example, 8 bits, 16 bits, 32 bits, 64 bits, and so
forth. Because each hexadecimal digit maps directly to four binary bits, these
data-path values are more easily represented in hexadecimal. This may explain
the decline in popularity of the octal system and the corresponding rise in
popularity of the hexadecimal system.

Representing Numbers Using Powers
An alternative way of representing numbers is by means of powers; for
example, IO3, where 10 is the base value and the superscripted 3 is known as
the power or exponent. We read 1 O3 as “Ten to the power of three. ” The power
ow many times the base value must be multiplied by itself; thus, lo3
represents 10 x 10 x 10. Any value can be used as a base (Figure 7- 17 ).
Any base to the power of one is equal to itself; for example, = 8. Strictly
speaking, a power of zero is not part of the series, but, by convention, any base
to the power of zero equals one; for example, 8’ = 1. Powers provide a conve-
nient way to represent column-weights in place-value systems (Figure 7- 18).