Operating instructions

NUMBER SYSTEMS
I NUMBER SYSTEMS IN GENERAL following is an example of a binary number:
To learn and understand new nunlber systems, it
is necessary to analyze principles which are true
of all number systems:
A number is expressed as the sum of terms.
The number 26 is the decimal equivalent of
binary 11010, so it is seen that the expansion
of a binary number by powers is a simple
method of converting from binary to decimal.
Each term is the product of a digit
times a
base raised to
a power.
The base of
a system is equal to the number
of digits in the system.
C. The Octal System
In any number system, the largest single digit
is always one less than the base. The base of the octal system is 8; the base
equals the total number of digits in the system
(0 through 7). The largest single digit is 7,
which is
1 less than the base. The following
is
an example of an octal number:
The rightmost or least significant digit counts
units. Each count in another column from the
right contains a multiple of the base.
Whenever any column holds the highest valued
digit of a particular number system, and one
is added to it, the column goes back to zero
and develops a carry to the next
most signifi-
cant column.
= 64 + 32 + 4 = 100
10
The number 100 is the decimal equivalent of
octal 144. Again it is seen that expansion of
a number by powers is
a simple method of
converting to decimal--this time it is octal
to decimal.
A. The Decimal System
Principles of a number system can be most
easily understood
by first relating them to an
example in a familiar system-- the decimal
system:
The following table illustrates various
numbers in three systems:
Decimal
Base 10
Octal Binary
Base 8 Base
2
--
The same number can be written as four
sums as follows:
The base of the decimal system is 10, and
the base equals the total number of digits in
the
systenl (0 through 9). The largest single
digit is 9, which is 1 less than ye base. It
rnyt be remembered that 10 = 10 and
10
= 1. The last principle listed above is
illustrated by the fact that when
1 is added
to the 9 of 4789, the 9 goes to zero, and the
number becomes 4790.
B. The Binary
System
The base of the binary systenl is 2; the
base equals the total number of digits in
the system
(0 through 1). The largest single
digit
is 1, which is 1 less than the base. The