a. General. Unfortunately the decimal system does not lend itself to convenient implementation in
digital systems. It is very difficult to design equipment so it works with ten different voltage levels, each
one representing one of the decimal characters 0 through 9. On the other hand, it is very easy to design
electronic circuits that operate with only two different voltage levels. For this reason the binary (base 2)
numbering system was developed. It is the basic system for operation of almost all digital systems,
although other systems are often used in conjunction with it.
b. Characteristics. The binary number system has two binary digits called bits which are assigned
the symbols 0 and 1. The binary system is also a positional value system. Each binary digit has its own
value or weight expressed as a power of two. An example of a binary number is 1011 which is read as
ONE-ZERO-ONE-ONE. The number 1301 is not a binary number, since 3 is not a binary digit. The
right-most digit is the least significant digit (LSD) and carries the least weight while the left-most digit is
the most significant digit (MSD) and carries the most weight.
c. Conversion of Binary.
(1) General. In computer work, it frequently happens that a value stated in one number system
has to be converted to that same value in another number system. Unless conversions are made
accurately, computer operations and output become meaningless.
(2) Positional Notation. By conversion agreement, a radix point is not written except when
necessary to separate the integral and fractional parts of an expression. In writing 100.00, only 100 is
written. The .00 portion is ordinarily discarded because it is meaningless and unnecessarily uses up
computer capacity. On the other hand, the radix point is needed in writing numbers such 11.1 because the
number contains both integral and fractional parts. If the radix point were omitted, the expression would
be understood as 111, which is entirely different from the original number. In fact it equals 11.1
multiplied by the radix. Further, moving the radix point one position to the left in 11.1 yields 1.11 and
this quantity is equal to 11.1 divided by the radix. This property of the radix point position is the basis of
the powers-of-ten method of simplifying the multiplication and division of decimal numbers. This
concept will be put to use later in this text to aid in converting from decimal to binary and decimal to
octal. Most digital computers are built to handle their internal operations in the binary system, while
reading in and out in the decimal system. This requires the computer to convert, with correct positional
notation, all of its inputs and output.
(a) The table in Figure 1-3 covers conversion of binary numbers to decimal numbers and
conversion of decimal numbers to binary numbers.