0.2 Binary Numbers
In preparation for the detailed study of computer arithmetic, we need to understand the binary number
system. Decimal numbers are converted from base 10 to base 2 in order to store numbers on a
computer and to simplify computer operations like addition and multiplication.

The decimal system
(427.325)10 = 4 ´ 102 + 2 ´ 101 + 7 ´ 100 + 3´ 10-1 + 2 ´ 10-2 + 5 ´ 10-5
The expression on the right contains power of 10 together the digits 0,1,2,3,4,5,6,7,8,9 . If
we admit the possibility of having an infinite number of digits to the right of the decimal point, the
any real number can be expressed in the manner just illustrated, with a sign ( + or - ) affixed to it.
Thus, for example, -p is
-p = -3.1415926535897932846264338
The last 8 written here stands for 8 ´ 10-26 .

The binary system
Only the two digits 0 and 1 are used. A typical number in the binary system can also be written
out in detail; for example
(1001.11101)
2
= 1´ 23 + 0 ´ 22 + 0 ´ 21 + 1´ 20 + 1´ 2-1 + 1´ 2-2 + 1´ 2-3 + 0 ´ 2-4 + 1´ 2-5
= (9.90625)10
This is the same real number as (9.90625)10 in decimal notation.
In general, binary numbers are expressed as
… b2b1b0·b-1b-2… ,
where each binary digit, or bit, is 0 or 1. The base 10 equivalent to the number is
… b2 22 + b1 21 + b0 20 + b-1 2-1 + b-2 2-2 …

Decimal to Binary
To convert decimal to binary, it is simple to break the number into integer and fractional parts and
convert each part separately.
Example:

(53.7)10 = (?)2
Binary to Decimal
To convert a binary number to decimal, it is again best to separate into integer and fractional
parts.
Example:
(110101.1011)2 = ?
(
)
Example: 0.1011
(
Example: 0.10110
)
2
2
=?
=?