Wednesday, January 4, 2017
Binary
to Decimal and Decimal to Binary Conversion
The number system that we are mostly familiar with
is a decimal number system. The decimal number system has a base of ten which
implies that there are ten numbers by the help of which we can represent any
number of the decimal family. The numbers are from 0 to 9. Representation is
such as (15)10 the base 10 is written as suffix or radix. If it is not written
then by default we must understand that it is a decimal number by default. On
the other hand, binary number invented by Gottfried Leibniz in 1679 has a base
of two. “Bi” means two so from there we can say that the base of the binary
system is 2. That is, only two numbers are sufficient to represent a number in
binary format. The numbers which are used in binary number system are 0 and 1.
Binary number can be represented by putting 2 in the prefix which denotes the
base. If the base is not given, then it is by default assumed to be a decimal
number .We have to be very careful in writing a binary number, a slight mistake
may result in a very serious error. For example, a binary number is written as
follows-
(00110)2.
Now a question may arise, why do we need binary
number? We have decimal number system which is familiar to all of us and most
of the persons do not understand binary. The answer is that any programmable
device or a processor can work in two modes either high or low. Here, high
denotes the supply is connected to that point and low denotes that the point is
grounded or is at a state of zero volts. This is called positive logic and in other
logic system the reverse is taken which is known as negative logic system.
Also, we can say that a high means that it performs
some function or work and low indicates that it has not performed any work. The
reverse may also be true if we take negative logic system. So, from the above
description we can say that it is much easier and convenient to use binary
number system in the computer instead of decimal and also conversion will be
required in order that the output result which is given in binary form should
be converted to decimal for the sake of the user.
Conversion from Binary to Decimal
Conversion of Integer Numbers
Expand the number given in binary form in the power
of 2 and sum the values, the result which we will get will be in the decimal
form. For example-
Convert Binary Number to Decimal Number
Conversion of Decimal Point Number to Decimal
This can also be done in the same way, however after
the decimal point the number should be multiplied with 2-1, 2-2 etc.
For example,
Conversion from Decimal to Binary
Integer Numbers
Divide the number by 2 and take only the remainder,
if division is completed than take only the remainder which gives the binary
number.
Example
integer numbers
So, the binary equivalent of (14)10 is (1110)2
After the dash (-) remainder is written.
Suppose we are converting the decimal number (87)10.
Now the conversion is shown below
∴ (87)10 = (1110101)2
For Fractional Numbers
In this case, the successive multiplication is done.
The number which is to be converted is multiplied with base or radix of binary
number which is 2. The integer part or the carry of the product is taken out
and the same process is repeated until we get an integer. For example-
The binary equivalent of (.95)10 is evaluated as
follows-
Since, we are not getting the integer value after
successive multiplication, we can approximate the value to be (.111110….)2.
Conversion of Negative Number
In case of a negative number we can go for 2’s
complement representation of a signed number.
Example- 9 = 0000 1001
1’s complement = 1111 0110.
Adding 1 we get = 11110111 which is the 2’s
complement representation of (-9).
