Binary Number System
Data is represented within the computer as a sequence of bits (which is
called binary code) and is processed using the laws of binary number
arithmetic. Computers can understand only the binary number system.
The binary system has only two digits 0 and 1. Computers use binary
system because the electrical devices can understand only ‘on’ (1) or ‘off’ (0) states.
Also electrical and magnetic storage devices having two stable states that are simple to construct. Though these elementary devicescould be combined to handle numbers with large bases, they are less
efficient than two state devices and are too complicated and costly.
Binary numbers can be constructed just like decimal numbers except
that the base is 2 instead of 10.
Thus, 646 (decimal) = 6 x 102 + 4 x 101 + 6 x 100
In the decimal system, the first place is for 1s, 2nd place is for 10s, and
the 3rd place is for 100. On the other hand, in the binary number system, the factor is 2 instead of 10. The first place is still for 1s, but the second is for 2s, the 3rd for 4s, the 4th for 8s and so on.
Examples
1. 1010101 (binary) = 1 x 26 + 0x25 +1x24+0x23+1x22+0x21+1x20
= 64+0+16+0+4+0+1= 85 (decimal)
2. 101011 (binary) = 1x25+0x24+1x23+0x22+1x21+1x20
= 32+0+8+0+2+1=43 (decimal)
3. 11111 (binary) = 1 x 24 + 1 x 23 +1 x 22 + 1 x 21 + 1x20
= 16+8+4+2+1=31 (decimal)
4. 10001011 (binary) = 1 x 27 + 0 x 26 + 0 x 25 + 0 x 24 + 1 x 23 + 0 x
22 + 1 X 21 + 1 X 20
= 128+0+0+0+8+0+2+1 = 139 (decimal)
It is seen from the examples above that while converting binary number
into decimal, the following rules have to be adopted:
1. Multiply the right most bit by 1.
2. Going right to left, multiply each succeeding bit by twice the factor
used with the previous bit.
3. Add the products according to the usual rules of decimal addition.
Conversion from Decimal to Binary:-
Converting decimal number to binary is slightly difficult than
converting binary to decimal. Normally a division procedure is used
which has the following rules :
1. Divide the decimal number by 2, save remainder.
2. If the quotient is 0, proceed to step 3, if the quotient is not 0,
replace the number with quotient and repeat step1.
3. The binary representation of the decimal number is the remainders
starting with the first remainder at the right.
Example-
The process of converting 47 into binary equivalent is shown in the
following table :
Thus, (47)10 = (101111)2
The equivalent of 77 in binary can be found out as follows
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