Number systems
Decimal Number System:-
The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands and so on.
Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position, and its value can be written as,
The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands and so on.
Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position, and its value can be written as,
Binary Number System
Characteristics of binary number system are as follows:
Example:-
Binary Number : 101012
Calculating Decimal Equivalent:-
Characteristics of binary number system are as follows:
- Binary number system uses two digits 0 and 1.
- Also called base 2 number system.
- Each position in a binary number represents a 0 power of the base (2).
- Last position in a binary number represents a x power of the base (2).
Example:-
Binary Number : 101012
Calculating Decimal Equivalent:-
Note : 101012 is normally written as 10101.
Octal Number System
Characteristics of octal number system are as follows:
Octal Number : 125708
Calculating Decimal Equivalent:-
Octal Number System
Characteristics of octal number system are as follows:
- Uses eight digits 0,1,2,3,4,5,6,7.
- Also called base 8 number system.
- Each position in an octal number represents a 0 power of the base (8). Example 80
- Last position in an octal number represents a x power of the base (8). Example 8x where x represents the last position - 1.
Octal Number : 125708
Calculating Decimal Equivalent:-
Note : 125708 is normally written as 12570.
Hexadecimal Number System
Characteristics of hexadecimal number system are as follows:
Hexadecimal Number : 19FDE16
Calculating Decimal Equivalent :-
Hexadecimal Number System
Characteristics of hexadecimal number system are as follows:
- Uses 10 digits and 6 letters, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
- Letters represents numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15.
- Also called base 16 number system
- Each position in a hexadecimal number represents a 0 power of the base (16). Example 160
- Last position in a hexadecimal number represents a x power of the base (16). Example 16x where x represents the last position - 1.
Hexadecimal Number : 19FDE16
Calculating Decimal Equivalent :-
Representation of a Binary Number
We saw above that in the decimal number system, the weight of each digit to the left increases by a factor of 10. In the binary number system, the weight of each digit increases by a factor of 2 as shown. Then the first digit has a weight of 1 ( 2^(0 )), the second digit has a weight of 2 ( 2^(1) ), the third a weight of 4 ( 2^(2) ), the fourth a weight of 8 ( 2^(3) ) and so on.So for example, converting a Binary to Decimal number would be:-
By adding together ALL the decimal number values from right to left at the positions that are represented by a “1” gives us: (256) + (64) + (32) + (4) + (1) = 35710 or three hundred and fifty seven as a decimal number.Then, we can convert binary to decimal by finding the decimal equivalent of the binary array of digits 1011001012 and expanding the binary digits into a series with a base of 2 giving an equivalent of 35710 in decimal .
Repeated Division-by-2 Method
Another name of this method is Double-Dabble method. We have seen above how to convert binary to decimal numbers, but how do we convert a decimal number into a binary number. An easy method of converting decimal to binary number equivalents is to write down the decimal number and to continually divide-by-2 (two) to give a result and a remainder of either a “1” or a “0” until the final result equals zero.
So for example :- Convert the decimal number 29410 into its binary number equivalent.
So, the binary equivalent of 29410 :- 1001001102
Another example :- Convert the decimal number 67510 into its binary number equivalent.
Another example :- Convert the decimal number 67510 into its binary number equivalent.
So, the binary equivalent of 67510 :- 10101000112
Another method to represent a decimal number into a binary form Each decimal digit is converted into a four bit binary.
( 325 )10 =( 0011 0010 0101 )
3 2 5
( 739 )10 =( 0111 0011 1001 )
7 3 5
BCD ADDITION
1. add bcd.
2. if sum is invalid bcd or if carry is generated in the addition , add ( 0110 ) equivalent to ( 6 ) to the sum.
1 1
3 2 9 - 0 0 1 1 0 0 1 0 1 0 0 1
+1 8 2 - 0 0 0 1 1 0 0 0 0 0 1 0
5 1 1 0 1 0 1 1 0 1 1- invalid bcd 1 0 1 1 - invalid bcd
+ 0 1 1 0 +0 1 1 0
0 1 0 1 0 0 0 1 0 0 0 1
5 1 1
Another method to represent a decimal number into a binary form Each decimal digit is converted into a four bit binary.
( 325 )10 =( 0011 0010 0101 )
3 2 5
( 739 )10 =( 0111 0011 1001 )
7 3 5
BCD ADDITION
1. add bcd.
2. if sum is invalid bcd or if carry is generated in the addition , add ( 0110 ) equivalent to ( 6 ) to the sum.
1 1
3 2 9 - 0 0 1 1 0 0 1 0 1 0 0 1
+1 8 2 - 0 0 0 1 1 0 0 0 0 0 1 0
5 1 1 0 1 0 1 1 0 1 1- invalid bcd 1 0 1 1 - invalid bcd
+ 0 1 1 0 +0 1 1 0
0 1 0 1 0 0 0 1 0 0 0 1
5 1 1
VALID AND INVALID BCD
Binary system complements
As the binary system has base r = 2. So the two types of complements for the binary system are 2's complement and 1's complement.
1's complement
The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's. This is called as taking complement or 1's complement. Example of 1's Complement is as follows.
2's complement
The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.
2's complement = 1's complement + 1
Example of 2's Complement is as follows.
The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.
2's complement = 1's complement + 1
Example of 2's Complement is as follows.