NUMBER SYSTEMS AND CODES - University of Houston
NUMBER SYSTEMS AND CODES
Topics to be covered:
• Number systems
– Number notations
– Arithmetic
– Base conversions
– Signed number representation
• Codes
– Decimal codes
– Error detection code
– Gray code
– ASCII code
1. Number Systems
The decimal (real), binary, octal hexadecimal number systems are used to represent information in digital systems. Any number system consists of a set of digits and a set of operators (+, (, (, (). The radix or base of the number system denotes the number of digits used in the system.
|Decimal (base 10) |0 1 2 3 4 5 6 7 8 9 |
|Binary (base 2) |0 1 |
|Octal (base 8) |0 1 2 3 4 5 6 7 |
|Hexadecimal (base 16) |0 1 2 3 4 5 6 7 8 9 A B C D E F |
|Decimal |Binary |Octal |Hexadecimal |
|00 |0000 |00 |0 |
|01 |0001 |01 |1 |
|02 |0010 |02 |2 |
|03 |0011 |03 |3 |
|04 |0100 |04 |4 |
|05 |0101 |05 |5 |
|06 |0110 |06 |6 |
|07 |0111 |07 |7 |
|08 |1000 |10 |8 |
|09 |1001 |11 |9 |
|10 |1010 |12 |A |
|11 |1011 |13 |B |
|12 |1100 |14 |C |
|13 |1101 |15 |D |
|14 |1110 |16 |E |
|15 |1111 |17 |F |
1. Number notations
A number can be represented in either positional notation or polynomial notation.
Positional notation
It is convenient to represent a number using positional notation. A positional notation is written as a sequence of digits with a radix point separating the integer and fractional part.
[pic]
where r is the radix, n is the number of digits of the integer part, and m is the number digits of the fractional part.
Polynomial notation
A number can be explicitly represented in polynomial notation.
[pic]
where rp is a weighted position and p is the position of a digit.
In decimal number system
[pic]
In binary number system
[pic]
In octal number system
[pic]
In hexadecimal number system
[pic]
2. Arithmetic
Addition
In binary number system,
|(101101)2 +(11101)2 : |1111 1 |
|+ | 101101 |
| | 11101 |
| |1001010 |
In octal system,
|(6254)8 +(5173)8 : |1 1 |
|+ | 6254 |
| | 5173 |
| |13447 |
In hexadecimal system,
|(9F1B)16 +(4A36)16 : | 1 1 |
|+ | 9F1B |
| | 4A36 |
| | D951 |
Subtraction
In binary number system,
|(101101)2 -(11011)2 : | 10 10 |
|- | 101101 |
| | 11011 |
| | 10010 |
In octal system,
|(6254)8 -(5173)8 : | 8 |
|- | 6254 |
| | 5173 |
| | 1061 |
In hexadecimal system,
|(9F1B)16 -(4A36)16 : | 16 |
|- | 9F1B |
| | 4A36 |
| | 54E5 |
Multiplication
In binary number system,
|(1101)2 ( (1001)2 : | |
|( | 1101 |
| | 1001 |
| | 1101 |
| | 0000 |
| | 0000 |
| |1101 |
| |1110101 |
Division
In binary number system,
|(1110111)2 ((1001)2 : | 1101 |
|1001 |1110111 |
| |1001 |
| | 1011 |
| | 1001 |
| | 1011 |
| | 1001 |
| | 10 |
3. Base conversions
Convert (100111010)2 to base 8
[pic]
or
[pic]
Convert (100111010)2 to base 10
[pic]
Convert (100111010)2 to base 16
[pic]
or
[pic]
Convert (372)8 to base 2
[pic]
Convert (372)8 to base 10
[pic]
Convert (372)8 to base 16
[pic]
Convert (9F2)16 to base 2
[pic]
Convert (9F2)16 to base 8
[pic]
Convert (9F2)16 to base 10
[pic]
Binomial expansion (series substitution)
To convert a number in base r to base p.
1) Represent the number in base p in binomial series.
2) Change the radix or base of each term to base p.
3) Simplify.
Convert base 10 to base r
Convert (174)10 to base 8
|8 |1 |7 |4 | |6 |LSB |
| |8 |2 |1 | |5 | |
| | |8 |2 | |2 |MSB |
| | | |0 | | | |
Therefore (174)10 = (256)8
Convert (0.275)10 to base 8
|8 |( |0.275 |( |2.200 |MSD |
|8 |( |0.200 |( |1.600 | |
|8 |( |0.600 |( |4.800 | |
|8 |( |0.800 |( |6.400 | |
|8 |( |0.400 |( |3.200 |LSD |
Therefore (0.275)10 = (0.21463()8
Convert (0.68475)10 to base 2
|2 |( |0.68475 |( |1. 3695 |MSD |
|2 |( |0.3695 |( |0.7390 | |
|2 |( |0.7390 |( |1.4780 | |
|2 |( |0.4780 |( |0.9560 | |
|2 |( |0.9560 |( |1.9120 |LSD |
Therefore (0.68475)10 = (0.10101()2
4. Signed Number Representation
There are 3 systems to represent signed numbers:
• Signed-magnitude
• 1's complement
• 2's complement
In binary number system
Signed-magnitude system In signed-magnitude systems, the most significant bit represents the number's sign, while the remaining bits represent its absolute value as an unsigned binary magnitude.
• If the sign bit is a 0, the number is positive.
• If the sign bit is a 1, the number is negative.
1's Complement system A 1's complement system represents the positive numbers the same way as in the signed-magnitude system. The only difference is negative number representations.
Let be N any positive integer number and [pic] be a negative 1's complement integer of N. If the number legnth is n bits, then [pic] For example in a 4-bit system, 0101 represents +5 and
[pic]
1010 represents (5
2's Complement System A 2's complement system is similar to 1's complement system, except that there is only one representation for zero.
Let be N any positive integer number and [pic] be a negative 2's complement integer of N. If the length of the number is n bits, then [pic] For example in a 4-bit system, 0101 represents +5 and
[pic]
1011 represents (5
Adding and subtracting signed numbers
Signed-magnitude system
|(a) |5 | | 0101 |
| |+2 | |+0010 |
| |7 | | 0111 |
|(b) |-5 | | 1101 |
| |-2 | |+1010 |
| |-7 | | 1111 |
|(c) |5 | | 0101 |
| |-2 | |+1010 |
| |3 | | 0011 |
|(d) |-5 | | 1101 |
| |+2 | |+0010 |
| |-3 | | 1011 |
1's complement system
|(a) |5 | | 0101 |
| |+2 | |+0010 |
| |7 | | 0111 |
|(b) |-5 | | 1010 |
| |-2 | |+1101 |
| |-7 | | 1 0111 |
| | | |1 |
| | | | 1000 |
|(c) |5 | | 0101 |
| |-2 | |+1101 |
| |3 | | 1 0010 |
| | | |1 |
| | | | 0011 |
|(d) |-5 | | 1010 |
| |+2 | |+0010 |
| |-3 | | 1100 |
2's complement system
|(a) |5 | | 0101 |
| |+2 | |+0010 |
| |7 | | 0111 |
|(b) |-5 | | 1011 |
| |-2 | |+1110 |
| |-7 | | 1 1001 |
|(c) |5 | | 0101 |
| |-2 | |+1110 |
| |3 | | 1 0011 |
|(d) |-5 | | 1011 |
| |+2 | |+0010 |
| |-3 | | 1101 |
Overflow conditions
Carry-in ( carry-out
| | | 0111 |
|5 | |0101 |
|+3 | |+0011 |
| -8 | | 1000 |
| | | 1000 |
|-5 | |1011 |
|-4 | |+1100 |
| 7 | | 1 0111 |
Carry-in = carry-out
| | | 0000 |
|5 | |0101 |
|+2 | |+0010 |
| 7 | | 0111 |
| | | 1110 |
|-6 | |1010 |
|-2 | |+1110 |
| -8 | | 1 1000 |
2. Codes
1. Decimal codes
|Decimal Digit |BCD |Excess-3 |2421 |
| |8421 | | |
|0 |0000 |0011 |0000 |
|1 |0001 |0100 |0001 |
|2 |0010 |0101 |0010 |
|3 |0011 |0110 |0011 |
|4 |0100 |0111 |0100 |
|5 |0101 |1000 |1011 |
|6 |0110 |1001 |1100 |
|7 |0111 |1010 |1101 |
|8 |1000 |1011 |1110 |
|9 |1001 |1100 |1111 |
2. Error detection code
Parity bit
|Odd Parity | |Even Parity |
|P |Message | |P |Message |
|1 |0000 | |0 |0000 |
|0 |0001 | |1 |0001 |
|0 |0010 | |1 |0010 |
|1 |0011 | |0 |0011 |
|0 |0100 | |1 |0100 |
|1 |0101 | |0 |0101 |
|1 |0110 | |0 |0110 |
|0 |0111 | |1 |0111 |
|0 |1000 | |1 |1000 |
|1 |1001 | |0 |1001 |
|1 |1010 | |0 |1010 |
|0 |1011 | |1 |1011 |
|1 |1100 | |0 |1100 |
|0 |1101 | |1 |1101 |
|0 |1110 | |1 |1110 |
|1 |1111 | |0 |1111 |
3. Gray code
|Decimal Equivalent |Binary Code |Gray Code |
|0 |0000 |0000 |
|1 |0001 |0001 |
|2 |0010 |0011 |
|3 |0011 |0010 |
|4 |0100 |0110 |
|5 |0101 |0111 |
|6 |0110 |0101 |
|7 |0111 |0100 |
|8 |1000 |1100 |
|9 |1001 |1101 |
|10 |1010 |1111 |
|11 |1011 |1110 |
|12 |1100 |1010 |
|13 |1101 |1011 |
|14 |1110 |1001 |
|15 |1111 |1000 |
4. ASCII code
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