Data Representation
Computer memory stores only sequences of 0's and 1's. In order for
data to be meaningful we must agree how to encode that data with 0's
and 1's. Unfortunately, there are several systems that are used.
- Unsigned Integers
- - Simply use binary
- - Using N bits, the integers 0..2^(N)-1 can be represented.
- Examples-17
- Signed Magnitude
- - Use the first bit for a sign (0 for +, 1 for -)
- - Use the remaining N-1 bits for the magnitude (absolute value), in
binary
- - For N bits, -(2^(N-1)-1) to +(2^(N-1)-1) can be represented.
- - Simple, but arithmetic circuits become awkward
- - Two different zeroes are created (+0 is 000..00 and -0 is 100..00)
- - Arithmetic is awkward eg. add 0101 with 1110. Since signs are
different, determine the larger whose sign will be kept for answer,
subtract larger from smaller. Not pretty
- -Examples: +15, -6
- Excess-N
- - To represent X, use the binary code for X+M
- - Excess-2^(N-1) is normal, but any value of M can be used
- - The code for zero is 100..00 in Excess-2^(N-1)
- - First bit is still a sign bit in Excess-2^(N-1), but 1 is + and 0
is -
- - Excess-2^(N-1) is almost the same as Twos Complement (differ in sign
only)
- - Excess-2^(N-1) with N bits can represent -2^(N-1) to +2^(N-1)-1
- Example: For 4 bits use excess-8 notation -8,-7,0,7
- Ones Complement
- - Positive numbers begin with a 0, otherwise use normal binary codes
- - Negative numbers use the complement (toggle all bits) of the code
for the corresponding positive number
- - For N bits, -(2^(N-1)-1) to +(2^(N-1)-1)
- - Two different zeroes are created (+0 is 000..00 and -0 is
111..11?)
- Example: 5, -5
- Twos Complement
- - Positive numbers begin with a 0, otherwise use normal binary codes
- - Negative numbers use the "twos complement" of the code
for the corresponding positive number
- - To "twos complement" something:
- 1. Toggle every bit and then add 1 to the result (in binary)
- OR
- 2A. Leave the lowest 1 and trailing zeroes alone. Toggle the other
bits.
- OR
- 2B. Toggle any bit which has a 1 bit to its right.
- - Zero is represented ONLY by 00...00 (the only code which does this).
- - The "twos complement" operation is self-inverting. Applied
to any number (positive or negative) it will negate it (exception: -2^(N-1)
cannot be negated).
- - The first bit is a sign bit (0 for +, 1 for -)
- - Using N bits -(2^(N-1)) to +(2^(N-1)-1) can be represented
- Example: 5,-5,4,-4 in all styles
The 4-bit codes for signed integers in these formats:
| Value |
Signed Magnitude |
Excess-7 |
Excess-8 |
Excess-9 |
Ones Complement |
Twos Complement |
| -9 |
---- |
---- |
---- |
0000 |
---- |
---- |
| -8 |
---- |
---- |
0000 |
0001 |
---- |
1000 |
| -7 |
1111 |
0000 |
0001 |
0010 |
1000 |
1001 |
| -6 |
1110 |
0001 |
0010 |
0011 |
1001 |
1010 |
| -5 |
1101 |
0010 |
0011 |
0100 |
1010 |
1011 |
| -4 |
1100 |
0011 |
0100 |
0101 |
1011 |
1100 |
| -3 |
1011 |
0100 |
0101 |
0110 |
1100 |
1101 |
| -2 |
1010 |
0101 |
0110 |
0111 |
1101 |
1110 |
| -1 |
1001 |
0110 |
0111 |
1000 |
1110 |
1111 |
| 0 |
1000 or 0000 |
0111 |
1000 |
1001 |
1111 or 0000 |
0000 |
| +1 |
0001 |
1000 |
1001 |
1010 |
0001 |
0001 |
| +2 |
0010 |
1001 |
1010 |
1011 |
0010 |
0010 |
| +3 |
0011 |
1010 |
1011 |
1100 |
0011 |
0011 |
| +4 |
0100 |
1011 |
1100 |
1101 |
0100 |
0100 |
| +5 |
0101 |
1100 |
1101 |
1110 |
0101 |
0101 |
| +6 |
0110 |
1101 |
1110 |
1111 |
0110 |
0110 |
| +7 |
0111 |
1110 |
1111 |
---- |
0111 |
0111 |
| +8 |
---- |
1111 |
---- |
---- |
---- |
---- |