Abstract
- There are 3 Variants, computer use the 2’s Complement (补码) variant to store integer
- Computers use 2’s Complement (补码) to store the integers, since it solves the flaws in Sign-and-Magnitude (原码) and 1’s Complement (反码)
- Below is a 4-bits Integer Encoding under the 3 Variants
3 Variants
Sign-and-Magnitude (原码)
- Represent positive and negative decimal integers in binary with the help of Sign Bit
- Divide the number of bits into 2 parts, one part for negative number, another part for positive number
Require extra Logic Gates
- The Adder logic gates cant be used for subtraction
- This can be avoided using 1’s Complement (反码) or 2’s Complement (补码)
1’s Complement (反码)
- For negative number, all bits are inverted except the Sign Bit which is 1 and positive number unchanged
- We can invert all bits except Sign Bit to obtain Sign-and-Magnitude (原码)
Reuse Logic Gates
- We re-can use Adder logic gates, without building extra logic circuit like a subtraction logic gates
Mathematically wrong definition of 0
- We have
+0(when all bits are 0) &-0(when all bits are 1) under this encoding, this can be solved by 2’s Complement (补码)
2’s Complement (补码)
- This builds on top of 1’s Complement (反码), the only difference is to add extra
1to the negative number in 1’s Complement (反码) to have only 1 binary encoding for 0 - We can calculate the decimal value by taking the value Sign Bit (negative) and perform addition with value of Magnitude (all are positive)
Integer Overflow
- When a value is out of the range that the Datatype can hold
Check for Integer Overflow
- If the sign of the two operand are the same, and the sign of the result is different. Then we have an Integer (整数) overflow
- Think about how can the sum of two positive integer result in a negative number
- And how the sum of two negative integer result in a positive number
- The sum of a negative and positive integer guaranteed no integer overflow
Terminologies
Sign Bit
- The first bit from left
- 0 denotes positive
- 1 denotes negative
Magnitude
- All the bits in the encoding except Sign Bit