Computer Arithmetic

Sign-magnitude representation

There are several alternative conventions used to represent negative as well as positive integers, all of which involve treating the most significant (leftmost) bit in the word as a sign bit. If the sign bit is 0, the number is positive: if the sign bit is 1, the number is negative.

The simplest form of representation that employs a sign bit is the sign-magnitude representation. In an n-bit word, the rightmost n - 1 bits hold the mag­nitude of the integer.

In general, if an n-bit sequence of binary digits an−1an−2...a1a0 size 12{a rSub { size 8{n - 1} } a rSub { size 8{n - 2} } "." "." "." a rSub { size 8{1} } a rSub { size 8{0} } } {} represented A, its value is

A=∑i=0n−2ai2i size 12{A= Sum cSub { size 8{i=0} } cSup { size 8{n - 2} } {a rSub { size 8{i} } 2 rSup { size 8{i} } } } {} If an−1=0 size 12{a rSub { size 8{n - 1} } =0} {}

A=−∑i=0n−2ai2i size 12{A= - Sum cSub { size 8{i=0} } cSup { size 8{n - 2} } {a rSub { size 8{i} } 2 rSup { size 8{i} } } } {} If an−1=1 size 12{a rSub { size 8{n - 1} } =1} {}

For example: +18= 0 0010010

- 18=1 0010010

There are several drawbacks to sign-magnitude representation. One is that addition and subtraction require a consideration of both the signs of the numbers and their relative magnitudes to carry out the required operation. Another drawback is that there are two representations of 0 (e.g 0000 0000 and 1000 00000).

This is inconvenient, because it is slightly more difficult to test for 0 (an operation performed frequently on computers) than if there were a single representation.

Because of these drawbacks, sign-magnitude representation is rarely used in implementing the integer portion of the ALU. Instead, the most common scheme is twos complement representation.

Twos Complement Representation

• Ones Complement

The ones complement of a number is represented by flipping the number's bits one at a time. For example, the value 01001001 becomes 10110110.

Suppose you are working with B-bit numbers. Then the ones complement for the number N is 2B size 12{2 rSup { size 8{B} } } {}-1 -N

• Twos complement

The twos complement of a number N in a B-bit system is 2B size 12{2 rSup { size 8{B} } } {}-N.

There is a simple way to calculate a twos complement value: invert the number's bits and then add 1.

For a concrete example, consider the value 17 which is 00010001 in binary. Inverting gives 11101110. Adding one makes this 11101111.

• Twos Complement Representation

Like sign magnitude, twos complement representation uses the most significant bit as a sign bit, making it easy to test whether an integer is positive or negative. It differs from the use of the sign-magnitude representation in the way that the other bits are interpreted.

Consider an n-bit integer. A, in twos complement representation. If A is positive, then the sign bit an-1 is zero. The remaining, bits represent the magnitude of the number in the same fashion as for sign magnitude:

A=∑i=0n−2ai2i size 12{A= Sum cSub { size 8{i=0} } cSup { size 8{n - 2} } {a rSub { size 8{i} } 2 rSup { size 8{i} } } } {} If an−1=0 size 12{a rSub { size 8{n - 1} } =0} {}

The number zero is identified as positive and therefore has a 0 sign bit and a mag­nitude of all 0s. We can see that the range of positive integers that may he repre­sented is from 0 (all of the magnitude bits are 0) through - 1 (all of the magnitude bits are 1). Any larger number would require more bits.

For a negative number A (A < 0), the sign bit, is one. The remain­ing n-1 bits can take on any one of an−12n−1 size 12{a rSub { size 8{n - 1} } 2 rSup { size 8{n - 1} } } {} values. Therefore, the range of negative integers that can be represented is from -1 to - 2n−1 size 12{2 rSup { size 8{n - 1} } } {}

This is the convention used in twos complement representation, yielding the following expression for negative and positive numbers:

A = − 2 n − 1 a n − 1 + ∑ i = 0 n − 2 a i 2 i size 12{A= - 2 rSup { size 8{n - 1} } a rSub { size 8{n - 1} } + Sum cSub { size 8{i=0} } cSup { size 8{n - 2} } {a rSub { size 8{i} } 2 rSup { size 8{i} } } } {}

The range of A is from - 2n−1 size 12{2 rSup { size 8{n - 1} } } {} to 2n−1 size 12{2 rSup { size 8{n - 1} } } {} -1.

Example: Using 8 bit to represent

+50= 0011 0010

-70=1011 1010

Converting between Different Bit Lengths

The rule for twos complement integers is to move the sign hit to the new leftmost position and fill in with copies of the sign bit. For positive numbers, fill in with zeros, and for negative numbers, till in with ones.

For example:

+18 = 00010010

+18 = 00000000 00010010

-18 = 10010010

-18 = 11111111 10010010

Hardware for Addition and Subtraction:

Hardware Implementation of Unsigned Binary Multiplication:

Execution of Example:

Flowchart for Unsigned Binary Multiplication:

Solution 1:

• Convert to positive if required
• Multiply as above
• If signs were different, negate answer

Solution 2:

• Booth’s algorithm:

Example of Booth’s Algorithm: