CS255 Sylabus

The sign-magnitude representation for signed integer values

How western civilization represents signed integer (whole) values

The western civilazation represents the positive integer (whole) values using the decimal number system:
And represents the negative integer (whole) values by prepending a negative sign before the decimal number representation:

This way to represent sign integer values is called:

The sign-magnitude representation
(Because the representation consists of a sign and a magnitude (= absolute value))

The mapping between the sign-magnitude representation and the signed values is as follows:

Sign- magnitude 0 representation: ... -4 -3 -2 -1 -0 1 2 3 ... --------------------------------------------- Value: •••• ••• •• • • •• •••

Short-comings of the sign-magnitude representation

There are 2 short-comings of the sign-magnitude representation:

There are 2 ways to represent (= write) the value ZERO in the sign-magnitude representation:

The meaning of arithmetic operation performed depends on the values in the operation

Example:

When we add: When we add: 4 4 + 3 + -3 --------- ---------- 7 1 We added 4 and 3 we subtracted 4 and 3 !

The sign-magnitude representation used in computers in the past

In the past, computers had used the sign-magnitude representation to store signed values

It is very straightforward to represent signed values in binary:

The left-most bit in the binary number represents the sign of the value
The remaining bits in the binary number are used to represent the magnitude (= absolute value)

Example: a sign-magnitude representation using 8 bits in binary:

00000111 represents 7 10000111 represents -7

Same short-comings are found in the sign-magnitude representation in binary

The same 2 short-comings is present in the sign-magnitude representation in binary:

The are 2 representations for the value ZERO:

The meaning of arithmentic operations depends on the value of the operands

Example:

  To compute:   7 + (-7)

                  00000111      (+7)
              +   10000111      (-7)
           ----------------
                  00000000

  We would not perform an addition
  Instead, we would perform a subtraction

Computers (in the past) that used the sign-magnitude representation for signed values must use more complex arithmetic circuits

Computers nowadays use the two's complement encoding method to represent signed integer numbers because of these advantages:

There is only 1 code representation for the value ZERO
The 2's complement code has a Mathematical property that when we add a negative value (such as + (-7)), we don't need to change the addition operation to a subtraction.

The best way to understand the two's complement encoding is to consider the decimal analogy: ten's complement encoding
- We will first discuss the 10s complement encoding
- Then we will discuss the 2s complement encoding.