Integers: signed magnitude

Signed magnitude

Reserve one particular bit for the sign.

Sign bit: If msb is 0, then positive value, if msb is 1, then negative value.

Converting base 10 to N-bit signed magnitude (SM):

1. Ignoring sign, convert base 10 value to binary.

2. If less than (N-1) bits, pad rest of bits to (N-1) with 0.

3. If negative, set msb to 1.

Converting N-bit SM to base 10:

1. Convert lower (N-1) bits to base 10.

2. If msb is 1, put minus sign in front of number.

Example:

Convert 3 in base 10 to 4-bit SM:

binary value: 11

using 3 bits: 011

with sign bit: 0011

Convert -3 in base 10 to 4-bit SM:

binary value: 11

using 3 bits: 011

with sign bit: 1011

What about 15?

binary value: 1111

However, this is already 4 bits, and the sign bit needs to be 0 for positive value.

Range of values for SM

How many possible values (i.e., representations)?

2^Nfor N bits

Maximum value

2^N-1 - 1 for (N-1) bits

Minimum value

-(2^N-1 - 1) for (N-1) bits

Total values, including 0:

2 *(2^N-1 -1) + 1 = 2^N - 1

What happened to the extra value?

There are actually 2 zeroes!

positive zero: N 0's

negative zero: 1 followed by (N-1) 0's

So, total number of representations is 2^N, but total number of values is 1 less.

Disadvantage of SM: 2 zeroes adds complexity to hardware.

Addition

It would be nice if we could use the same hardware for signed or unsigned addition.

However, that is not the case for signed magnitude.

Consider adding -1 and -1 in 4-bit SM

1001 + 1001 = 0010 in 4 bits, since the carry from the msb is lost.

This gives a result of +2, which is incorrect.

One way around this is to ignore the sign bit in doing the addition,

then use the sign bit for the result.

However, that doesn't work if the sign bits are different.

The sign bit of the result will depend on the relative magnitudes.

The magnitude of the result will be the difference in the magnitudes.

For example, (-5) + 3 is -2, but 5 + (-3) is +2.

Negation

Negating a value means giving it the opposite sign.

This is easy with SM, since we simply flip the sign bit.

negative value: sign bit 1 --> 0

positive value: sign bit 0 --> 1