Lecture 06 - Signed numbers

Goals

• Learn how to represent signed numbers
• Learn the process of converting back and forth between two's compliment and decimal
• Learn about two's compliment addition

negative numbers

We are, of course, missing out on half of all integers -- we have nothing less than 0! we call this unsigned binary

How can we represent signed numbers? If we are working on paper, then we can just put a negative sign in front, but in hardware we don't get a negative sign

there are a collection of different approaches I will tell you about two

• signed magnitude which seems like the obvious way
• two's compliment which is the way we actually do it

signed magnitude

we could just reserve the first bit as a sign bit

this is called signed magnitude

there are a couple of reasons we aren't fond of this

• we get two zeros: 000 (0) and 100 (-0)
• comparisons between two numbers require a lot of extra logic
• addition gets harder because we need extra logic to check which operation we are actually doing

two's compliment

two's compliment is based on a simple process that we use to negate a number

• flip all the bits (run them all through a NOT operation)
• add 1 to the result (ignoring any carry out)

this works in either direction -- making a positive number negative and a negative number positive

In order for this to make any sense, it is essential that we know how many bits our numbers are

Example: the 4-bit binary representation of 6 is 0110 to find the representation of -6

• flip the bits: 1001
• add 1: 1010

-6 = 1010 in 4-bit two's compliment

comparing two's compliment and signed magnitude

Note that we still end up with the high-order bit being 1 when the number is negative

conversions

We use the same process if we want to convert a two's compliment number to decimal

• if the number is positive
  • convert the number like it was unsigned
• if the number is negative
  • negate it
    • flip all the bits
    • add 1
  • convert the now positive number like it was unsigned
  • negate the result (add a minus sign)

Example: finding the value of the 4-bit two's compliment number 1110

• negate it
  • flip the bits: 0001
  • add 1: 0010
• convert it to decimal: 2
• negate the result: -2

Math with two's compliment

One of the nice features of two's compliment is that because the representation is designed to be cyclic, addition just works

 1111
  1111 (-1)
+ 0011 ( 3)
----------
  0010 ( 2)

Note that we have ignored the carry out of the last column

That is not to say that we can't sum up to numbers we can't represent

 0111
  0111 (7)
+ 0111 (7)
----------
  1110 (-2)

Just as we saw with unsigned numbers -- the result is more positive than we can represent, so we have overflow

This happens in the other direction as well

 10
  1010 (-6)
+ 1001 (-7)
----------
  0011 ( 3)

Here our result is to negative for us to be able to represent the number we call this underflow

somewhat confusingly, we sometimes just refer to "Exceeding our ability to represent the result" generically as overflow.

The key to telling when we have a good result or when we have underflow or overflow is to look at the carry in and the carry out of the leftmost column. If they match, the value is correct, if they don't match then we had underflow or overflow.

Mechanical level

vocabulary underflow

Skills

• convert from decimal to two's compliment
• convert from two's compliment to decimal
• add two two's compliment numbers together