eolas/zk/Signed_and_unsigned_numbers.md

tags
binary binary-encoding

Signed and unsigned numbers

In order to represent negative integers in binary we use signed numbers. Signed binary is basically binary where negative integers can be represented. Unsigned binary is standard binary without negative integers.

In order to represent negative integers alonside positive integers a natural approach is to divide the available encoding space / word length into two subsets: one for representing non-negative integers and one for representing negative integers.

The primary method for doing this is to use two's complement. This method makes signed numbers possible in a way that complicates the hardware implementation of the binary arithmetic operations as little as possible.

Two's complement

Signed numbers can be implemented in binary in a number of ways. The differences come down to how you choose to encode the negative integers. A common method is to use "two's complement".

The two's complement of a given binary number is its negative equivalent

For example the two's complement of 0101 (binary 5) is 1011. There is a simple algorithm at work to generate the complement for 4-bit number:

1. Take the unsigned number, and flip the bits. In other words: invert the values, so 0 becomes 1 and 1 becomes 0.
2. Add one

To translate a signed number to an unsigned number you flip them back and still add one:

Formal expresssion: 2^n - x

The heuristic account of deriving two's complement above can be formalised as follows:

in a binary system that uses a word size of n bits, the two's complement binary code that represents negative x is taken to be the code that represents 2^n - x

Let's demonstrate this, again using -5 as the value we wish to encode.

Applying the formula to a 4-bit system, negative 5 can be derived as follows:

2^4 -5

16 -5 = 11

11 = 1011

So basically we carry out the subtraction against the word length and then convert the decimal result to binary.

We can also confirm the output by noting that when the binary form of the number and its negative are added the sum will be 0000 if we ignore the overflow bit:

1011 + 0101 = 0000

(This makes sense if we recall that earlier we derived the complement by inverting the bits.)

Advantages

The chief advantage of the two's complement technique of signing numbers is that its circuit implementation is no different from the adding of two unsigned numbers. Once the signing algorithm is applied the addition can be passed through an adder component without any special handling or additional hardware.

Let's demonstrate this with the following addition:

7 + -3 = 4

First we convert 7 to binary: 0111.

Then we convert -3 to unsigned binary, by converting 3 to its two's complement

0011 \rArr 1100 \rArr 1101

Then we simply add the binary numbers regardless of whether they happen to be positive or negative integers in decimal:

0111 \ 1101 \ ====\ 0100

Which is 4. This means the calculation above would be identical whether we were calculating 7 + -3 or 7 + 13.

The ease by which we conduct signed arithmetic with standard hardware contrasts with alternative approaches to signing numbers. An example of another approach is signed magnitude representation. A basic implemetation of this would be to say that for a given bit-length (6, 16, 32...) if the most significant bit is a 0 then the number is positive. If it is 1 then it is negative.

This works but it requires extra complexity to in a system's design to account for the bit that has a special meaning. Adder components would need to be modified to account for it.

Considerations

A limitation of signed numbers via two's complement is that it reduces the total informational capacity of a 4-bit number. Instead 16 permutations of bits giving you sixteen integers you instead have 8 integers and 8 of their negative equivalents. So if you wanted to represent integers greater than decimal 8 you would need to increase the bit length.