For example, in a 16-bit sign-magnitude system, the pattern 1000000011111111 represents the number and the pattern 0000000000000101 represents +5.

The other technique for representing both positive
and negative integers is known as * two's complement*. It has two
compelling advantages over the sign-magnitude representation, and
is now universally used for integers, but as we will see below
sign-magnitude is still used to represent real numbers. The two's
complement method is based on the fact that binary arithmetic in
fixed-length words is actually arithmetic over a finite cyclic
group. If we ignore overflows for a moment, observe what happens
when we add 1 to the largest possible number in an **n**-bit
system (this number is represented by a string of **n** 1's):

The result is a
pattern with a leading 1 and **n** 0's. In an **n**-bit
system only the low order **n** bits of
each result are saved, so this sum is functionally
equivalent to 0. Operations that lead to sums with very large
values ``wrap around'' to 0, i.e. the system is a finite cyclic
group. Operations in this group are defined by arithmetic
modulo 2.