The set of numbers of the form a + bÖ2 where a
and b are integers form a mathematical structure called a
ring. It is easy to show that R is closed for addition, that 0
belongs to R and is the additive identity and that every number in
R has an additive inverse which is in R. Also addition of
numbers in R is associative so this is an additive
group.
What about multiplication?
Again it is easy show that R is closed for multiplication, that 1
belongs to R and is the multiplicative identity and that multiplication
of numbers in R is associative. However it is also easy
to find a counter example to show that not every number in R has a
multiplicative inverse which is in R
(Try this for youself, for example
look for an inverse for (2 + 3Ö2) and you will find that it would have
to be
but this number is not in R.
NB. R is the set of numbers
a + bÖ2 where a and b are integers).
So we can add, subtract and multiply these numbers. If u, v and
w are in the set R it is easy to show that the distributive
property holds: u(v + w) = uv + uw. So we have some of the same
structure as the arithmetic of real numbers but without division.
This structure is called a ring.