Challenge Level

The set of numbers of the form $a + b\sqrt 2$ 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 $(2 + 3\sqrt 2)$ and you will find that it would have to be $({-1\over 7} + {3\sqrt 2\over 14})$ but this number is not in $R$. NB. $R$ is the set of numbers $a + b\sqrt 2$ 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 .