# Thread: Help with a multiplication table...

1. ## Help with a multiplication table...

I need help with a multiplication table for $\displaystyle Z_{2}[i] = {\{a+bi|a,b \in Z}\}$

I know the elements are {0, 1, i, i+1} for addition, are the multiplication elements the set \{0}?

Also, I can't figure out how to determine $\displaystyle i*i$ & $\displaystyle i(i+1)$

I have only had experience with modular arithmetic in $\displaystyle {Z}$

any explaination would be appreciated. Thanks in advance.

2. Originally Posted by ginafara
I need help with a multiplication table for $\displaystyle Z_{2}[i] = {\{a+bi|a,b \in Z}\}$

I know the elements are {0, 1, i, i+1} for addition, are the multiplication elements the set \{0}?

Also, I can't figure out how to determine $\displaystyle i*i$ & $\displaystyle i(i+1)$

I have only had experience with modular arithmetic in $\displaystyle {Z}$

any explaination would be appreciated. Thanks in advance.
The multiplicative group is the same as the group itself. (Otherwise it would not be closed.)

As so $\displaystyle ii^2$, what is -1 in your group? It is the additive inverse of 1, so
$\displaystyle 1 + x = 0$

$\displaystyle 1 + a + ib = 0$

$\displaystyle a = -1, b = 0$

Since this is based on $\displaystyle Z_2$,
$\displaystyle x = -1 \equiv 1$

You can do the same thing with $\displaystyle i(i + 1) = -1 + i \equiv 1 + i$.

-Dan