1. ## Complex Numbers

Put the following expressions into the shape $\displaystyle a+ib$ with $\displaystyle a,b \in \mathbb{R}$.

(a) $\displaystyle i^3$

(b) $\displaystyle (1-i)i$

(a) $\displaystyle -i$

(b) $\displaystyle 1+i$

I know that the real and imaginary parts of a+ib are a and b respectively. But I don't know how to arrive at this answers, can anyone please explain? Thanks.

2. Originally Posted by demode
Put the following expressions into the shape $\displaystyle a+ib$ with $\displaystyle a,b \in \mathbb{R}$.

(a) $\displaystyle i^3$

(b) $\displaystyle (1-i)i$

(a) $\displaystyle -i$

(b) $\displaystyle 1+i$

I know that the real and imaginary parts of a+ib are a and b respectively. But I don't know how to arrive at this answers, can anyone please explain? Thanks.
Remember the defintion of i is $\displaystyle i^2=-1$

For the first $\displaystyle i^3=i^2\cdot i$

For the 2nd just distribute and use the property above.

3. Originally Posted by TheEmptySet
Remember the defintion of i is $\displaystyle i^2=-1$

For the first $\displaystyle i^3=i^2\cdot i$

For the 2nd just distribute and use the property above.

Thank you, that helps a lot. But I don't understand why $\displaystyle i^2=-1$! When I look at the imaginary-real axis, $\displaystyle i$ on the imaginary axis corresponds to 1 on the real axis. That's why I thought $\displaystyle i^3 = 1^3$.

4. Originally Posted by demode
Thank you, that helps a lot. But I don't understand why $\displaystyle i^2=-1$! When I look at the imaginary-real axis, $\displaystyle i$ on the imaginary axis corresponds to 1 on the real axis. That's why I thought $\displaystyle i^3 = 1^3$.
$\displaystyle i^2 = -1$ by DEFINITION of imaginary and complex numbers.

This is because they needed to create a number (i.e. out of their imaginations) that $\displaystyle = \sqrt{-1}$. They called this number $\displaystyle i$.

So if $\displaystyle i = \sqrt{-1}$ then that must mean $\displaystyle i^2 = -1$.