# Proof

• Jan 29th 2008, 04:07 PM
Ideasman
Proof
For each $\displaystyle k \in \mathbb{Z}$, prove:

$\displaystyle i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i$

Okay.. so I need help with this. I thought the best way to prove these is by induction? So is induction something like:

Basis step, let k = 1.

i^{4*1} = i^{4} = 1 and hence it works

Now assume its true for $\displaystyle i^{4k}$. Prove that it's true for $\displaystyle i^{4(k+1)}$

$\displaystyle i^{4(k+1)} = i^{4k}i^{4}$

We know that i^{4k} is 1, and i^{4} is 1, and hence it's true?

Something along those lines?
• Jan 29th 2008, 04:12 PM
Jhevon
Quote:

Originally Posted by Ideasman
For each $\displaystyle k \in \mathbb{Z}$, prove:

$\displaystyle i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i$

Okay.. so I need help with this. I thought the best way to prove these is by induction? So is induction something like:

Basis step, let k = 1.

i^{4*1} = i^{4} = 1 and hence it works

Now assume its true for $\displaystyle i^{4k}$. Prove that it's true for $\displaystyle i^{4(k+1)}$

$\displaystyle i^{4(k+1)} = i^{4k}i^{4}$

We know that i^{4k} is 1, and i^{4} is 1, and hence it's true?

Something along those lines?

yes, that method seems fine to me, though i'd word it differently