Imaginary unit

The imaginary unit, $i=\sqrt{-1}$ , is the fundamental component of all complex numbers. In fact, it is a complex number itself. It has a magnitude of 1, and can be written as $1 \mathrm{cis} \left(\frac{\pi}{2}\right)$ .

Problems

Introductory

Find the sum of $i^1+i^2+\ldots+i^{2006}$

Solutions

Introductory

Let's begin by computing powers of $i$ .
$i^1=\sqrt{-1}$

$i^2=\sqrt{-1}\cdot\sqrt{-1}=-1$

$i^3=-1\cdot i=-i$

$i^4=-i\cdot i=-i^2=-(-1)=1$

$i^5=1\cdot i=i$

We can now stop because we have come back to our original term. This means that the sequence i, -1, -i, 1 repeats. Note that this sums to 0. That means that all sequences $i^1+i^2+\ldots+i^{4k}$ have a sum of zero (k is a natural number). Since $2006=4\cdot501+2$ , the original series sums to the first two terms of the powers of i, which equals $-1+i$ .

Page

Toolbox

Search

Imaginary unit

Contents

Problems

Introductory

Solutions

Introductory

See also