*This sixth article of the mathematical journey through open source, introduces you to the imaginary numbers through octave.*

Yes, we have done the powerful matrix math with *octave*, without even thinking of how it is internally done. Now, on the same note, let’s continue to explore one of the most fascinating or rather most imaginary stuff of math.

## i Fun

Yes what else the imaginary numbers, numbers which really don’t exist but we try to visualize them as points on a 2-D plane with the real part on x-axis and the imaginary part on y-axis. So to plot the imaginary *i*, with a blue star (*), issue the following command at the *octave* prompt:

```
$ octave -qf
octave:1> plot(i, "b*")
octave:2>
```

and here pops the display window as shown in Figure 3.

What is this i? Just square root of -1. Simple! Right? Remember, we got an error trying *sqrt(-1)* in *bc*. *octave* is neat. It will answer you politely with an *i*. Still confused. Just check out the following:

```
$ octave -qf
octave:1> i
ans = 0 + 1i
octave:2> i * i
ans = -1
octave:3> sqrt(-1)
ans = 0 + 1i
octave:4> i^3
ans = -0 - 1i
octave:5>
```

*i* Puzzle

And now comes the puzzle part. What is the imaginary part of *i ^{i}*? It’s zero. What? Is it a real number? Yes it is. It is in fact equal to

*e*. Check out for yourself:

^{-π/2}```
$ octave -qf
octave:1> i^i
ans = 0.20788
octave:2> exp(-pi/2)
ans = 0.20788
octave:3>
```

Yes, you don’t really need to bother about how. octave just gets you there. And that’s fun with mathematics without getting into mathematics.

*i*: just a number

If you are not puzzled by this, great! In fact, as the imaginary numbers are numbers, just special numbers, in *octave*, they can be used in any operations you may think of with numbers. Try out *sqrt*, *exp*, *log* – addition, subtraction, multiplication, division are just trivial ones. And more fun would be with the *octave* power of vectors & matrices. Watch out:

```
$ octave -qf
octave:1> sqrt(i)
ans = 0.70711 + 0.70711i
octave:2> exp(i)
ans = 0.54030 + 0.84147i
octave:3> log(i)
ans = 0.00000 + 1.57080i
octave:4> [i # Press <Enter> here
> i] * [i i]
ans =
-1 -1
-1 -1
octave:5>
```

## Is *e*^{iθ} really *cos θ + i sin θ*?

^{iθ}

A very simple check. Let’s plot both the curves. Let *θ* (theta) be set to values from 0 to *π* (pi) with say intervals of 0.01, and then let’s plot the two curves in blue and red:

```
$ octave -qf
octave:1> th=0:0.01:pi;
octave:2> plot(th, exp(i*th), "b*;exp;", th, cos(th)+i*sin(th), "r^;cs;")
octave:3>
```

Figure 4 shows the plot with the 2 curves coinciding exactly with each other.

Equipped with *i*, let’s play out with more puzzles, as we move on. Get ready with *octave* controls.