Complex number.

“complex” as opposed to “simple” ?

“number” for what ?

Not for counting !

Not for measuring ! We’ll see about that !

“Square root of -1”, maybe, if that means anything at all !

Who needs the “i” ? It’s not essential.

Here goes…..

They say that (a+ib)(p+iq) = ap – bq + (bp + aq)i

But only if i is the square root of -1.

Getting rid of the i

Let us put the a and the b in a+ib together in brackets, as (a,b), and call this “thing” a “pair”.

This gets rid of the (magic) i straightaway.

Let us define an operation * to combine pairs:

(a,b)*(p,q) = (ap-bq, bp+aq)

This is the “pair” version of the “multiplication of complex numbers”.

It’s more interesting to read this as “(a,b) is applied to (p,q)”, and even better if we think of (p,q) as a “variable” and “apply (a,b)” as a function.

Ok, so we will write (x,y) instead of (p,q), and then

(a,b)*(x,y) = (ax-by, bx+ay)

Let us call the output of the “apply (a,b)” function the pair (X,Y)

Then

X = ax-by

Y = bx+ay

Now we can see this as a transformation of points in the plane:

The function “apply (a,b)” sends the point (x,y) to the point (X,Y)

Looking at some simple points we see that

(1,0)*(x,y) = (x,y)….no change at all

(-1,0)*(x,y) = (-x,-y)…the “opposite” of (x,y),

so doing (-1,0)* again gets us back to no change at all.

(0,1)*(x,y) = (-y,x)….which you may recognize as a rotation through 90 deg.

and doing (0,1)* again we get

(0,1)*(0,1)*(x,y) = (0,1)*(-y,x) = (-x,-y)….a rotation through 180 deg.

So with a bit of faith we can see that (0,1)*(0,1) is the same as (-1,0), and also that (-1,0)*(-1,0) = (1,0)…check it!

Consequently we have a system in which there are three interesting operations:

(1,0)* has no effect, it is like multiplying by 1

(-1,0)* makes every thing negative, it is like multiplying by -1, and

(0,1)*(0,1)* has the same effect as (-1,0)*

So we have found something which behaves like the square root of -1, and it is expressed as a pair of ordinary numbers.

It is then quite reasonable to give the name “i” to this “thing”, and use “i squared = -1”.

And generally, a complex number can be seen as a pair of normal (real) numbers, and bye-bye the magic !

When you think about it a fraction also needs two numbers to describe it.

Next post : matrix representation of “apply (a,b) to (x,y)”.