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)”.

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Where did you get the negative in ax-by?

Starting from the algebraic version a+ib and accepting that if the symbol is meaningful then i squared is equal to -1 and accepting that formally we can rewrite (a+ib)(x+iy) as ax+(ib)(iy)+a(ix)+(ib)y then the second term can be rearranged as i times i times b times y , which is i squared times by.

So the second term becomes -by.

The purpose of all of this is to show that there is something which when applied twice has a reversal or negating effect. At the end of the day What I have presented shows that this something is not a normal number (it needs two normal numbers to describe it), but since the two number system obeys all the rules of the single number system we can give this thing a name and work with its name as if it were a normal number.

So we can use the name “i” for the pair (0,1)

That’s what I thought you were doing, which is very nicely thought out. I am thinking of it from teaching students, who would not understand why the first pair has a negative, without a negative multiple. I would assume then (although that might be dangerous!) that students would need to have an introduction to/some knowledge of i, -i?

Cleargrace (I like that name), this is in reply to your second comment.

The thing is that the square roots of negative numbers keep popping up, with quadratic equations firstly. The question is – what to do about it? ignore them? Why should we? My feeling is that for example root(-9) can be written as 3*root(-1), so the only worrisome thing is root(-1). I would keep it in this form and avoid the ‘i’ for a short time anyway. There is no harm in using the ‘i’ in the conventional way for doing ‘sums’ and algebra. But there is with quite a few students a lingering suspicion that it’s all a big fiddle. This is when the “pair of normal numbers” treatment will be understood and appreciated. Then if time permits you can do the same thing for fractions, where (a,b) replaces the magic of a/b, and for negative numbers, where (a,b) replaces the weirdness of the negative sign, and a signed number is defined as the result of a subtraction.

I have just realised what the Queen of Hearts was trying to do in Alice in Wonderland. Lewis Carroll was a mathematician, and one of the most important things in math is the assertion “Symbols mean what I say they mean!”.