# How can I practice Jean-Robert Argand idea of the rotation of a square root of -1

I am studying complex numbers and I really need an intuition on how they work. I found the following video of Mathematician named Adrien Douady

He explains complex numbers in a algebraic geometry way, as he mentions using his protractor and turning it by 180 degrees to represent multiplication by (-1). But then at 4:26 he mentiond a mathematician Robert Argand http://en.wikipedia.org/wiki/Jean-Robert_Argand, and that he had a great Idea. "He said to himself, since multiplying by -1 is a 180 degree rotation, the square root is 1/2 of 180 degree rotation. How is this? how can I practice this? from what philophy of mathematics does this come from?

Thank and forgive me if I sound to dumb.

For vectors on the plane, the mapping $v\mapsto c\cdot v$ is zooming the vector by scalar $c$, if $c\in\Bbb R$.

When we compose two such maps, $f:=v\mapsto c\cdot v$ and $g:=v\mapsto d\cdot v$, we have that $g\circ f=v\mapsto d\cdot c\cdot v$.

In particular, if $c=d$, we get $v\mapsto c^2\cdot v$.

The mapping $s:v\mapsto -v$ can be indeed viewed as the rotation by $180^\circ$. Let us find its square root(s)!

Both rotations $r_{1,2}$ by $\pm90^\circ$ are such that $r_1\circ r_1=s=r_2\circ r_2$.

This can be set in a correspondence with the complex imaginary units $\pm i$.

This way, complex numbers can be identified with 'zooming-and-rotating' transformations of the plane:

• The complex number $1$ corresponds to the identity, i.e. zoom by $1$ and rotate by $0^\circ$.
• $0$ corresponds to the degenerated zooming by $0$, i.e. the mapping $v\mapsto 0$
• A positive real number $c$ corresponds to the zooming by $c$ (i.e. $v\mapsto c\cdot v$) and no rotation (i.e. rotating by $0^\circ$)
• A negative real number $-c$ (with $c>0$) corresponds to the zooming by $c$ and rotating by $180^\circ$.
• The imaginary unit $i$ corresponds to rotation by $+90^\circ$ (and no zooming, i.e. zoom by $1$).

Addition of complex numbers corresponds to 'pointwise addition' of these transformation: $$(f+g):=v\mapsto f(v)+g(v)$$ (There's an important point hidden here: try to prove that $f+g$ is again a 'zooming-and-rotating' transformation.)

Multiplication of complex numbers corresponds to composition of these transformation.

• So complex numbers come about when c=d and we get c^2? those are vectors right? (I am trying to understand sentence 2,3 of your answer) Commented May 16, 2013 at 11:37
• Errh... Start a clear page for your thoughts... I talked about $c^2$ because 'square root' (of a rotation) was mentioned.. I have updated my answer. say if anything is unclear. Commented May 16, 2013 at 11:51
• So if I understand well, and linguistically speaking, one place we need complex numbers is because of the rotation and the zoom? Because we need to move back and forth a point that need's to be scalled negativly or positivly without loosing its information and there was a gap with negative roots? So mathematicomechanically it was not allowd since the invention of the i? Commented May 16, 2013 at 12:16
• yes, something like that. We don't really need complex numbers. They just exist. The first place where complex numbers are mathematically needed, is the formula for the roots of cubic polynomials. If all the $3$ roots are real, then we have to move out to $\Bbb C$ for applying the formula (but imaginary parts will finally cancel each other). Commented May 16, 2013 at 12:20
• REALLY thank you. I think I got a good working pattern in my mind on were we need this. Can you direct me to some good books were I can read more about complex numbers? thank you again Commented May 16, 2013 at 12:26