We'll begin our discussion of complex numbers with a simple picture
of the real number line. Zero in the centre, $1,2,3,\dots$ to the
right, $-1,-2,-3,\dots$ to the left. Think of each real number not
as a point on the line, but as the corresponding vector, as the arrow
emanating from $0$ and ending at, say, $2$. Then real number addition
is vector addition; to add two numbers, say, $1$ and $2$, place
the tail of one at the tip of the other. The sum is the vector ending
at the final tip, $1+2=3$. This works for negative numbers too; $2+(-3)=-1$,
$2+(-2)=0$. Multiplication is rescaling; to multiply a vector/number
by $3$, you scale up its length by a factor of $3$, and leave the
direction the same; $2\rightarrow6$, and $-2\rightarrow-6$. To multiply
by a negative number $v$; you still rescale by the length of $v$,
but you also reverse direction, rotating around $180^{\circ}$, just
like the negative vectors are all $180^{\circ}$ away from their positive
counterparts. So $-3\cdot-2=6$.

We can extend the law of addition to all vectors in the plane containing
the real ones. Given two vectors of whatever lengths in whatever directions,
you add them by putting tail to tip and completing the triangle. This
plus this equals that, this plus this equals that.

To multiply a vector by a real number (a real vector), we use the
same rule as before; if the real number is positive, pointing horizontally
to the right, then you just rescale its target by its length, this
times $2$ is that. If the real number is negative, pointing horizontally
to the left, it rescales its target by its length, and then rotates
it by $180^{\circ}$. To multiply by an arbitrary vector in the plane,
say this one, you rescale by its length, which we call its \textbf{magnitude},
and rotate counterclockwise by the angle it forms with the positive
reals, with the horizontal, which we call its \textbf{argument}. So
multiplying by this doubles lengths, and rotates $45^{\circ}$ counterclockwise;
\[
(\nearrow)\cdot(\uparrow)=(\nwarrow)\]


Multiplying by this halves lengths and rotates $300^{\circ}$ counterclockwise
(or $60^{\circ}$ clockwise), taking this to that, and this to that.
Mulitplying by $1$, the unit vector to the right, fixes the length,
and rotates by $0^{\circ}$, ie, leaves everything the same. So $1$
is the multiplicative identity. The unit vector pointing north is
notable, multiplying by it just rotates things $90^{\circ}$. In particular,
$(\uparrow)\cdot(\uparrow)=(\leftarrow)$, which is $-1$. We usually
write $i$ for $\uparrow$, and we've just shown that $i^{2}=-1$.

Complex numbers are vectors in the plane, with addition given by vector
addition, and multiplication given by dilation and rotation. Real
numbers form the horizontal axis, and imaginary numbers form the vertical
axis. Why does it make sense to call these arrows {}``numbers''?
Because they satisfy all the basic rules of arithmetic: