Why is $i/i$ equal to 1 I never got a chance to take complex analysis in college so I decided to study it on my own. In the beginning of the books they start by proving some properties of complex numbers however I noticed that most of the proofs involve (usually implicitly) the assumption that $i/i = 1$ It is not obvious to me that this should be true so I am trying to prove it myself.
I want to prove that $i/i = 1$ starting with the definition that $i ^ {2} = -1 $ my first thought was to simply apply complex division like so.
$ \frac{0 + i}{0 + i} = \frac{0 + i}{0 + i} \left (\frac{0 - i}{0 - i} \right ) = \frac {0 - 0i +i0 - (i \cdot i)}{0 - 0i +i0 - (i \cdot i)} = \frac {1}{1} = 1$
However, it occurs to me that in multiplying by the conjugate divided by itself, I am implicitly assuming that $\frac{i}{i} = 1$ which is of course circular reasoning.
How should I approach this?
 A: The notation $x/y$ is just another way of writing $xy^{-1}$. By definition, $yy^{-1}=y^{-1}y=1$. So $i/i=ii^{-1}=1$.
This all follows from the definition of inverses and the notation we use for them. There is nothing to prove, really.
A: Your proof would be as simple as citing an axiom.
$i * 1 = i$  because 1 is the multiplicative identity.
dividing both sides of the equation by $i$ gives us:
$1= \frac{i}{i}$
A: You should ask yourself the following: what does "division" mean, anyway?  I would say this: once you know how multiplication works, the quotient $a/b$ means the number $x$ for which $bx = a$.  For example, I would say that
$$
20/4 = 5
$$
because $x = 5$ is the unique solution to
$$
4x = 20
$$
Now: what should $i/i$ be?  It should be the value of $x$ for which
$$
i = i\,x
$$

There's an interesting technical issue here: how do we know that $x$ is the only number for which $i\,x = i$?  Here's a justification I like: if we "combine like terms" we can rewrite the equation as
$$
i(x - 1) = 0
$$
Now, we can use the following fact: if $ab = 0$, then either $a = 0$ or $b = 0$.  Since $i \neq 0$, it must be the case that $(x-1) = 0$, which is to say that $x = 1$.
Of course, the statement "if $ab = 0$, then either $a = 0$ or $b = 0$" should be proven.  However, I find it very intuitively believable, so I'm content to leave it at that.
A: Think of complex multiplication as rotation and scale of the number being multiplied.
Every real number number multiplied by $i$ rotates $90\deg$ counterclockwise in the complex plane. Because division is just reversed multiplication, dividing by $i$ rotates the number clockwise. When you start with $i$ the point $0+i$ lays in the $y$-plane of the complex plane $1$ units of distance from the origin, if you rotate it clockwise you get the correct answer on the real line, $1$.
A: You can see the identity in this way: $$\displaystyle\frac{1}{i}=-i,$$
in this way you obtain $-i\cdot i=-(-1)=1.$
