The Imaginary number Sometimes one writes $i^2=-1$ to characterize the imaginary number and sometimes as the root of $-1$. So when I resolve the first equation, I get for the imaginary number two roots of $-1$ and thus two complex numbers. I am confused. 
Is it a question of definition or are the both definitions the same?
Thanks for your comment.
 A: Often people write that $i=\sqrt{-1}$ however this doesn't really mean anything as $\sqrt\cdot$ is only defined for non negative real numbers. One can think of the definition of $i$ like this:

Start with the real numbers with the normal operations of addition and multiplication. Add to the reals a symbol $i$ and the rule that $i^2=-1$.

From this we can write numbers as a linear combination $a+bi$ for real $x$ and $y$ and we can derive the rules for complex addition and multiplication by the definition of $i$
It is true that in the complex numbers there are two solutions to the equation $z^2+1=0$ however we can't define $i$ by saying "start with the equation and find a solution" because no solution exists in the reals.
The other thing to note is that most things to do with the complex numbers have a lot of symmetry to do with conjugation, If you imagine drawing everything in reverse on an argand diagram (so up becomes down, anti-clockwise becomes clockwise and such like) then you will find that nothing actually changes and in that sense the choice of $i$ as opposed to $-i$ is not only not a real choice but also one that doesn't matter.
A: When you write $\sqrt2$, are you worried that the equation $x^2=2$ has two solutions ?
