Why does $\mathrm{abs}(i)=1$ I plugged into google calculator (because that is what I had on hand) $\mathrm{abs}(i)$ and it was said that equaled $1$. Is that true? And if so can I have a proof that it is true?
Note:
As a HS student taking AP Calculus my math proofs are not very advanced but I do understand the non textbook proof (mathematical proof) so if you would answer with a less formal proof I would very much appreciate it.
 A: The absolute value of a complex number $a+bi$ (with $a$ and $b$ real) is defined as $\sqrt{a^2+b^2}$.  In the case of $i=0+1i$, we get $|i|=\sqrt{0^2+1^2}=\sqrt{1}=1$.
One motivation for this definition is that if you think of $a+bi$ as representing the point $(a,b)$ in the plane, then $\sqrt{a^2+b^2}$ is the distance from this point to the origin $(0,0)$ (which represents the complex number $0+0i=0$).  So the absolute value of a complex number measures "how far it is from $0$", just like the absolute value of a real number.
For some discussion of further motivation behind this definition, you may be interested in this answer of mine.
A: This is simply the definition of $|z|$ for complex $z$. If $z=a+bi$ with $a,b$ real, then $$|z|=\sqrt{a^2+b^2}.$$ This is the distance of $z$ to $0$ using the standard Euclidean metric.  
When $b=0$, you get the standard real absolute value:
$$\sqrt{a^2}=|a|.$$
One nice feature if this definition is that if $w,z$ are complex numbers, then:
$$|wz|=|w|\cdot|z|.$$
This is not 100% obvious from the definition, and there are reasons related to the "geometry" of complex multiplication that explain this.
A: For a complex number $z$, you can think of $|z|$ referring to the distance between $(0,0)$ and $z$ on a complex plane. (you can think of absolute value as the distance between the origin and your number. e.g. $-3$ and $3$ are both $3$ away from 0)
We find the distance using Pythagorean theorem:
$$
| a + bi | = \sqrt{a^2+b^2}
$$
so if we add our values:
$$
\begin{align}
&\sqrt{0^2 + 1^2} \\
=&\sqrt{1} \\
=&1
\end{align}
$$
A: Every complex number $z$ can be uniquely written as $z=re^{i\theta}$ for some $r\in\mathbb R, r\geq 0$ and $\theta\in[0,2\pi)$. The norm of $z$ is defined as $|z|=r$.
Now because $i=1\cdot e^{i\pi}$, we have that $|i|=1$.
A: The modulus of a complex number is the square root of the product with its conjugate. This gives:
$$\lvert i\rvert=\sqrt{i(-i)}=\sqrt 1.$$
A: absolute value/ complex modulus is related to the distance you have to travel between the point and the origin.
