Elementary complex numbers question I've just started learning about the quaternions and it's raised some interesting questions for me about the complex numbers.
The $+$ sign in the expression $a+bi$ is confusing me. Usually when we use the $+$ sign it represents an operation combining two elements of the same space; $3+2=5$, $+$ is the operation of combining these two integers into a new integer. With the complex numbers the + sign in for example $1+i$ doesn't represent an operation. How could you combine an (strictly) imaginary number and a real number?
Apologies for the poorly expressed question,  but any help would be greatly appreciated.
 A: $a+bi$ represents the addition of a real number $a$ and an imaginary number $bi$. One can also represent a complex number as a point in $\mathbb{R}^2$: $(a,b)$, where $a$ is the real component and $b$ is the imaginary component. We can write these representations as
$$
a+bi=a(1,0)+b(0,1)=(a,b)
$$
This is why we generally talk about the real line and the complex plane.
$\hspace{3.2cm}$
Addition of complex numbers is handled as addition of vectors in $\mathbb{R}^2$. Multiplication of complex numbers is performed using the identity $i^2=-1$, so that the product of two complex numbers yields another complex number.

Clarification:
The complex numbers are an extension field of the reals (of degree 2). $\mathbb{R}$ can be embedded in $\mathbb{C}$ as $\{(x,0):x\in\mathbb{R}\}$. $1+0i\in\mathbb{C}$ is not the same number as $1\in\mathbb{R}$ since they live in different sets.
$\mathbb{R}$ can be embedded in $\mathbb{C}$, but it is not a subset of $\mathbb{C}$; however, the embedding of $\mathbb{R}$ as $\{(x,0):x\in\mathbb{R}\}$ is an isomorphism, and so often people say that $\mathbb{R}\subset\mathbb{C}$ because of this.
A: Some would view the complex number $1$ as $1+0i$ and the complex number $i$ as $0+1i$, so it's always viewed as a pair of real numbers, and then addition is addition of pairs, so that when you add $1+0i$ to $0+1i$ you get $1+1i$.
You can also view it geometrically: A complex number is an arrow pointing from the origin to a point in the plane, and then addition is completing a parallelogram.
Viewed in that way, multiplication is rotation and dilation. The simplest instance is that multiplying by $i$ means rotating $90^\circ$ counterclockwise and by $-i$ is $90^\circ$ clockwise.
Similar question troubled the brilliant physicist Edwin Jaynes concerning tensor algebras: How can you add a vector and a scalar?  He thought it seemed as bad as adding meters to kilograms.
A: There is a difference between interpreting '+' as a sign or as a binary operation. The sign '+' although often omitted, serves to distinguish a number, say $+x$ from its additive inverse $-x$. So viewed in this way the sign '+' or '-' is used in the same way as one would use $x^{-1}$ to indicate a unique inverse relating to a particular binary operation. A complex number, as also indicated in the other answers here, really consist of two components, and might as well be written $(a,b)$ - so writing it as $a+bi$, the '+' is merely a sign, and not a binary operation between the two components. You might as well also write $ib+a$: although this is not the convention, this is not "wrong".
It should be noted that complex numbers are not the same as $\mathbb{R}^2$ specifically in terms of how $i$ is defined and multiplication is defined (you can multiply the real part with the imaginary part of another complex number, etc.), as described in other answers here - and that is why a complex number is not usually denoted as a pair in brackets - so as not to confuse with $\mathbb{R}^2$, I think. 
