Section 2.2 Algebra with Complex Numbers: Rectangular Form
The form of the complex number in Section 2.1:
\begin{equation}
z=x+iy\tag{2.2.1}
\end{equation}
is called the rectangular form, to refer to rectangular coordinates.
We will now extend the definitions of algebraic operations from the real numbers to the complex numbers. For two complex numbers \(z_1 = x_1 + i y_1\) and \(z_2 = x_2 + i y_2\text{,}\) we define
Addition and Subtraction.
The sum of \(z_1\) and \(z_2\) is given by
\begin{align}
z_1 + z_2 \amp = (x_1 + i y_1)+(x_2 + i y_2)\notag\\
\amp = (x_1 + x_2)+ i (y_1 +y_2)\tag{2.2.2}
\end{align}
Notice that all we have done is add the real parts of the complex numbers and separately added the imaginary parts. You should be able to convince yourself, with a diagram in the complex plane, that this definition is the same as the parallelogram rule for addition of vectors. The rule for subtraction of complex numbers follows as a straightforward extension.
Multiplication.
To define multiplication, we need a new rule, \(i^2=-1\text{.}\) We say that "\(i\) is a square root of minus one". This rule has no analogy for vectors in two dimensions and gives us additional algebraic structure that these vectors do not have.
Now we define multiplication in an obvious way by using the distributive rule of multiplication (i.e. we can "FOIL" everywhere).
\begin{align}
z_1 z_2 \amp = (x_1 +i y_1)(x_2 + iy_2)\notag\\
\amp = x_1 y_1 + x_1 i y_2 +i y_1 x_2 + (i)^2 y_1 y_2\notag\\
\amp = (x_1 x_2 - y_1 y_2)+i (x_1 y_2 +x_2 y_1)\tag{2.2.3}
\end{align}
It is conventional to rearrange the terms in the product into standard form, i.e. so that the real terms are all together and the pure imaginary terms are all together.
Informal Description of Properties.
We see (and/or have assumed) that addition and multiplication of complex numbers is commutative, associative, and distributive. This means that you can do algebra with complex numbers exactly as you are used to; just remember that whenever you see \(i^2\) to replace it with \(-1\text{.}\)
