Link to Current (updated) notes

Complex Numbers and Operations in the Complex Plane

With $\imath \equiv \sqrt{-1}$ , the complex numbers can be defined as the space of numbers spanned by the vectors:

$$\left( \begin{array}{c} 1\ 0 \end{array} \right) \left( \begin{array}{c} 0\ \imath \end{array} \right)$$ (08-1)

so that any complex number can be written as

$$z = x \left( \begin{array}{c} 1\ 0 \end{array} \right) + y\left( \begin{array}{c} 0\ \imath \end{array} \right)$$ (08-2)

or just simply as

$$z = x + i y$$ (08-3)

where $x$ and $y$ are real numbers. Re$z \equiv x$ and Im$z \equiv y$ .

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-08)

(html Lecture-08)

(xml+mathml Lecture-08)

Operations on complex numbers

Complex Plane and Complex Conjugates

Because the complex basis can be written in terms of the vectors in Equation 8-1, it is natural to plot complex numbers in two dimensions--typically these two dimensions are the ``complex plane'' with $(0,\imath)$ associated with the $y$ -axis and $(1,0)$ associated with the $x$ -axis.

The reflection of a complex number across the real axis is a useful operation. The image of a reflection across the real axis has some useful qualities and is given a special name--``the complex conjugate.''

Figure: Plotting the complex number $z$ in the complex plane: The complex conjugate ($\bar{z}$ ) is a reflection across the real axis; the minus ($-z$ ) operation is an inversion through the origin; therefore $-(\bar{z}) = \bar{(-z)}$ is equivalent to either a reflection across the imaginary axis or an inversion followed by a reflection across the real axis.
The real part of a complex number is the projection of the displacement in the real direction and also the average of the complex number and its conjugate: Re$z = (z + \bar{z})/2$ . The imaginary part is the displacement projected onto the imaginary axis, or the complex average of the complex number and its reflection across the imaginary axis: Im$z = (z - \bar{z})/(2\imath)$ .