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Chapter 2: COMPLEX NUMBERS

# Introduction

If I needed to describe the distance between two cities, I could provide
an answer consisting of a single number in miles, kilometers, or some
other unit of linear measurement. However, if I were to describe how to
travel from one city to another, I would have to provide more
information than just the distance between those two cities; I would
also have to provide information about the *direction* to travel, as well.

The kind of information that expresses a single dimension, such as linear distance, is called a *scalar*
quantity in mathematics. Scalar numbers are the kind of numbers you've
used in most all of your mathematical applications so far. The voltage
produced by a battery, for example, is a scalar quantity. So is the
resistance of a piece of wire (ohms), or the current through it (amps).

However, when we begin to analyze alternating current circuits, we find
that quantities of voltage, current, and even resistance (called *impedance*
in AC) are not the familiar one-dimensional quantities we're used to
measuring in DC circuits. Rather, these quantities, because they're
dynamic (alternating in direction and amplitude), possess other
dimensions that must be taken into account. Frequency and phase shift
are two of these dimensions that come into play. Even with relatively
simple AC circuits, where we're only dealing with a single frequency, we
still have the dimension of phase shift to contend with in addition to
the amplitude.

In order to successfully analyze AC circuits, we need to work with
mathematical objects and techniques capable of representing these
multi-dimensional quantities. Here is where we need to abandon scalar
numbers for something better suited: *complex numbers*. Just like
the example of giving directions from one city to another, AC quantities
in a single-frequency circuit have both amplitude (analogy: distance)
and phase shift (analogy: direction). A complex number is a single
mathematical quantity able to express these two dimensions of amplitude
and phase shift at once.

Complex numbers are easier to grasp when they're represented
graphically. If I draw a line with a certain length (magnitude) and
angle (direction), I have a graphic representation of a complex number
which is commonly known in physics as a *vector*: (Figure below)

*A vector has both magnitude and direction.*

Like distances and directions on a map, there must be some common frame
of reference for angle figures to have any meaning. In this case,
directly right is considered to be 0^{o}, and angles are counted in a positive direction going counter-clockwise: (Figure below)

*The vector compass*

The idea of representing a number in graphical form is nothing new. We all learned this in grade school with the "number line:" (Figure below)

*Number line.*

We even learned how addition and subtraction works by seeing how lengths (magnitudes) stacked up to give a final answer: (Figure below)

*Addition on a "number line".*

Later, we learned that there were ways to designate the values *between* the whole numbers marked on the line. These were fractional or decimal quantities: (Figure below)

*Locating a fraction on the "number line"*

Later yet we learned that the number line could extend to the left of zero as well: (Figure below)

*"Number line" shows both positive and negative numbers.*

These fields of numbers (whole, integer, rational, irrational, real,
etc.) learned in grade school share a common trait: they're all *one-dimensional*.
The straightness of the number line illustrates this graphically. You
can move up or down the number line, but all "motion" along that line
is restricted to a single axis (horizontal). One-dimensional, scalar
numbers are perfectly adequate for counting beads, representing weight,
or measuring DC battery voltage, but they fall short of being able to
represent something more complex like the distance *and* direction between two cities, or the amplitude *and*
phase of an AC waveform. To represent these kinds of quantities, we
need multidimensional representations. In other words, we need a number
line that can point in different directions, and that's exactly what a
vector is.

**REVIEW:**- A
*scalar*number is the type of mathematical object that people are used to using in everyday life: a one-dimensional quantity like temperature, length, weight, etc. - A
*complex number*is a mathematical quantity representing two dimensions of magnitude and direction. - A
*vector*is a graphical representation of a complex number. It looks like an arrow, with a starting point, a tip, a definite length, and a definite direction. Sometimes the word*phasor*is used in electrical applications where the angle of the vector represents phase shift between waveforms.

## Related Content:

Video lecture: The Complex Numbers