Polar and rectangular notation
In order to work with these complex
numbers without drawing vectors, we first need some kind of standard
mathematical notation. There are two basic forms of complex number
notation: polar and rectangular.
Polar form is where a complex number is
denoted by the length (otherwise known as the magnitude,
absolute value, or modulus) and the angle of its
vector (usually denoted by an angle symbol that looks like this: ∠). To
use the map analogy, polar notation for the vector from New York City to
San Diego would be something like "2400 miles, southwest." Here are two
examples of vectors and their polar notations:
Standard orientation for vector angles in
AC circuit calculations defines 0o as being to the right
(horizontal), making 90o straight up, 180o to the
left, and 270o straight down. Please note that vectors angled
"down" can have angles represented in polar form as positive numbers in
excess of 180, or negative numbers less than 180. For example, a vector
angled ∠ 270o (straight down) can also be said to have an
angle of -90o. The above vector on the right (5.4 ∠ 326o)
can also be denoted as 5.4 ∠ -34o.
Rectangular form, on the other hand, is
where a complex number is denoted by its respective horizontal and
vertical components. In essence, the angled vector is taken to be the
hypotenuse of a right triangle, described by the lengths of the adjacent
and opposite sides. Rather than describing a vector's length and
direction by denoting magnitude and angle, it is described in terms of
"how far left/right" and "how far up/down."
These two dimensional figures (horizontal
and vertical) are symbolized by two numerical figures. In order to
distinguish the horizontal and vertical dimensions from each other, the
vertical is prefixed with a lower-case "i" (in pure mathematics) or "j"
(in electronics). These lower-case letters do not represent a physical
variable (such as instantaneous current, also symbolized by a lower-case
letter "i"), but rather are mathematical operators used to
distinguish the vector's vertical component from its horizontal
component. As a complete complex number, the horizontal and vertical
quantities are written as a sum:
The horizontal component is referred to
as the real component, since that dimension is compatible with
normal, scalar ("real") numbers. The vertical component is referred to
as the imaginary component, since that dimension lies in a
different direction, totally alien to the scale of the real numbers.
The "real" axis of the graph corresponds
to the familiar number line we saw earlier: the one with both positive
and negative values on it. The "imaginary" axis of the graph corresponds
to another number line situated at 90o to the "real" one.
Vectors being two-dimensional things, we must have a two-dimensional
"map" upon which to express them, thus the two number lines
perpendicular to each other:
Either method of notation is valid for
complex numbers. The primary reason for having two methods of notation
is for ease of longhand calculation, rectangular form lending itself to
addition and subtraction, and polar form lending itself to
multiplication and division.
Conversion between the two notational
forms involves simple trigonometry. To convert from polar to
rectangular, find the real component by multiplying the polar magnitude
by the cosine of the angle, and the imaginary component by multiplying
the polar magnitude by the sine of the angle. This may be understood
more readily by drawing the quantities as sides of a right triangle, the
hypotenuse of the triangle representing the vector itself (its length
and angle with respect to the horizontal constituting the polar form),
the horizontal and vertical sides representing the "real" and
"imaginary" rectangular components, respectively:
To convert from rectangular to polar,
find the polar magnitude through the use of the Pythagorean Theorem (the
polar magnitude is the hypotenuse of a right triangle, and the real and
imaginary components are the adjacent and opposite sides, respectively),
and the angle by taking the arctangent of the imaginary component
divided by the real component:
- REVIEW:
- Polar
notation denotes a complex number in terms of its vector's length and
angular direction from the starting point. Example: fly 45 miles ∠ 203o
(West by Southwest).
- Rectangular
notation denotes a complex number in terms of its horizontal and
vertical dimensions. Example: drive 41 miles West, then turn and drive
18 miles South.
- In rectangular notation, the first
quantity is the "real" component (horizontal dimension of vector) and
the second quantity is the "imaginary" component (vertical dimension
of vector). The imaginary component is preceded by a lower-case "j,"
sometimes called the j operator.
- Both polar and rectangular forms of
notation for a complex number can be related graphically in the form
of a right triangle, with the hypotenuse representing the vector
itself (polar form: hypotenuse length = magnitude; angle with respect
to horizontal side = angle), the horizontal side representing the
rectangular "real" component, and the vertical side representing the
rectangular "imaginary" component.
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