The expression of numerical quantities is something we tend
to take for granted. This is both a good and a bad thing in the study of
electronics. It is good, in that we're accustomed to the use and
manipulation of numbers for the many calculations used in analyzing
electronic circuits. On the other hand, the particular system of notation
we've been taught from grade school onward is not the system used
internally in modern electronic computing devices, and learning any
different system of notation requires some re-examination of deeply
ingrained assumptions.
First, we have to distinguish the difference between numbers and the
symbols we use to represent numbers. A number is a mathematical
quantity, usually correlated in electronics to a physical quantity such as
voltage, current, or resistance. There are many different types of numbers.
Here are just a few types, for example:
WHOLE NUMBERS: |
1, 2, 3, 4, 5, 6, 7, 8, 9 . . . |
|
INTEGERS: |
-4, -3, -2, -1, 0, 1, 2, 3, 4 . . . |
|
IRRATIONAL NUMBERS: |
π (approx. 3.1415927), e (approx. 2.718281828), |
square root of any prime |
|
REAL NUMBERS: |
(All one-dimensional numerical values, negative and positive, |
including zero, whole, integer, and irrational numbers) |
|
COMPLEX NUMBERS: |
3 - j4 , 34.5 ∠ 20o |
Different types of numbers find different application in the physical
world. Whole numbers work well for counting discrete objects, such as the
number of resistors in a circuit. Integers are needed when negative
equivalents of whole numbers are required. Irrational numbers are numbers
that cannot be exactly expressed as the ratio of two integers, and the ratio
of a perfect circle's circumference to its diameter (π) is a good physical
example of this. The non-integer quantities of voltage, current, and
resistance that we're used to dealing with in DC circuits can be expressed
as real numbers, in either fractional or decimal form. For AC circuit
analysis, however, real numbers fail to capture the dual essence of
magnitude and phase angle, and so we turn to the use of complex numbers in
either rectangular or polar form.
If we are to use numbers to understand processes in the physical world,
make scientific predictions, or balance our checkbooks, we must have a way
of symbolically denoting them. In other words, we may know how much money we
have in our checking account, but to keep record of it we need to have some
system worked out to symbolize that quantity on paper, or in some other kind
of form for record-keeping and tracking. There are two basic ways we can do
this: analog and digital. With analog representation, the quantity is
symbolized in a way that is infinitely divisible. With digital
representation, the quantity is symbolized in a way that is discretely
packaged.
You're probably already familiar with an analog representation of money,
and didn't realize it for what it was. Have you ever seen a fund-raising
poster made with a picture of a thermometer on it, where the height of the
red column indicated the amount of money collected for the cause? The more
money collected, the taller the column of red ink on the poster. |