It is imperative to understand that the type of numeration
system used to represent numbers has no impact upon the outcome of any
arithmetical function (addition, subtraction, multiplication, division,
roots, powers, or logarithms). A number is a number is a number; one plus
one will always equal two (so long as we're dealing with *real*
numbers), no matter how you symbolize one, one, and two. A prime number in
decimal form is still prime if it's shown in binary form, or octal, or
hexadecimal. π is still the ratio between the circumference and diameter of
a circle, no matter what symbol(s) you use to denote its value. The
essential functions and interrelations of mathematics are unaffected by the
particular system of symbols we might choose to represent quantities. This
distinction between *numbers* and *systems of numeration* is
critical to understand.
The essential distinction between the two is much like that between an
object and the spoken word(s) we associate with it. A house is still a house
regardless of whether we call it by its English name *house* or its
Spanish name *casa*. The first is the actual thing, while the second is
merely the symbol for the thing.
That being said, performing a simple arithmetic operation such as
addition (longhand) in binary form can be confusing to a person accustomed
to working with decimal numeration only. In this lesson, we'll explore the
techniques used to perform simple arithmetic functions on binary numbers,
since these techniques will be employed in the design of electronic circuits
to do the same. You might take longhand addition and subtraction for
granted, having used a calculator for so long, but deep inside that
calculator's circuitry all those operations are performed "longhand," using
binary numeration. To understand how that's accomplished, we need to review
to the basics of arithmetic. |