The Romans devised a system that was a substantial
improvement over hash marks, because it used a variety of symbols (or
ciphers) to represent increasingly large quantities. The notation for 1
is the capital letter I. The notation for 5 is the capital letter
V. Other ciphers possess increasing values:
X = 10
L = 50
C = 100
D = 500
M = 1000
If a cipher is accompanied by another cipher of equal or lesser value to
the immediate right of it, with no ciphers greater than that other cipher to
the right of that other cipher, that other cipher's value is added to the
total quantity. Thus, VIII symbolizes the number 8, and CLVII
symbolizes the number 157. On the other hand, if a cipher is accompanied by
another cipher of lesser value to the immediate left, that other cipher's
value is subtracted from the first. Therefore, IV symbolizes
the number 4 (V minus I), and CM symbolizes the
number 900 (M minus C). You might have noticed that ending
credit sequences for most motion pictures contain a notice for the date of
production, in Roman numerals. For the year 1987, it would read:
MCMLXXXVII. Let's break this numeral down into its constituent parts,
from left to right:
M = 1000
+
CM = 900
+
L = 50
+
XXX = 30
+
V = 5
+
II = 2
Aren't you glad we don't use this system of numeration? Large numbers are
very difficult to denote this way, and the left vs. right / subtraction vs.
addition of values can be very confusing, too. Another major problem with
this system is that there is no provision for representing the number zero
or negative numbers, both very important concepts in mathematics. Roman
culture, however, was more pragmatic with respect to mathematics than most,
choosing only to develop their numeration system as far as it was necessary
for use in daily life.
We owe one of the most important ideas in numeration to the ancient
Babylonians, who were the first (as far as we know) to develop the concept
of cipher position, or place value, in representing larger numbers. Instead
of inventing new ciphers to represent larger numbers, as the Romans did,
they re-used the same ciphers, placing them in different positions from
right to left. Our own decimal numeration system uses this concept, with
only ten ciphers (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) used in "weighted"
positions to represent very large and very small numbers.
Each cipher represents an integer quantity, and each place from right to
left in the notation represents a multiplying constant, or weight,
for each integer quantity. For example, if we see the decimal notation
"1206", we known that this may be broken down into its constituent
weight-products as such:
1206 = 1000 + 200 + 6
1206 = (1 x 1000) + (2 x 100) + (0 x 10) + (6 x 1)
Each cipher is called a digit in the decimal numeration system,
and each weight, or place value, is ten times that of the one to the
immediate right. So, we have a ones place, a tens place, a
hundreds place, a thousands place, and so on, working from right
to left.
Right about now, you're probably wondering why I'm laboring to describe
the obvious. Who needs to be told how decimal numeration works, after you've
studied math as advanced as algebra and trigonometry? The reason is to
better understand other numeration systems, by first knowing the how's and
why's of the one you're already used to.
The decimal numeration system uses ten ciphers, and place-weights that
are multiples of ten. What if we made a numeration system with the same
strategy of weighted places, except with fewer or more ciphers?
The binary numeration system is such a system. Instead of ten different
cipher symbols, with each weight constant being ten times the one before it,
we only have two cipher symbols, and each weight constant is twice
as much as the one before it. The two allowable cipher symbols for the
binary system of numeration are "1" and "0," and these ciphers are arranged
right-to-left in doubling values of weight. The rightmost place is the
ones place, just as with decimal notation. Proceeding to the left, we
have the twos place, the fours place, the eights place,
the sixteens place, and so on. For example, the following binary
number can be expressed, just like the decimal number 1206, as a sum of each
cipher value times its respective weight constant:
11010 = 2 + 8 + 16 = 26
11010 = (1 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (0 x 1)
This can get quite confusing, as I've written a number with binary
numeration (11010), and then shown its place values and total in standard,
decimal numeration form (16 + 8 + 2 = 26). In the above example, we're
mixing two different kinds of numerical notation. To avoid unnecessary
confusion, we have to denote which form of numeration we're using when we
write (or type!). Typically, this is done in subscript form, with a "2" for
binary and a "10" for decimal, so the binary number 110102 is
equal to the decimal number 2610.
The subscripts are not mathematical operation symbols like superscripts
(exponents) are. All they do is indicate what system of numeration we're
using when we write these symbols for other people to read. If you see "310",
all this means is the number three written using decimal numeration.
However, if you see "310", this means something completely
different: three to the tenth power (59,049). As usual, if no subscript is
shown, the cipher(s) are assumed to be representing a decimal number.
Commonly, the number of cipher types (and therefore, the place-value
multiplier) used in a numeration system is called that system's base.
Binary is referred to as "base two" numeration, and decimal as "base ten."
Additionally, we refer to each cipher position in binary as a bit
rather than the familiar word digit used in the decimal system.
Now, why would anyone use binary numeration? The decimal system, with its
ten ciphers, makes a lot of sense, being that we have ten fingers on which
to count between our two hands. (It is interesting that some ancient central
American cultures used numeration systems with a base of twenty. Presumably,
they used both fingers and toes to count!!). But the primary reason that the
binary numeration system is used in modern electronic computers is because
of the ease of representing two cipher states (0 and 1) electronically. With
relatively simple circuitry, we can perform mathematical operations on
binary numbers by representing each bit of the numbers by a circuit which is
either on (current) or off (no current). Just like the abacus with each rod
representing another decimal digit, we simply add more circuits to give us
more bits to symbolize larger numbers. Binary numeration also lends itself
well to the storage and retrieval of numerical information: on magnetic tape
(spots of iron oxide on the tape either being magnetized for a binary "1" or
demagnetized for a binary "0"), optical disks (a laser-burned pit in the
aluminum foil representing a binary "1" and an unburned spot representing a
binary "0"), or a variety of other media types.
Before we go on to learning exactly how all this is done in digital
circuitry, we need to become more familiar with binary and other associated
systems of numeration. |