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.
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