Control Decimal Vs Binary Numeration

Counting from zero to twenty using four different kinds of numeration systems: hash marks, Roman numerals, decimal, and binary:

System:    Hash Marks               Roman     Decimal     Binary
-------    ----------               -----     -------     ------
Zero       n/a                       n/a         0          0
One        |                          I          1          1
Two        ||                         II         2          10
Three      |||                        III        3          11
Four       ||||                       IV         4          100
Five       /|||/                      V          5          101
Six        /|||/ |                    VI         6          110
Seven      /|||/ ||                   VII        7          111
Eight      /|||/ |||                  VIII       8          1000
Nine       /|||/ ||||                 IX         9          1001
Ten        /|||/ /|||/                X          10         1010
Eleven     /|||/ /|||/ |              XI         11         1011
Twelve     /|||/ /|||/ ||             XII        12         1100
Thirteen   /|||/ /|||/ |||            XIII       13         1101
Fourteen   /|||/ /|||/ ||||           XIV        14         1110
Fifteen    /|||/ /|||/ /|||/          XV         15         1111
Sixteen    /|||/ /|||/ /|||/ |        XVI        16         10000
Seventeen  /|||/ /|||/ /|||/ ||       XVII       17         10001
Eighteen   /|||/ /|||/ /|||/ |||      XVIII      18         10010
Nineteen   /|||/ /|||/ /|||/ ||||     XIX        19         10011
Twenty     /|||/ /|||/ /|||/ /|||/    XX         20         10100

Neither hash marks nor the Roman system are very practical for symbolizing large numbers. Obviously, place-weighted systems such as decimal and binary are more efficient for the task. Notice, though, how much shorter decimal notation is over binary notation, for the same number of quantities. What takes five bits in binary notation only takes two digits in decimal notation. This raises an interesting question regarding different numeration systems: how large of a number can be represented with a limited number of cipher positions, or places? With the crude hash-mark system, the number of places IS the largest number that can be represented, since one hash mark "place" is required for every integer step. For place-weighted systems of numeration, however, the answer is found by taking base of the numeration system (10 for decimal, 2 for binary) and raising it to the power of the number of places. For example, 5 digits in a decimal numeration system can represent 100,000 different integer number values, from 0 to 99,999 (10 to the 5th power = 100,000). 8 bits in a binary numeration system can represent 256 different integer number values, from 0 to 11111111 (binary), or 0 to 255 (decimal), because 2 to the 8th power equals 256. With each additional place position to the number field, the capacity for representing numbers increases by a factor of the base (10 for decimal, 2 for binary).

An interesting footnote for this topic is the one of the first electronic digital computers, the Eniac. The designers of the Eniac chose to represent numbers in decimal form, digitally, using a series of circuits called "ring counters" instead of just going with the binary numeration system, in an effort to minimize the number of circuits required to represent and calculate very large numbers. This approach turned out to be counter-productive, and virtually all digital computers since then have been purely binary in design. To convert a number in binary numeration to its equivalent in decimal form, all you have to do is calculate the sum of all the products of bits with their respective place-weight constants. To illustrate:

Convert 110011012  to decimal form:
bits =         1  1  0  0  1  1  0  1             
.              -  -  -  -  -  -  -  -
weight =       1  6  3  1  8  4  2  1
(in decimal    2  4  2  6
notation)      8    

The bit on the far right side is called the Least Significant Bit (LSB), because it stands in the place of the lowest weight (the one's place). The bit on the far left side is called the Most Significant Bit (MSB), because it stands in the place of the highest weight (the one hundred twenty-eight's place). Remember, a bit value of "1" means that the respective place weight gets added to the total value, and a bit value of "0" means that the respective place weight does not get added to the total value. With the above example, we have:

12810  + 6410  + 810  + 410  + 110  = 20510

If we encounter a binary number with a dot (.), called a "binary point" instead of a decimal point, we follow the same procedure, realizing that each place weight to the right of the point is one-half the value of the one to the left of it (just as each place weight to the right of a decimal point is one-tenth the weight of the one to the left of it). For example:

Convert 101.0112  to decimal form:
.
bits =         1  0  1  .  0  1  1               
.              -  -  -  -  -  -  -  
weight =       4  2  1     1  1  1   
(in decimal                /  /  /
notation)                  2  4  8

410  + 110  + 0.2510  + 0.12510  = 5.37510

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Control System of Numeration

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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