Decimal versus binary numeration

Let’s count 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

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Systems of numeration

The Romans devised a system that was a substantial improvement over hash marks, because itused a variety of symbols (or ciphers) to represent increasingly large quantities. The notationfor 1 is the capital letter I. Thenotation for 5 is the capital letter V. Other ciphers possessincreasing values:

X = 10

L = 50

C = 100

D = 500

M = 1000
Systems of numeration

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

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Numbers and symbols

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:
(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 6 < 20

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