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Numbers are certainly the most abstract codes we deal with on a regular basis. When we see the number

we don't immediately need to relate it to anything. We might visualize 3 apples or 3 of something else, but we'd be just as comfortable learning from context that the number refers to a child's birthday, a television channel, a hockey score, or the number of cups of flour in a cake recipe. Because our numbers are so abstract to begin with, it's more difficult for us to understand that this number of apples

doesn't necessarily have to be denoted by the symbol

Much of this chapter and the next will be devoted to persuading ourselves that this many apples

can also be indicated by writing

Let's first dispense with the idea that there's something inherently special about the number ten. That most civilizations have based their number systems around ten (or sometimes five) isn't surprising. From the very beginning, people have used their fingers to count. Had our species developed possessing eight or twelve fingers, our ways of counting would be a little different. It's no coincidence that the word digit can refer to fingers or toes as well as numbers or that the words five and fist have similar roots.

So in that sense, using a base-ten, or decimal (from the Latin for ten), number system is completely arbitrary. Yet we endow numbers based on ten with an almost magical significance and give them special names. Ten years is a decade; ten decades is a century; ten centuries is a millennium. A thousand thousands is a million; a thousand millions is a billion. These numbers are all powers of ten:

101 =10

102 =100

103 =1000 (thousand)

104 =10,000

105 =100,000

106 =1,000,000 (million)

107 =10,000,000

108 =100,000,000

109 =1,000,000,000 (billion)

Most historians believe that numbers were originally invented to count things, such as people, possessions, and transactions in commerce. For example, if someone owned four ducks, that might be recorded with drawings of four ducks:

Eventually the person whose job it was to draw the ducks thought, "Why do I have to draw four ducks? Why can't I draw one duck and indicate that there are four of them with, I don't know, a scratch mark or something?"

And then there came the day when someone had 27 ducks, and the scratch marks got ridiculous:

Someone said, "There's got to be a better way," and a number system was born.

Of all the early number systems, only Roman numerals are still in common use. You find them on the faces of clocks and watches, used for dates on monuments and statues, for some page numbering in books, for some items in an outline, and—most annoyingly—for the copyright notice in movies. (The question "What year was this picture made?" can often be answered only if one is quick enough to decipher MCMLIII as the tail end of the credits goes by.)

Twenty-seven ducks in Roman numerals is

The concept here is easy enough: The X stands for 10 scratch marks and the V stands for 5 scratch marks.

The symbols of Roman numerals that survive today are

The I is a one. This could be derived from a scratch mark or a single raised finger. The V, which is probably a symbol for a hand, stands for five. Two V's make an X, which stands for ten. The L is a fifty. The letter C comes from the word centum, which is Latin for a hundred. D is five hundred. Finally, M comes from the Latin word mille, or a thousand.

Although we might not agree, for a long time Roman numerals were considered easy to add and subtract, and that's why they survived so long in Europe for bookkeeping. Indeed, when adding two Roman numerals, you simply combine all the symbols from both numbers and then simplify the result using just a few rules: Five I's make a V, two V's make an X, five X's make an L, and so forth.

But multiplying and dividing Roman numerals is difficult. Many other early number systems (such as that of the ancient Greeks) are similarly in-adequate for working with numbers in a sophisticated manner. While the Greeks developed an extraordinary geometry still taught