Date: 08/12/2003 at 17:00:24 From: Benjamin Subject: Decimal system How universal is math, and in what ways is it merely an artificial language? I realized the other day that while math is universal in function, the decimal system is just an artificial form of expression. For example suppose there were instead a "heximal system," 1 2 3 4 5 6 10. The decimal system is so engrained in us from such an early age, that this idea of an alternate base number seems absurd at a first glance. In any case, my question is why the decimal system? Why not 12 as the base number since it is divisible by more, and I would imagine would be more docile in math functions? Thanks, Benjamin
Date: 08/12/2003 at 17:57:03 From: Doctor Warren Subject: Re: Decimal system Hi Benjamin, Congratulations on your very astute observation. You are absolutely correct that numbers exist independently of the numeral system used to represent them. The number fifteen is an abstract concept that exists independently of its decimal representation (the numeral one followed by the numeral five). You can dream up any base you'd like, and represent your numbers in that base. For example, the base commonly called "octal" is composed of the symbols (numerals): 0 1 2 3 4 5 6 7 (there are no such symbols as 8 or 9 in octal.) There are also bases with MORE than ten symbols, such as "hexadecimal." The symbols used in hexadecimal are, by convention: 0 1 2 3 4 5 6 7 8 9 A B C D E F The symbols A through F mean the numbers "ten" through "fifteen." Another common base is called "binary," and has only two symbols: 0 1 Binary is used by all digital machines, like your computer. A digital switch is either on or off, and you can represent numbers in binary by strings of ones and zeros, just as you can represent numbers in decimal by strings of decimal numerals. The standard place system we use in decimal numbers is commonly used with other bases, too. For example the symbols "123" mean the following things in decimal, octal, and hexadecimal: decimal: 1 2 3 1*10^2 + 2*10^1 + 3*10^0 octal: 1 2 3 1*8^2 + 2*8^1 + 3*8^0 hexadecimal: 1 2 3 1*16^2 + 1*16^1 + 1*16^0 In other words, the symbols "123" means the number "one hundred twenty-three" in decimal, but they mean the number "eighty-three" in octal, and they mean "two hundred ninety-one" in hexadecimal. (Try adding 1*64 + 2*8 + 3*1 and 1*256 + 1*16 + 3*1.) You're right, the decimal system is very much ingrained in our notion of numbers. When I say the number "eighty-three," I don't necessarily mean the numeral 8 followed by the numeral 3; what I mean is the number that follows eighty-two and precedes eighty-four. There is no way for us to name a number, however, except in terms of numerals in some specific base, like decimal. You can do all your arithmetic the same way in different bases, too. For example, if you want to add two numbers, consider the following "rule:" decimal: 9 + 1 = 10 octal: 7 + 1 = 10 hexadecimal: F + 1 = 10 If you want to try your hand at doing math in different bases, the Windows calculator (Start Menu->Programs->Accessories->Calculator) can do math in decimal, octal, hexadecimal, and binary in its "scientific mode" (View menu->Scientific). As for your second question: why base 10? The simple answer is that we have ten fingers. Some other cultures have used number systems with more "symmetries." The Babylonians used a base-60 system, largely because of the number of ways you can divide 60. Here's a great page on the history of mathematics in different cultures: http://www-gap.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html Take a look at the numerals of the ancient Babylonians. - Doctor Warren, The Math Forum http://mathforum.org/dr.math/
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