Uses of Bases Other Than Base 10Date: 09/28/2004 at 12:38:10 From: Julie Subject: different math bases & when/why they're used Why are there different math bases and what would they be used for? I know that in the US we go by base 10, because of our 10 fingers, but I don't know who would use different bases and why that would be important for us to know. Date: 09/28/2004 at 14:22:35 From: Doctor Douglas Subject: Re: different math bases & when/why they're used Hi Julie. Thanks for writing to the Math Forum. It's true that for most people, the decimal (base-10) system is familiar, easy to use, and works well for most everyday purposes. It is a reasonable compromise between the number of different digits (in the sense of "symbol", {0,1,2,3,4,5,6,7,8,9}, and in the compactness of notation. And it is natural for us because we humans have ten digits (in the sense of "fingers"). But there are other base systems that can and have been used. The Mayans and Babylonians used base-20 and base-60 systems. These might have been more convenient for certain mathematical operations. For example, we now use 360 degrees in a circle. Why 360 and not 100 or 1000? Certainly the number 360 is conveniently divisible by many small numbers, such as 2,3,4,5,6,12,15, etc., and 100 and 1000 are less convenient in this regard. Similarly, the English system of measurement units uses 12 inches in a foot, and 36 inches in a yard. If all we ever did was in base-10, then those of us in the United States probably would have converted to the metric system a long time ago. Even if we did convert to metric for most things, it's likely that the measurement of time would still retain the concepts of day and year, which aren't related to each other by a factor of 10, but by astronomical quantities (the rotation of the earth and the revolution of the earth about the sun) that don't come in a convenient ratio of 10. The now widespread use of cheap computing power and storage has made the compactness issue less important, and so they use the binary number system, based on the two digits {0,1}, and its closely related variants of octal (eight digits: {0,1,2,3,4,5,6,7}) and hexadecimal (sixteen digits: {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}). You and I might write out the number two thousand and four as 2004, and here is how a computer might represent them: 2004 (decimal) = 7D4 (hexadecimal) = 3724 (octal) = 11111010100 (binary). For a computer, the last form leads to the simplest circuitry necessary to manipulate the number, while the hexadecimal form is compact to write in text (so you might see this in compression algorithms, or compact text representations of programs), and of course the decimal form is the form most familiar to us. There are also fractional bases, of which the natural base is the most important: e = 2.71828..., the Base of Natural Logarithms http://mathforum.org/dr.math/faq/faq.e.html. You might also come across a hybrid base, where different digits use different bases. You might think that this is weird, but there is one very familiar example--a digital clock: ##:## AM wx yz Here the digits wx represent the hours and the digits yz represent the minutes. You can see that the digit z simply counts through the numbers zero through nine, so that it is a base-10 digit. But the digit y only takes values from zero through five, so that it is a base-6 digit. When taken together as a single unit, the combination yz represents a number from zero to fifty-nine (base-60), which naturally enough means that there are 60 minutes in each hour. Similarly, the digit x is a base-10 digit by itself, but the combination wx is a base-12 object. And don't forget the "A" in AM or PM--that too is a digit (base-2) where the two symbols are not numerals, but letters: A and P. So depending on how you look at it, there are two or three number bases being used simultaneously in the simple process of telling time--yet it doesn't seem that unfamiliar to us. There is another place where I've seen hybrid bases is the recording of innings in baseball games: for example, a starting pitcher might throw 5 innings and get 2 more batters out before leaving the game. An inning is three outs per side, so the old way of saying this is that the pitcher went "five and two-thirds" innings, but now many stat boxes report this as "5.2 innings". The number that comes after 5.2 is 6.0, then 6.1, etc. Note that the two digits in this quantity represent different things: one represents the number of complete innings and the digit after the decimal point represents the number of additional outs. And 3 outs (not 10) make a complete inning for one team. For much more information about number bases (how to represent numbers in them, and convert between them), please check out our archives, where there is lots of additional information. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/ |
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