Arithmetic vs. Exponential IncreasesDate: 05/06/99 at 17:49:50 From: Heather Buck Subject: Arithmetic versus Mathematics To Whom it May Concern; I have a question that is perhaps as much a matter of English as it is of mathematics. The question arises from something I am reading that says "....the work produced... will increase exponentially rather than arithmetically." My first response was - exponents surely are part of arithmetic, and therefore this sentence does not make sense. Then I started wondering if 'arithmetic' meant something different from 'mathematics', and indeed it does. The dictionary says 'arithmetic' is the simplest branch of mathematics, dealing with computation of figures. Then I began to wonder if exponentiation could be considered as computation, because it is essentially a glorified form of multiplication. What do you think? Heather Buck Date: 05/07/99 at 17:00:08 From: Doctor Peterson Subject: Re: Arithmetic versus Mathematics The real issue here is to interpret the phrase "increase exponentially rather than arithmetically." I was displeased that my dictionary likewise doesn't give a clear definition of what this means, but it did have an entry for "arithmetic sequence," which suggests the idea. This is really a matter of mathematics more than English, but it ought to be clearly stated in dictionaries anyway! You may have heard of arithmetic (accent on "met") and geometric sequences. An arithmetic sequence is a sequence of numbers that increase by addition: the difference between successive terms is a constant. For example, 1, 3, 5, 7 is an arithmetic sequence with constant difference 2; each term is 2 more than the one before. On the other hand, a geometric sequence is one which increases by multiplication: the ratio of successive terms is a constant. For example, 1, 2, 4, 8 is a geometric sequence with constant ratio 2; each term is 2 times the previous one. To confuse the matter a bit, there is another pair of names for such sequences. The terms of an arithmetic sequence are said to grow "linearly," or in a straight line, because if you graph them that's what they look like. If the first term is a and the difference is k, the "n"th term will be s[n] = a + kn The terms of a geometric sequence are said to grow "exponentially"; that is, the "n"th term can be calculated as s[n] = a * k^n where a is the first term, k is the constant ratio, "*" means multiplication and "^" means an exponent. So we can describe the growth of some number as growing "arithmetically" (or "linearly") if it increases by a certain amount every year, or as growing "geometrically" (or "exponentially") if it doubles in a certain period. And that's what the original quotation was talking about. Exponential growth gets faster and faster; linear (arithmetical) growth is slow and steady. Now as to how arithmetic and mathematics are related, what you found was probably more or less right, but it's irrelevant to the question! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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