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### Arithmetic vs. Exponential Increases

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Date: 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
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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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Associated Topics:
High School Definitions
High School Exponents
High School Sequences, Series
Middle School Definitions
Middle School Exponents

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