Arithmetic vs. Exponential Increases

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

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/