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Roughing It More Rigorously

Date: 12/02/2010 at 01:19:42
From: Alex
Subject: Are there rigid definitions of "approx. equal" notations?

I'm a physics major, and I'm using LaTeX to make a comprehensive study
guide for my class' final exam. But something that's bugged me is that
there seem to be so many different ways to write "approximately equal 
to" -- and not just in LaTeX.

I've seen all of these symbols used:

   a tilde
   a double tilde (wavy equals sign)
   an equals sign with a tilde over it
   an equals sign with a dot over it
   &c

Are there differences among them? Is there one preferred by mathematicians
or the scientific community?

I think this is a simple case where a more rigid definition needs to be
put into place. Perhaps one could indicate the same order of magnitude or
number of digits? For example,

   n ~ m if (1 + (floor(log(n)))) = (1 + (floor(log(m))))

Thank you for your help!



Date: 12/06/2010 at 00:09:33
From: Doctor Vogler
Subject: Re: Are there rigid definitions of 

Hi Alex,

Thanks for writing to Dr. Math. 

I too have seen all of those ways to denotate "approximately equal to."
They do not have different meanings, but rather reflect the notational
preference of the writer. And each of them is usually used for imprecise
definitions (rather than mathematical definitions) of "approximately equal
to," especially when the author is a physicist or otherwise not a
mathematician.

That said, there *are* rigorous (precise) definitions of "approximately
equal to" and they apply in different situations. For example, the same
number of digits might not be practical if you don't know whether the
number you are approximating is close, for example, to the number 999. But
something very close to what you are describing would be written by
mathematicians using something like this:

  |log(n) - log(m)| < 0.5,

That expression is almost the same as having the same number of digits,
but isn't affected by being close to where that number changes (which
usually makes it easier to prove that it's true), and is very precise. You
can probably imagine similar ways to make precise statements about how
close n is to m. When n and m are functions of some other variable (like
distance to a center of mass, for example), then mathematicians often use
Big-O notation to show the main terms and precisely indicate which terms
are omitted; see

    http://en.wikipedia.org/wiki/Big_O_notation 

But the decision of which term is "main" depends on a limit, so the
approximation is good close to the limiting value and less good as you get
farther away.

It is also common to write something like ...

  n < m < n + 1/(n + 3)^2

... to indicate how m is approximately equal to m in a precise way.

In a particular book/paper/proof/etc., the author may find it convenient
to stick with a single definition of "approximately equal to" and use the
tilde (or one of the other notations) to mean this definition. No such
convention, however, is universal, and so any author who uses such a
convention is obligated to say precisely what he means by that notation.
In the absence of such a statement by the author, you can safely assume
that he or she probably means the informal statement, which is precisely
what you mean when you say in plain English "approximately equal to."

If you have any questions about this or need more help, please write back
and show me what you have been able to do, and I will try to offer further
suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Definitions
High School Logs

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