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/
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