Symbol for Irrational Numbers?

Date: 09/23/2002 at 00:43:42
From: Rebekah
Subject: Symbol for irrational numbers?

I've asked everyone in my math department, and nobody seems to know 
the answer to this seemingly simple question:  What is the standard 
symbol used to represent the irrational numbers?  

One of my professors said he thought that it was Q-bar. Is this true?  
Thank you for your time.

Rebekah

Date: 09/24/2002 at 01:19:12
From: Doctor Mike
Subject: Re: Symbol for irrational numbers?

Hi Rebekah,   
   
Let's look first at groups of numbers that DO have common symbols.  
There is I for the integers, and N for the natural numbers, and R is 
common for the real numbers. The letter Q from quotient is common for 
the rationals.  The set of integers is SO important that they have 
another common letter Z devoted to them, from the word Zahl in German, 
which means number. 
  
Now that we have some examples of letters that DO stand for 'things', 
consider what all those 'things' have in common. They are all 
mathematical systems, in the sense that they are sets of mathematical 
objects that you can combine using at least one mathematical 
operation, with the result of those operations still being within the 
system. If you are at the college level, you have been or soon will be 
introduced to algebraic systems such as Groups and Fields. One of the 
most important properties of such a system is "closure," which means 
that combining system elements by system operation(s) keeps you inside 
the system. Examples. Adding 2 integers or multi- plying 2 integers 
still yields an integer. Same for reals and rationals. The system is 
"closed" in the sense that you can't "get out" by using the system 
operations.
  
The irrationals, however, are NOT such a system. If you multiply the 
two irrationals 2*pi and 1/pi together you get the result 2, which not 
only is not irrational, it is a natural or counting number. The set of 
irrationals is not even closed with respect to addition. Example: (pi-
1) + (6-pi) equals 5. My point is that it should be expected for the 
irrationals not to have a common letter devoted to them, because the 
irrationals do not form a mathematical system in the sense just 
described.

But now to get on with trying to answer your question. Your 
professor's suggestion makes sense because a bar over a symbol 
sometimes signifies complement, and within the reals, the irrationals 
are the complement of the set of rationals. 
   
In set theory you have a set operaion of "minus," where A-B is the set 
of all members of A that are not in B.  That's sort of an A "take 
away" B. Using this operation the irrationals are R-Q. This is similar 
to the Q-bar idea.
  
If I were writing a math paper and had to come up with something, I 
would probably just choose some letter and clearly introduce how I am 
using it: "Let S represent the set of irrational numbers, ...."   

- Doctor Mike, The Math Forum
  http://mathforum.org/dr.math/