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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?

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.

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/
```
Associated Topics:
College Modern Algebra
High School Basic Algebra

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