Below is the relationship between a real number and a decimal number.
is this correct  or not?
A decimal number is a  real number multiplied by ten powered +ve or -ve integer.

Either case please prove it with an appropriate example.
 thanks in advance.

curious,

There are different kinds of numbers  and  there are 
different kinds of notations for numbers.

Example from the animal kingdom:
  Some animals are  mammals and they may be rodents or horses , etc

   A particular breed of horse can be described as 
   The Blue Horse = English version of Le Cheval Bleu. 

   Here, two languages are example of two different notations for the
same thing. Usually, a thing described in one  notation can be
translated to another notation, but it is not always so: there are
things which require a special notation.


Different kinds of numbers are described here:
http://en.wikipedia.org/wiki/Number 

Different notations for numbers are e.g. decimal or fractions
        scientific or engineering notations a re special cases of decimal.

Notations are described here:

http://en.wikipedia.org/wiki/Scientific_notation
http://en.wikipedia.org/wiki/Decimal

All real numbers can be written as 
 "infinite decimal expansion"
http://en.wikipedia.org/wiki/Decimal

 The word 'infinite' is important here; it means that number of
decimals needed MAY be infinite; it does not have to be.

Example: number   1/4 in decimal notation is  0.25  (same as 0.25000
...  of course) and so (if we ignore the zeros) this number can be
expressed with two digits only (one two and one five).

         number 1/3 would require infinite number of decimal places(
.3333 ...    with three repeated infinite number of times).

Now, a number  in a decimal notation  can be also expressed in a
scientific notation:

  Example    3456000.00     can be written as     3.45 E6  i.e as  3.45 10^6  

   On a computer   10^6  i.e ten to the power of 6  is easier written
as E6  - it means the same thing.

  All the numbers we used in the examples are real numbers.

So:  answer to"  is it correct ?" 
                 1)   "decimal number = function(real number)" or  
                 2)   "Real number    = function(Decimal number)" in all cases?

 is
      1) Yes; meaning: every real  number can be expressed in decimal
notation (if we allow infinite expansion)
      2) Yes; meaning: decimal number ( written  with En multiplier or
not) , represents a real number.

 Is that clear? If so, please rate the explanation.


Hedgie

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Request for Answer Clarification by curious_indian-ga on 13 Dec 2005 07:12 PST

You have mentioned "if we allow infinite expansion" in your
explanation : Is it norm to allow infinite expansion?

---

Clarification of Answer by hedgie-ga on 13 Dec 2005 19:39 PST

Is it norm to allow infinite expansion?

  Yes. I mentioned it since elementary courses of arithmetics do not stress
       it enough. 
       As fubini correctly comments,  it is necessary to represent all 
       real numbers. The numbers which can be represented by finite decimal
       expansion are sometimes called 'decimal numbers' as mentioned in 
       wikipedia. They are a  subset of rational numbers; since use of that term
       can cause confusion with 'decimal notation' it is not recommended.

I assume by decimal numbers that you are simply referring to a number
with a decimal point in it.  These numbers are a subset of the real
numbers.  Some numbers cannot be expressed as decimals (or fractions),
such as the square root of 2.

It's a bit more complex than that.

We can say that every decimal number is a real number. In fact, in
classical theories a real number seems to be almost defined by it's
property of being able to be expressed as a number of decimals. In
fact, Georg Cantor charachterized the real numbers by considering the
open unit interval (0,1), and listing all the numbers in that interval
as,

0.abcdefg... 

where each letter is a digit from zero to nine. Of course that
continues forever, it doesn't end when we run out of letters.

Thus, we find that every number that is not a complex number can be
represented with just a decimal number. Complex numbers can be
represented as A+Bi where A is the "real" component and B is the
imaginary component.

The rational numbers are any number that can be expressed as an
interger divided by another integer. This is the most exotic set
before the reals. The real numbers add in irrational numbers and other
transfinite numbers like the sqare root of two or Pi. The only way to
represent these numbers (which are by far the "largest" subset of the
real numbers) is with decimals. So while ansel001 is correct in that
we can't express irrational numbers with decimals, this is because
they are non-repeating and non-terminating. They are the closest we
can do for such numbers. They also carry the property where you can
always just use the decimal out to however many places you need, so
it's fine for most people.

I'm not sure where you got your defenition for a decimal, but I don't
know if it would be characterized as correct. The reason being is that
there are plenty of decimal numbers that are also real numbers. If you
define decimal numbers to be all the real numbers between zero and
one, then the defenition could work. However, I don't see the utility
doing such a thing. The reason has already been stated, all decimal
numbers are real numbers, so there isn't a big reason to differentiate
them.

For example, .5 is both 5*(10^-1) and it is just .5, you don't need to
justify it's place in the real number line.

On the other hand, you run into problems when you consider
tracindental numbers like Pi. What real number would you multiply 10
to the something by to get it? You can't just say you would take the
real number 31,415,926,535... and multiply it by 10 to the negative
infinity minus one (that's not a well-formed statement by any means).
Thus, I think it's better to just leave decimal numbers to be defined
as being real numbers.

fubini-ga and ansel001-ga

I am trying to resolve the statments credibility . i.e. can each
decimal be defined as a function of real where the equation is
DECIMAL= REAL*10^(+/-)(integer)
which need not be true for REAL = DEMIAL*(something)
thanks
curious_indian

No, as what I said about the number Pi (or any irrational number for that reason.

In fact, I just realized that there is an extremely trivial case in
which your formula does hold. If we take the decimal number to be
equal to the real number, than 10 to the zero power will be one, so
the formula holds. But if you don't restrict the decimal number to
being the real number, it gets shot full of holes.

If you want more specfic help with your question you need to define
exactly what you mean by a decimal number. It's not a defined math
term that I've seen, so you'll have to supply us with that. You'll
also need to tell us if you really mean real numbers, or natural
numbers. If you really mean real numbers, then yes your formula holds
vacuously because you can always just define your real number to be
equal to your "decimal number".

Your question is still not well-formed.

fubini-ga

real number and decimal number is as defined in wikepedia.
http://en.wikipedia.org/wiki/Decimal
http://en.wikipedia.org/wiki/Real_number

to re-address the question:
which is correct "decimal number = function(real number)" or  
                 "Real number    = function(Decimal number)" in all cases?
please support the answer with a simple explanation  

thanks
curious_indian

It is normal to allow for infinite expansion. If it weren't then we
couldn't express the bulk of the real numbers in that method.

thanks Everyone. The answer is satisfactory.
 thanks

Just a minor comment - fubini said
"The rational numbers are[...] the most exotic set before the reals."
In between there are also the alebraic numbers, which contain some 
irrational numbers (like sqrt(2)) but not others (like pi).

In terms of the original question, the Wikipedia entry mentioned later
does in fact provide a definition of "decimal number" (for those who
didn't look it up) - it is a number of the form a/b where a is an
integer and b is a power of 10. So, the answer is no, curious_indian's
statement was not correct. It should have been: "A decimal number is
an INTEGER multiplied by ten to the power of another integer - where
both integers can be positive, zero or negative."
The key difference is that we are replacing "real number" with "integer".

Yes Manuka 

Clarification of Answer by hedgie-ga on 13 Dec 2005 19:39 PST
says
"The numbers which can be represented by finite decimal
       expansion are sometimes called 'decimal numbers' as mentioned in 
       wikipedia."

In case  you did not noticed.  I discourage use of that definition
because it is confusing to many people.
Hedgie