Wrapping my head around different base-number systems this is my first post on this forum, I'm interested in mathematics but don't have any education beyond the high-school level in the subject, so go easy on me.
What I know right now: the base-10 system is a system of number notation that was more or less arbitrary selected over other number systems... if we were to change all the numbers in an equation to a different system, we could calculate the correct answer in that system.  There is no reason why the base 10 system is any more correct than the base 15 or base 5 system.  If this isn't true, please correct me.
Question 1: Firstly, is it possible to have a number system that is below 1?  I would assume no, because to do so would require some sort of notation capable of expressing a fraction, which would require a base-system at least equal to one or greater.  Am I wrong?  is it theoretically possible?
Question 1.5:  Is asking if you can have a base-number system that is not a whole number system an ironic question?  Like is asking if you can have a base 4.5 system a silly question because by saying "4.5" im assuming that I mean four and a half, and the half would be half of ten which implies a base 10 system? Is it possible to have a number-system that is between two integers?  Which leads to my third question
Question 2: what is it for a number to be a "whole number" when we relate the idea of being "whole" to the number system being used to express it?  Are some numbers whole in some systems but not whole in others?
Question 3:  If some numbers are whole in some systems but not in others, what implications does this have on prime numbers??
Thanks, Sam
 A: Regarding number bases less than $1$ (Question 1), suppose you have a base-one-tenth number system. The first few integers in this system would be
$$1,2,3,4,5,6,7,8,9,0.1,1.1,2.1,3.1,4.1,5.1,6.1,7.1,8.1,9.1,0.2$$
The reason $0.1$ is the next integer after $9$ is that the base
of this number system, $b$, has the value we would normally call
$\frac{1}{10}$ (writing in base ten), 
and the first place after the decimal point has place value $b^{-1}$.
And of course $\left(\frac 1n\right)^{-1} = n$,
for any $n$.
This is kind of a silly example since you can just write all your numbers
backwards and put the decimal point between the ones' place and the
tens' place. So it is not too surprising that I've never seen this
system explained before.
Moving on to Question 1.5,
a much more interesting system is the base-$\phi$ number system.
The number $\phi$ is also known as the Golden Ratio:
$\phi = \frac12(1 + \sqrt 5)$.
It turns out that all the powers of $\phi$ have the form
$\frac12(a + b\sqrt 5)$ for some integers $a$ and $b$,
and lots of nice cancellation can occur.
So if we allow only the digits $0$ and $1$ 
in this number system (these are the only non-negative integers
less than $\phi$), we can write the first few integers as
$$1, 10.01, 100.01, 101.01, 1000.1001, 1010.0001$$
But regarding Questions 3 and 4:
whatever system it is written in, a whole number is a whole number.
Since we're so used to using only whole-number-based systems,
the numbers written in base-one-tenth and base-$\phi$ in the lists above
(the "first few integers" in each base)
may not look like whole numbers, but nevertheless
that's what every single one of them is.
Or to put it another way,
each whole number is produced by adding $1$ to the number before it.
That is (essentially) the way the whole numbers are defined.
The numbering system you use may have some bizarre effects on how
you "carry" digits when adding $1$ to the previous whole number
causes the ones' place to "roll over", but as long as you do the
arithmetic operation correctly you will get the same actual number
as you would have gotten in any other base.
Likewise any other mathematical facts involving integers or fractions
are exactly the same in an number base; they may just look different
due to the different numerical notation.
A: I'll try to answer to the $3^{rd}$ question. 
Numbers are used to measure things and to count things. We can't mix the two concepts. To count we use integers and they are integer independently from the base we use to write them. Moreover integers are defined WITHOUT any representation (Peano axioms). Positive integers are defined as the set $\mathbb{X}$ who satisfies the following axioms:


*

*$\mathbb{X}$ contains at least one element $\overline{x}$;

*for any $x\in \mathbb{X}-{\overline{x}}$ there exist one and only one $s(x)\ne x:\;s(x)\in \mathbb{X}$;

*$s\left(\overline{x}\right)$ does not exist;

*if $x,\;y\in \mathbb{X},\;x\ne y$ then $s(x)\ne s(y)$;

*if a subset $A$ of $\mathbb{X}$ contains $\overline{x}$ and $s(a)$ for any $a\in A$ then $A$ is $\mathbb{X}$.


As you can see positive integers are  even not represented.
Maybe that what you are thinking when you say of non integer representations  is the measure of something wrt some measure unit, which is all different subject and which, by the way, many people confuse with counting.
Hope this helps.
A: The number $x$ representation is power series: 
$x=\sum_{n=1}^{\infty}a_n b^n$ 
where $b$ is the base number and $\{a_n\}$ is a series that represent the number. Usually when $b$ is an integer we limit $a_n$ to be in the group $T=\{0,1,2,...b-1\}$ this way all the integer number can be represented. So if we want to represent all the numbers we expand the representation:
$x=\sum_{n=-\infty}^{\infty}a_n b^n$  and using a dot to denote the place of $a_0$. For example $100.1$ for $b=2$ represent $4.5$. This way all the rational numbers can be represented.
1) so, is it possible to use $b=0.5$ ? Yes, it is like flipping the representation around the dot, because $0.5^n=2^{-n}$, and the number 4.5 will be represented as $1.001$. Notice that we choose $T=\{0,1\}$ 
1.5) What if we choose $b=4.5$ and $T=\{0,0.5,1,1.5,...4\}$, and using the symbols $\{0,1,2,3,4,5,6,7,8\}$ then $20$ will represent $4.5$, and $11$ will represent $2.75$. This way without the dot, we represent not only the integer but also a part of the rational numbers. In that number-system part of the not integer numbers represents in integer notation.
2) The "Whole number" is defined regardless of its representation(number-system). It is related to the unit of the field you working with, we using the field of the real numbers $\mathbb{R}$ and its unit is the number 1. The field definition and definition of its unit is independent on the number representation. 
We define a series over the filed with unit 1: $x_0 = 0$,
$x_n = x_{n-1}+1$ 
x is a "whole number" if $x \in \{x_n\}$
3) Prime numbers, as well indepented of the number representation method. For example 5 is always will be a prime. even if we remove all the odd number from integer representation by choosing $T=\{0,2,4,6,8\}$ 5 is still there:
$0 \to 0$
$1 \to 2$
$2 \to 4$
$3 \to 6$
$4 \to 8$
$5 \to 20$
$20$ cannot be divided by $4$ since the decimal result  $5$ which is not an integer in this number system. So in this system $20$ is prime, it actually 5 in the new field the we uncontiously difined, it is $2\mathbb{R}$ with unit 2
A: Okay, so first I will mention a different way to think about bases. So for example, say you want to write 471 in base 5. So you start with a "471" in the units place:
471

Now, since we're in base 5, every 5 in the units place is worth 1 in the next place. So we can take 470 out of the units place and turn it into 94 in the next place to the left (Imagine this as repeatedly taking 5 out of the units place and adding 1 to the next place, 94 times). So our number is now:
94 1

Now, take out a 90 and add 18 to the next place to the left:
18 4 1

And once more:
3 3 4 1

So 471 is 3341 in base 5.
Now for question 1.5 (I'll get back to 1 later.) You give the example of base 9/2. What that would mean is that 9 in one place is worth 2 in the place to the left. So let's use our example of 471 again. We start with just a 471, as before:
471

Now, we can repeatedly take 9 out of the units place and add 2 to the next place. We do this 52 times, so we subtract 468 from the units place and add 104 to the next place:
104 3

Now you can take out 99 and put a 22 on the next place:
22 5 3

Take out an 18:
4 4 5 3

So 471 in base 10 is 4453 in base 9/2. Note that digits can be up to 8 in base 9/2. Now, I'll show an example of addition in base 9/2. Say you wanted to calculate 35 + 47, where all numbers are in base 9/2. First, just add all the digits pairwise:
   3   5
+  4   7
--------
   7  12

Now, noticing that 12 is more than 9, we can subtract a 9 and add 2 to the next place. So the number becomes 9 3. But now, notice again that you have a 9, so you can subtract it and add 2 to the next place again. So the final result is 203. So in base 9/2, 35 + 47 = 203.
For question 1, basically what you do when the base is less than 1, e.g. 1/5, is that when you take 5 out of one place, you add 1 to the place to the right instead of the place to the left. So you basically get the reverse of the number in base 5. For example, 48839 in base 5 is 9.3884 in base 1/5.
And for questions 2 and 3. Base systems are just a way of representing numbers, for our own convenience. But the way you represent a number does not change the number itself. So 471 in base 10 and 3341 in base 5 are still the same number. The base representations are just like a name we assign to a number. If you change the name of a number, you don't change its properties. So the concept of a whole number is unchanged. However, the way you tell if a number is a whole number may be different. In base 10 (or in any integer base), you can easily tell if a number is whole by making sure it has nothing after the decimal point (and no negative sign). In base 9/2, for example, it is not as easy to tell. 35, for example, is not whole in base 9/2. (it is equal to 37/2 in base 10). 
