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
Best Answer
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.