As we have different numeral bases in number system such as base 2(binary), base 10(decimal) etc. As binary (base 2) is smallest among all, is there a base value that is maximum?I was trying to search the total number of bases available in our number system?
I was checking for the same on wiki, but could not find a total count. I am not sure if there exists such total count on bases but just wanted to clarify with the community?
On
Any natural number greater than zero can be a base, so there are infinitely many. Once you choose a base $b$, you can express any natural number $n$ as $a_0+b\cdot a_1+b^2\cdot a_2 + \ldots$ where all the $a$'s are in the range $0$ to $b-1$. Base $10$ is what we are used to, base $2$ is common in computer usage, but the others work fine. There is just no motivation to use them. You can also have negative bases. You can express $-37_{10}=1\cdot (-10)^2+7\cdot (-10)^1+7\cdot (-10)^0=177_{-10}$. Arithmetic is no harder in these, but the numbers are longer, as this example shows. It has two digits in base $10$, but three in base $-10$. Some people have thought about fractional bases, some about real number bases. Only the reals change the number of bases, driving it up to $\mathfrak c$, the power of the continuum. If you work in base $\pi$ and allow digits up to $4$ you can represent all numbers (some with more than one representation) but $5_{10}$ becomes an unending decimal. Aw c'mon.
You can have a base in each positive natural number. There are countably infinitely many such numbers, so there are countably infinitely available bases.
(Think about it this way: for every positive integer, there exists a unique base in which that integer is $10$).