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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?

mrsan22
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    Each integer $b>1$ is a base. Some of them just have names (e.g., decimal, binary, hexadecimal, etc.; see https://en.wikipedia.org/wiki/Radix) – parsiad Sep 17 '16 at 01:57
  • Any real number except -1, 0 and 1 can serve as a base. – MasB Sep 17 '16 at 02:16
  • I thought so..that there exists infinitely many bases and that led me to this search.. I was trying to see if a number is palindrome in any base(starting with 2) but then I got stuck at a point, where I had to provide the max value for base(something like stopping condition). I was not able to decide that what should be the max value for a base against which I can check a given number is palindrome and if it's not then I can conclude that the given input is not palindrome in any base. – mrsan22 Sep 17 '16 at 02:17
  • Then you should ask your real question. Once the base gets larger than the number, the number is represented by a single digit, which is guaranteed to be a palindrome. Your stopping condition becomes when the base is greater than the number. The answer to your question is no, there is no number which is a palindrome in any base. if you express $n$ in base $n$, it is $10$, which is not a palindrome. – Ross Millikan Sep 17 '16 at 02:37
  • Thanks for clarification. I will update my post with this question as well. – mrsan22 Sep 17 '16 at 18:16

2 Answers2

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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 $10$).

Newb
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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.

Ross Millikan
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