# Base of number system in Math

i would like to know that how many base number system are there in Math? And how can we convert to each other (Ex convert from base2 to base10,.....). Please feel free to help me.

• If you're asking how many exist, there are infinitely many: base 1, base 2, base 3, base 4,.... You get the idea. You could create a short list of the most commonly used ones I suppose, but that's an entirely conventional/subjective matter. – David H Jun 23 '14 at 1:49
• Thank you very much,i understand.so i would like to know how can we convert from base2 to base8?please feel free to help me. – user159457 Jun 23 '14 at 2:01
• i am sorry if you are difficult to understand my question,because my English is poor. – user159457 Jun 23 '14 at 2:03
• You're English is just fine. As to your question, you can always ask WolframAlpha to do the conversion for you. I realize that's not an explanation of how the conversion is actually computed, but I'll let someone else answer that. – David H Jun 23 '14 at 2:13
• From base 2 to 8 is easy. Because $8 = 2^3$ just group the binary digits in groups of 3's and turn those groups into a normal base 10 number by doing base 2 to base 10: Ex: $1101011001_2$ = 001 101 011 001 = 1 5 3 1 = $1531_8$ Ex2: $11001011011_2$ = 011 001 011 011 = 3 1 3 3 = $3133_8$ – Dane Bouchie Jun 23 '14 at 2:38

## 2 Answers

To change base (or better, to express a number on some base) the basic algorithm is to get the remainders when dividing successively by the base, giving the digits in reverse order.

To get e.g. 1234 in base 16, you divide by 16: $$1234 / 16 = 77, \text{ remainder } 2 \\ 77 / 16 = 12, \text{ remainder } 13$$ So the result is CD2 (C is 12, D is 13).

You might want to check here

• Thank you very much Vonbrand. – user159457 Jun 23 '14 at 3:54

The way your question is phrased leaves something to be desired, so I would like to modify your question before answering it.

I am starting from square one and agree that we have a system of counting numbers with certain properties. What is the idea behind representing our counting numbers with a 'base'? How many different bases are there?

The idea is finding strings (or words) that describe all numbers in a compact and systematic way. In our Base Ten system, you can even read out the string; for example, if you see "1,230" you can say

"One thousand and two hundred and thirty".

The method used is to define exponentiation of integers > 1 and then exploit it by using certain facts. So, you can actually have a base for each integer > 1, but of course you have to be practical in your choice. Since humans have 10 fingers, it was kind of natural for us to adopt base 10.

A miraculous side result of all this is that not only do we have a nice way to represent integers, but you can discover rules that allow you to perform addition and multiplication by directly manipulating and playing around with the representations.

The best way to elucidate the whole matter is by delving into the theory, but I am not sure if this is an appropriate place.