I know it is simple and possible to convert any base to any base. First, convert any base to decimal and then decimal to any other base. However, I had done this before for range 2 to 36 but never done for 2 to 46.
I don't understand what I will put after 36, because 36 means 'z' (1-10 are decimal numbers then the 26 characters of the alphabet).
Please explains what happens after 36.
The symbols you use for digits are arbitrary. For example base64 encoding uses 'A' to represent the zero valued digit and '0' represents the digit with the value 52. In base64 the digits go through the alphabet A-Z, then the lower case alphabet a-z, then the traditional digits 0-9, and then usually '+' and '/'.
One base 60 system used these symbols: 
So the symbols used are arbitrary. There's nothing that 'happens' after 36 except what you say happens for your system.
Every base has a purpose. Usually we do base conversion to make complex computations simpler.
Here are some most popular bases used and their representation.
2-binary numeral system
used internally by nearly all computers, is base two. The two digits are 0 and 1, expressed from switches displaying OFF and ON respectively.
8-octal system
is occasionally used in computing. The eight digits are 0–7.
10-decimal system
the most used system of numbers in the world, is used in arithmetic. Its ten digits are 0–9.
12-duodecimal (dozenal) system
is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
16-hexadecimal system
is often used in computing. The sixteen digits are 0–9 followed by A–F.
60-sexagesimal system
originated in ancient Sumeria and passed to the Babylonians. It is still used as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes and hours).
64-Base 64
is also occasionally used in computing, using as digits A–Z, a–z, 0–9, plus two more characters, often + and /.
256-bytes
is used internally by computers, actually grouping eight binary digits together. For reading by humans, bytes are usually shown in hexadecimal.
The octal, hexadecimal and base-64 systems are often used in computing because of their ease as shorthand for binary. For example, every hexadecimal digit has an equivalent 4 digit binary number.
Radices are usually natural numbers. However, other positional systems are possible, e.g. golden ratio base (whose radix is a non-integer algebraic number), and negative base (whose radix is negative).
Your doubt is whether we can convert any base to any other base after base exceeds 36 ( # of Alphabets + # of digits = 26+ 10= 36)
Taking example of 64-Base It uses A–Z(Upper case)(26), a–z(lower case)(26), 0–9(10), plus 2 more characters. This way the constraint of 36 is resolved.
As we have (26+26+10+2)64 symbols in 64-base for representation, we can represent any number in 64 base. Similarly for more base they use different symbols for representation.
With number systems, you are allowed to play god.
What you need to understand is, that symbols are completely arbitrary. There is no god-given rule for "what comes after 36". You are free to define whatever you like.
To encode numbers with a certain base, all you need is the following:
Naturally, there's an infinite amount of possibilities to create such a symbol table for a certain base:
Θ
ェ
す
)
０
・
＿
ｏ
や
ι
You could use this, to encode numbers with base 10. Θ being the zero-element, ェ being the one, etc.
Of course, your peers would not be too happy if you started using the above symbol table. Because the symbols are arbitrary, we need conventions. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is a convention, as are the symbols we use for hexadecimal, binary, etc. It is generally agreed upon what symbol table we use for what basis, that is why we can read the numbers someone else writes down.
The important thing to remember is that all numbers are symbolic of a value. Thus if you wanted to do that, you could just make a list containing the values at each position. After base 36, you simply run out of characters you can make a logical sequence out of. For example, if you used the Cambodian Alphabet with 70 odd characters, you could do base 80.
Here is the complete code I have written, hope this will help.
import java.util.Scanner;
/*
* author : roottraveller, nov 4th 2017
*/
public class BaseXtoBaseYConversion {
BaseXtoBaseYConversion() {
}
public static String convertBaseXtoBaseY(String inputNumber, final int inputBase, final int outputBase) {
int decimal = baseXToDecimal(inputNumber, inputBase);
return decimalToBaseY(decimal, outputBase);
}
private static int baseXNumeric(char input) {
if (input >= '0' && input <= '9') {
return Integer.parseInt(input + "");
} else if (input >= 'a' && input <= 'z') {
return (input - 'a') + 10;
} else if (input >= 'A' && input <= 'Z') {
return (input - 'A') + 10;
} else {
return Integer.MIN_VALUE;
}
}
public static int baseXToDecimal(String input, final int base) {
if(input.length() <= 0) {
return Integer.MIN_VALUE;
}
int decimalValue = 0;
int placeValue = 0;
for (int index = input.length() - 1; index >= 0; index--) {
decimalValue += baseXNumeric(input.charAt(index)) * (Math.pow(base, placeValue));
placeValue++;
}
return decimalValue;
}
private static char baseYCharacter(int input) {
if (input >= 0 && input <= 9) {
String str = String.valueOf(input);
return str.charAt(0);
} else {
return (char) ('a' + (input - 10));
//return ('A' + (input - 10));
}
}
public static String decimalToBaseY(int input, int base) {
String result = "";
while (input > 0) {
int remainder = input % base;
input = input / base;
result = baseYCharacter(remainder) + result; // Important, Notice the reverse order here
}
return result;
}
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
System.out.println("Enter : number baseX baseY");
while(true) {
String inputNumber = scanner.next();
int inputBase = scanner.nextInt();
int outputBase = scanner.nextInt();
String outputNumber = convertBaseXtoBaseY(inputNumber, inputBase, outputBase);
System.out.println("Result = " + outputNumber);
}
}
}
