Any Base To Any Base Algorithm
The Any Base To Any Base Algorithm is a strategy for converting numbers between different base numeral systems. Numeral systems, or number systems, are methods for representing numbers with symbols, and the base of a numeral system is the number of distinct symbols employed. The most common numeral system is the decimal system (base 10), which uses ten symbols (0-9), while other popular numeral systems include binary (base 2) and hexadecimal (base 16). The algorithm allows us to convert a number from one base to another without first converting it to decimal or any other intermediate base, making it a versatile and efficient tool for working with numbers in different bases.
The first step in the Any Base To Any Base Algorithm is to express the number in its original base as a sum of powers of that base, also known as its "polynomial form". For instance, the number 1011 in binary can be written as (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0). Once this step is complete, we can convert the number to the desired target base by dividing the sum by the target base and obtaining the remainder. The remainder becomes the least significant digit (rightmost digit) of the converted number, and we continue the process with the quotient until the quotient is zero. The remainders collected at each step form the digits of the number in the target base, with each remainder's position in the sequence corresponding to its significance. The Any Base To Any Base Algorithm is particularly useful in computer science and mathematics, as it simplifies the process of working with different numeral systems and enables efficient conversions for various applications.
package Conversions;
import java.util.Arrays;
import java.util.HashSet;
import java.util.InputMismatchException;
import java.util.Scanner;
/**
* Class for converting from "any" base to "any" other base, when "any" means from 2-36.
* Works by going from base 1 to decimal to base 2. Includes auxiliary method for
* determining whether a number is valid for a given base.
*
* @author Michael Rolland
* @version 2017.10.10
*/
public class AnyBaseToAnyBase {
/**
* Smallest and largest base you want to accept as valid input
*/
static final int MINIMUM_BASE = 2;
static final int MAXIMUM_BASE = 36;
public static void main(String[] args) {
Scanner in = new Scanner(System.in);
String n;
int b1, b2;
while (true) {
try {
System.out.print("Enter number: ");
n = in.next();
System.out.print("Enter beginning base (between " + MINIMUM_BASE + " and " + MAXIMUM_BASE + "): ");
b1 = in.nextInt();
if (b1 > MAXIMUM_BASE || b1 < MINIMUM_BASE) {
System.out.println("Invalid base!");
continue;
}
if (!validForBase(n, b1)) {
System.out.println("The number is invalid for this base!");
continue;
}
System.out.print("Enter end base (between " + MINIMUM_BASE + " and " + MAXIMUM_BASE + "): ");
b2 = in.nextInt();
if (b2 > MAXIMUM_BASE || b2 < MINIMUM_BASE) {
System.out.println("Invalid base!");
continue;
}
break;
} catch (InputMismatchException e) {
System.out.println("Invalid input.");
in.next();
}
}
System.out.println(base2base(n, b1, b2));
in.close();
}
/**
* Checks if a number (as a String) is valid for a given base.
*/
public static boolean validForBase(String n, int base) {
char[] validDigits = {'0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E',
'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V',
'W', 'X', 'Y', 'Z'};
// digitsForBase contains all the valid digits for the base given
char[] digitsForBase = Arrays.copyOfRange(validDigits, 0, base);
// Convert character array into set for convenience of contains() method
HashSet<Character> digitsList = new HashSet<>();
for (int i = 0; i < digitsForBase.length; i++)
digitsList.add(digitsForBase[i]);
// Check that every digit in n is within the list of valid digits for that base.
for (char c : n.toCharArray())
if (!digitsList.contains(c))
return false;
return true;
}
/**
* Method to convert any integer from base b1 to base b2. Works by converting from b1 to decimal,
* then decimal to b2.
*
* @param n The integer to be converted.
* @param b1 Beginning base.
* @param b2 End base.
* @return n in base b2.
*/
public static String base2base(String n, int b1, int b2) {
// Declare variables: decimal value of n,
// character of base b1, character of base b2,
// and the string that will be returned.
int decimalValue = 0, charB2;
char charB1;
String output = "";
// Go through every character of n
for (int i = 0; i < n.length(); i++) {
// store the character in charB1
charB1 = n.charAt(i);
// if it is a non-number, convert it to a decimal value >9 and store it in charB2
if (charB1 >= 'A' && charB1 <= 'Z')
charB2 = 10 + (charB1 - 'A');
// Else, store the integer value in charB2
else
charB2 = charB1 - '0';
// Convert the digit to decimal and add it to the
// decimalValue of n
decimalValue = decimalValue * b1 + charB2;
}
// Converting the decimal value to base b2:
// A number is converted from decimal to another base
// by continuously dividing by the base and recording
// the remainder until the quotient is zero. The number in the
// new base is the remainders, with the last remainder
// being the left-most digit.
// While the quotient is NOT zero:
while (decimalValue != 0) {
// If the remainder is a digit < 10, simply add it to
// the left side of the new number.
if (decimalValue % b2 < 10)
output = Integer.toString(decimalValue % b2) + output;
// If the remainder is >= 10, add a character with the
// corresponding value to the new number. (A = 10, B = 11, C = 12, ...)
else
output = (char) ((decimalValue % b2) + 55) + output;
// Divide by the new base again
decimalValue /= b2;
}
return output;
}
}
