base conversion

yoshd — Thu, 01 Oct 2026 11:22:44 +0000

I write python code of a base conversion from any base system into other any base system, so I break down the code into some processes;

base-B to base-10
base-10 to base-B
base-B to base-B'

overview

base conversion (also known as radix conversion) is translating a number expressed in one numerical system into an equivalent value in another numerical system

# these numbers are same meaning
13   # base 10( known as decimal )
1101 # base 2 ( known as binary )

base-B → base-10

Here, B is an integer greater than 1 (for example, 2).
To convert a number written in base-B into base-10, multiplying each digit by $B^{p}$ , where $p$ is the positional power starting at 0 from rightmost integer digit.

In python, this conversion is executed by the built-in function int()

base10 = int('1101', 2) # '1101' is written in base-2 
print(base10) # 13

base-10 → base-B

First, base-10 → base-2

On the other hand, when you change decimal into the other base-2 system, you repeatedly divide the number by 2 and record the remainders.

13 = 2 * 6 + 1
6  = 2 * 3 + 0
3  = 2 * 1 + 1
1  = 2 * 0 + 1
=> 1101 
   collecting the remainders from bottom to top

This process can be implemented in Python as follows.

digits = []
decimal_value = 13
while decimal_value > 0:
    remainder = decimal_value % 2 
    digits.append(remainder)
    decimal_value //= 2 

print(''.join(reverse(digits))) # 1101

Generalizing to base-B

To convert a base-10 into an arbitary base-B system, 2 parts need to be changed.

Replace divisor 2 into base_to
Convert remainders greater than or equal to 10 into characters(A,B,C,...) using _digit_to_char() in the final code.

# Excerpt from the final code
digits = []
while decimal_value > 0:
    remainder = decimal_value % self.base_to # 2→other digit number
    digis.append(self._digit_to_char(remainder)) #
    decimal_value //= self.base_to # 2→other digit number

base-B → base-B'

B and B' are integer > 1.
generalization of the base convertion process, it can be break down into 2 processes;

conversion base-B into base-10
conversion base-10 into base-B'

# base_from, base_to are meaning base-B, base-B' 
def convert(self) -> str:
    # 1. conversion base-B into base-10
    decimal_value = int(self.number, self.base_from)
    # 2. conversion base-10 into base-B'
    if self.base_to == 10:
        return str(decimal_value)
    else:
        return self._decimal_to_base(decimal_value, self.base_to)

complete code for converting from base-B into base-B'

class RadixConverter:
    def __init__(self, number: str, base_from: int, base_to: int):
        self.number = number
        self.base_from = base_from
        self.base_to = base_to

    def convert(self) -> str:
        # Convert the number from base_from to decimal
        decimal_value = int(self.number, self.base_from)
        # Convert the decimal value to base_to
        if self.base_to == 10:
            return str(decimal_value)
        else:
            return self._decimal_to_base(decimal_value, self.base_to)

    def _decimal_to_base(self, decimal_value: int, base: int) -> str:
        if decimal_value == 0:
            return '0'
        digits = []
        while decimal_value > 0:
            remainder = decimal_value % base
            digits.append(self._digit_to_char(remainder))
            decimal_value //= base
        return ''.join(reversed(digits))

    def _digit_to_char(self, digit: int) -> str:
        if digit < 10:
            return str(digit)
        else:
            return chr(ord('A') + digit - 10)

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