Binary to Decimal to Hexadecimal - How and Why

Why?

Computers store data as binary (base 2). Programmers like to read and write binary as hexadecimal (base 16). And humans use decimal (base 10). Therefore, as a human programmer who needs to talk to computers, you need to know a little bit about all three systems.

Binary and Decimal

First, I'll walk you through how to convert between binary and decimal. Here's a table of each place in base 2 for one byte of data:

1	1	1	1	1	1	1	1
2^7	2^6	2^5	2^4	2^3	2^2	2^1	2^0
128	64	32	16	8	4	2	1

Binary to Decimal

To convert from binary to decimal by hand, simply write the powers of 2 beneath and add the numbers from right to left.

1	0	1	1	0	0	1	1
2^7	2^6	2^5	2^4	2^3	2^2	2^1	2^0
128	64	32	16	8	4	2	1

1 + 2 + 16 + 32 + 128 = 179

Let's say we have a binary string in Python code. How could we write a loop that will go through the string from right to left and add powers of 2?

def bin_to_dec(bin_string):
    decimal = 0
    # Initialize our rightmost power of 2 (2^0)
    power_of_2 = pow(2, 0);
    # Iterate through the reversed binary string (R -> L)
    for b in bin_string[::-1]:
        # If the digit is 1
        if int(b) == 1:
            # Add the power of 2 to our decimal number
            decimal += power_of_2
        # Double the power of 2 for the next place
        power_of_2 *= 2
    # Output the result
    return decimal

# Prints 179
print(bin_to_dec("10110011")) 

Decimal to Binary

There are multiple ways to convert from decimal to binary. I will teach you the way of dividing and taking the remainder. You divide by 2 and prepend the remainder to your binary string until the number you are dividing is 0.

179 / 2 = 89R1
_______1
 89 / 2 = 44R1
______11
 44 / 2 = 22R0
_____011
 22 / 2 = 11R0
____0011
 11 / 2 =  5R1
___10011
  5 / 2 =  2R1
__110011
  2 / 2 =  1R0
_0110011
  1 / 2 =  0R1
10110011

Here's the python code for that:

def dec_to_bin(decimal):
    bin_string = ""
    power_of_2 = pow(2, 0)
    while decimal > 0:
        # The remainder of dividing by 2 is the next bin digit
        bin_digit = decimal % 2
        # Divide the decimal by two (floor division)
        decimal //= 2
        # Prepend digit to string w/string concatenation
        bin_string = str(bin_digit) + bin_string
        # Double the power of 2 for the next place
        power_of_2 *= 2
    # Output the result
    return bin_string

# Prints "10110111"
print(dec_to_bin(179))

Generalizing: Base N to Decimal and Back

By the way, these approaches of converting bases works for any base system, including hexadecimal.

The code below uses Python's ord() and chr() functions which are used to convert ASCII letters to numbers and back.

def n_to_dec(n_string, n):
    decimal = 0
    power_of_n = pow(n, 0);
    for b in n_string[::-1]:
        # A = 10, B = 11, C = 12 ...
        b = ord(b) - 65 + 10 if not b.isdigit() else int(b)
        if int(b) != 0:
            decimal += int(b) * power_of_n
        power_of_n *= n
    return decimal

# Prints 255
print(n_to_dec("FF", 16))

def dec_to_n(decimal, n):
    n_string = ""
    power_of_n = pow(n, 0)
    while decimal > 0:
        n_digit = decimal % n
        # 10 = A, 11 = B, 12 = C ...
        n_digit = chr(65 - 10 + n_digit) if n_digit >= 10 else n_digit
        decimal //= n
        n_string = str(n_digit) + n_string
        power_of_n *= n
    return n_string

# Prints FF
print(dec_to_n(255, 16))

Hexadecimal and Binary

Sometimes, it is more useful to communicate in binary than decimal because you are talking about bytes of data. For example, a color's RGB values are each limited to a range of 0 to 255 because they each use a byte of data. However, writing out all of those 1's and 0's is tedious, so hexadecimal is used as a shorthand, where each nybble (half a byte, or 4 bits) is represented with a character from the set 0123456789ABCDEF. For example:

0011 = 3
0110 = 6
1100 = C (12)

You can take any binary number and convert it to hexadecimal by taking each section of 4 bits right to left and converting those.

10100101 (bin)

1010	0101
A	5

A5 (hex)

Conclusion

Of course, there are built in functions in languages like Python's bin() and hex() that will do these conversions, but it's useful to know how the process works yourself so you can do a quick conversion when needed. Now, the next time you see a color written as #FFFFFF, you will know that each RGB value is 255 and the lights are all on (making white). Hope you found this helpful!