The math behind converting from any base to any base without going through base 10?

Question

I've been looking into the math behind converting from any base to any base. This is more about confirming my results than anything. I found what seems to be my answer on mathforum.org but I'm still not sure if I have it right. I have the converting from a larger base to a smaller base down okay because it is simply take first digit multiply by base you want add next digit repeat. My problem comes when converting from a smaller base to a larger base. When doing this they talk about how you need to convert the larger base you want into the smaller base you have. An example would be going from base 4 to base 6 you need to convert the number 6 into base 4 getting 12. You then just do the same thing as you did when you were converting from large to small. The difficulty I have with this is it seems you need to know what one number is in the other base. So I would of needed to know what 6 is in base 4. This creates a big problem in my mind because then I would need a table. Does anyone know a way of doing this in a better fashion.

I thought a base conversion would help but I can't find any that work. And from the site I found it seems to allow you to convert from base to base without going through base 10 but you first need to know how to convert the first number from base to base. That makes it kinda pointless.

Commenters are saying I need to be able to convert a letter into a number. If so I already know that. That isn't my problem however. My problem is in order to convert a big base to a small base I need to first convert the base number I have into the base number I want. In doing this I defeat the purpose because if I have the ability to convert these bases to other bases I've already solved my problem.

Edit: I have figured out how to convert from bases less than or equal to 10 into other bases less than or equal to 10. I can also go from a base greater than 10 to any base that is 10 or less. The problem starts when converting from a base greater than 10 to another base greater than 10. Or going from a base smaller than 10 to a base greater than 10. I don't need code I just need the basic math behind it that can be applied to code.

Answer 1

This seems a very basic question to me, so excuse me if I lecture you a bit. The most important point for you to learn here is that a number is not its digit representation. A number is an abstract mathematical object, whereas its digit representation is a concrete thing, namely a sequence of symbols on a paper (or a sequence of bits in compute memory, or a sequence of sounds which you make when you communicate a number). What is confusing you is the fact that you never see a number but always its digit representation. So you end up thinking that the number is the representation.

Therefore, the correct question to ask is not "how to I convert from one base to another" but rather "how do I find out which number is represented by a given string of digits" and "how do I find the digit representation of a given number".

So let us produce two functions in Python, one for converting a digit representation to a number, and another for doing the opposite. Note: when we run the function Python will of course display the number it got in base 10. But this does not mean that the computer is keeping numbers in base 10 (it isn't). It is irrelevant how the computer represents the numbers.

def toDigits(n, b):
    """Convert a positive number n to its digit representation in base b."""
    digits = []
    while n > 0:
        digits.insert(0, n % b)
        n  = n // b
    return digits

def fromDigits(digits, b):
    """Compute the number given by digits in base b."""
    n = 0
    for d in digits:
        n = b * n + d
    return n

Let us test these:

>>> toDigits(42, 2)
[1, 0, 1, 0, 1, 0]
>>> toDigits(42, 3)
[1, 1, 2, 0]
>>> fromDigits([1,1,2,0],3)
42

Armed with conversion functions, your problem is solved easily:

def convertBase(digits, b, c):
    """Convert the digits representation of a number from base b to base c."""
    return toDigits(fromDigits(digits, b), c)

A test:

>>> convertBase([1,1,2,0], 3, 2) 
[1, 0, 1, 0, 1, 0]

Note: we did not pass through base 10 representation! We converted the base $b$ representation to the number, and then the number to base $c$. The number was not in any representation. (Actually it was, the computer had to represent it somehow, and it did represent it using electrical signals and funky stuff that happens in chips, but certainly those were not 0's and 1's.)

Answer 2

I think the best way to understand this is in discussion with an alien (at least as an analogy).

Definition $x$ is a number in base $b$ means that $x$ is a string of digits $<b$.

Examples The string of digits 10010011011 is a number in base 2, the string 68416841531 is a number in base 10, BADCAFE is a number in base 16.

Now Suppose I grew up on the planet QUUX where everyone is taught to work in $q$ for their whole lives, and I meet you who is used to base $b$. So you show me a number, and what do I do? I need a way to interpret it:

Definition I can interpret a number in base $b$ (Note: $b$ is a number in base $q$) by the following formula

$$\begin{array}{rcl} [\![\epsilon]\!] &=& 0 \\ [\![\bar s d]\!] &=& [\![\bar s]\!] \times b + d \end{array}$$

where $\epsilon$ denotes the empty string, and $\bar s d$ denotes a string ending in the digit $d$. See my proof that addition adds for an introduction to this notation.

So what's happened here? You've given me a number in base $b$ and I've interpreted it into base $q$ without any weird philosophy about what numbers truly are.

Key The key to this is that the $\times$ and $+$ I have are functions that operate on base $q$ numbers. These are simple algorithms defined recursively on base $q$ numbers (strings of digits).

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This may seem a bit abstract since I've been using variables rather than actual numbers throughout. So let's suppose you are a base 13 creature (using symbols $0123456789XYZ$) and I am used to base 7 (which is much more sensible) using symbols $\alpha \beta \gamma \delta \rho \zeta \xi$.

So I've seen your alphabet and tabulated it thus:

$$\begin{array}{|c|c||c|c||c|c|} \hline 0 & \alpha & 1 & \beta & 2 & \gamma \\ 3 & \delta & 4 & \rho & 5 & \zeta \\ 6 & \xi & 7 & \beta\alpha & 8 & \beta\beta \\ 9 & \beta\gamma & X & \beta\delta & Y & \beta\rho \\ & & Z & \beta\zeta & & \\ \hline \end{array}$$

So I know that you work in base $\beta\xi$, and I know what base 7 number any digit you write corresponds to.

Now if we were discussing physics and you were telling me about fundamental constants (say) $60Z8$ so I need to interpret this:

$$\begin{array}{rcl} [\![60Z8]\!] &=& \xi (\beta\xi)^3 + \alpha (\beta\xi)^2 + \beta \zeta (\beta\xi) + \beta\beta \\ \end{array}$$

So I start by multiplying out $\beta \zeta \times \beta\xi$ but this is grade school stuff for me, I recall:

Quux multiplication table

$$\begin{array}{|c|cccccc|} \hline \\ \times & \beta & \gamma & \delta & \rho & \zeta & \xi \\ \hline \beta & \beta & \gamma & \delta & \rho & \zeta & \xi \\ \gamma & \gamma & \rho & \xi & \beta\beta & \beta\delta & \beta\zeta \\ \delta & \delta & \xi & \beta\gamma & \beta\zeta & \gamma\beta & \gamma\rho \\ \rho & \rho & \beta\beta & \beta\zeta & \gamma\gamma & \gamma\xi & \delta\delta \\ \zeta & \zeta & \beta\delta & \gamma\beta & \gamma\xi & \delta\rho & \rho\gamma \\ \xi & \xi & \beta\zeta & \gamma\rho & \delta\delta & \rho\gamma & \zeta\beta \\ \beta\alpha & \beta\alpha & \gamma\alpha & \delta\alpha & \rho\alpha & \zeta\alpha & \xi\alpha \\ \hline \end{array}$$

so to find $\beta \zeta \times \beta\xi$ I do:

$$\begin{array}{ccc} & \beta & \zeta \\ \times & \beta & \xi \\ \hline & \xi & \gamma \\ & \rho & \\ \beta & \zeta & \\ \hline \delta & \beta & \gamma \\ \gamma & & \\ \end{array}$$

so I've got this far

$$\begin{array}{rcl} [\![60Z8]\!] &=& \xi (\beta\xi)^3 + \alpha (\beta\xi)^2 + \beta \zeta (\beta\xi) + \beta\beta \\ &=& \xi (\beta\xi)^3 + \alpha (\beta\xi)^2 + \delta \beta \gamma + \beta\beta \\ \end{array}$$

Now I need to perform the addition using the algorithm which was mentioned before:

$$\begin{array}{ccc} \delta & \beta & \gamma \\ & \beta & \beta \\ \hline \delta & \gamma & \delta \\ \end{array}$$

so

$$\begin{array}{rcl} [\![60Z8]\!] &=& \xi (\beta\xi)^3 + \alpha (\beta\xi)^2 + \beta \zeta (\beta\xi) + \beta\beta \\ &=& \xi (\beta\xi)^3 + \alpha (\beta\xi)^2 + \delta \beta \gamma + \beta\beta \\ &=& \xi (\beta\xi)^3 + \alpha (\beta\xi)^2 + \delta \gamma \delta \\ \end{array}$$

and continuing this way I get $$[\![60Z8]\!] = \delta\beta\gamma\gamma\gamma\rho.$$

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In summary: If I have my own conception of number in terms of base $q$ strings of digits, then I have way to interpret your numbers from base $b$ into my own system, based on the fundamental arithmetic operations - which operate natively in base $q$.

Answer 3

To convert a number $n$ from base A to base B you do the following:

while n!=0:
  digit[i++] = n mod B
  n = n div B

Since you can do arithmetic in any base, in particular in base A, you can compute n mod B and n div B without converting to any intermediate base.

Answer 4

We just covered something very much like what you are looking for in Calc I recently. I don't have my textbook handy, but here's a link to a website that might help you.

If you have access to a logarithm function that is.