4.4 Converting an Integer to a String in Any Base

Suppose you want to convert an integer to a string in some base between binary and hexadecimal. For example, convert the integer 10 to its string representation in decimal as "10", or to its string representation in binary as "1010". While there are many algorithms to solve this problem, including the algorithm discussed in the stack section, the recursive formulation of the problem is very elegant.

Let’s look at a concrete example with the number 769 to convert using base 10. Suppose we have a sequence of characters corresponding to the first 10 digits, like var convString = '0123456789'. It is easy to convert a number less than 10 to its string equivalent by looking it up in the sequence. For example, if the number is 9, then the string is convString[9] which gives us '9'. If we can arrange to break up the number 769 into three single-digit numbers, 7, 6, and 9, then converting it to a string is simple. A number less than 10 sounds like a good base case.

Knowing what our base is suggests that the overall algorithm will involve three components:

1. Reduce the original number to a series of single-digit numbers.
2. Convert the single digit-number to a string using a lookup.
3. Concatenate the single-digit strings together to form the final result.

The next step is to figure out how to change state and make progress toward the base case. Since we are working with an integer, let’s consider what mathematical operations might reduce a number. The most likely candidates are division and subtraction. While subtraction might work, it is unclear what we should subtract from what. Integer division with remainders gives us a clear direction. Let’s look at what happens if we divide a number by the base we are trying to convert to.

Using integer division to divide 769 by 10, we get 76 with a remainder of 9. This gives us two good results. First, the remainder is a number less than our base that can be converted to a string immediately by lookup. Second, we get a number that is smaller than our original and moves us toward the base case of having a single number less than our base. Now our job is to convert 76 to its string representation. Again we will use integer division plus remainder to get results of 7 and 6 respectively. Finally, we have reduced the problem to converting 7, which we can do easily since it satisfies the base case condition of $$n<base$$, where $$base=10$$.

Here is the JavaScript code that implement the algorithm outlined above for any base between 2 and 16:

function toStr(n, base) {
   var convertString = '0123456789ABCDEF';
   if (n < base) {
      return convertString[n];
   }
   else {
      return toStr(Math.floor(n/base), base) + convertString[n%base];
   }
}

console.log( toStr(1453, 16) );

Notice that in line 3 we check for the base case where n is less than the base we are converting to. When we detect the base case, we stop recursing and simply return the string from the convertString sequence. In line 6 we satisfy both the second and third laws–by making the recursive call and by reducing the problem size–using division.