Non-integer base of numeration

Following the steps above, we can create a β-expansion for a real number ${\displaystyle n\geq 0}$ (the steps are identical for an ${\displaystyle n<0}$, although n must first be multiplied by −1 to make it positive, then the result must be multiplied by −1 to make it negative again).

First, we must define our k value (the exponent of the nearest power of β greater than n, as well as the amount of digits in ${\displaystyle \lfloor n_{\beta }\rfloor }$, where ${\displaystyle n_{\beta }}$ is n written in base β). The k value for n and β can be written as:

${\displaystyle k=\lfloor \log _{\beta }(n)\rfloor +1}$

After a k value is found, ${\displaystyle n_{\beta }}$ can be written as d, where

${\displaystyle d_{j}=\lfloor (n/\beta ^{j}){\bmod {\beta }}\rfloor ,\quad n=n-d_{j}*\beta ^{j}}$

for k − 1 ≥  j > −∞. The first k values of d appear to the left of the decimal place.

This can also be written in the following pseudocode:

function toBase(n, b) {
	k = floor(log(b, n)) + 1
	precision = 8
	result = ""

	for (i = k - 1, i > -precision-1, i--) {
		if (result.length == k) result += "."
		
		digit = floor((n / b^i) mod b)
		n -= digit * b^i
		result += digit
	}

	return result
}

[1]

Note that the above code is only valid for ${\displaystyle 1<\beta \leq 10}$ and ${\displaystyle n\geq 0}$, as it does not convert each digits to their correct symbols or correct negative numbers. For example, if a digit's value is 10, it will be represented as 10 instead of A.

• JavaScript:[1]

  function toBasePI(num, precision = 8) {    
      let k = Math.floor(Math.log(num)/Math.log(Math.PI)) + 1;
      if (k < 0) k = 0;
  
      let digits = [];
  
      for (let i = k-1; i > (-1*precision)-1; i--) {
          let digit = Math.floor((num / Math.pow(Math.PI, i)) % Math.PI);
          num -= digit * Math.pow(Math.PI, i);
          digits.push(digit);
  
          if (num <= 0)
              break;
      }
  
      if (digits.length > k)
          digits.splice(k, 0, ".");
  
      return digits.join("");
  }
  

• JavaScript:[1]

  function fromBasePI(num) {
      let numberSplit = num.split(/\./g);
      let numberLength = numberSplit[0].length;
  
      let output = 0;
      let digits = numberSplit.join("");
  
      for (let i = 0; i < digits.length; i++) {
          output += digits[i] * Math.pow(Math.PI, numberLength-i-1);
      }
  
      return output;
  }
  

Base √2 behaves in a very similar way to base 2 as all one has to do to convert a number from binary into base √2 is put a zero digit in between every binary digit; for example, 1911₁₀ = 11101110111₂ becomes 101010001010100010101_√2 and 5118₁₀ = 1001111111110₂ becomes 1000001010101010101010100_√2. This means that every integer can be expressed in base √2 without the need of a decimal point. The base can also be used to show the relationship between the side of a square to its diagonal as a square with a side length of 1_√2 will have a diagonal of 10_√2 and a square with a side length of 10_√2 will have a diagonal of 100_√2. Another use of the base is to show the silver ratio as its representation in base √2 is simply 11_√2. In addition, the area of a regular octagon with side length 1_√2 is 1100_√2, the area of a regular octagon with side length 10_√2 is 110000_√2, the area of a regular octagon with side length 100_√2 is 11000000_√2, etc…

In the golden base, some numbers have more than one decimal base equivalent: they are ambiguous. For example: 11_φ = 100_φ.

There are also some numbers in base ψ are also ambiguous. For example, 101_ψ = 1000_ψ.

With base e the natural logarithm behaves like the common logarithm as ln(1_e) = 0, ln(10_e) = 1, ln(100_e) = 2 and ln(1000_e) = 3.

The base e is the most economical choice of radix β > 1 (Hayes 2001), where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.

Base π can be used to more easily show the relationship between the diameter of a circle to its circumference, which corresponds to its perimeter; since circumference = diameter × π, a circle with a diameter 1_π will have a circumference of 10_π, a circle with a diameter 10_π will have a circumference of 100_π, etc. Furthermore, since the area = π × radius², a circle with a radius of 1_π will have an area of 10_π, a circle with a radius of 10_π will have an area of 1000_π and a circle with a radius of 100_π will have an area of 100000_π.[2]