Burton MacKenzie

What if 7 was the same quantity as 34? And 142.001529 as well? That seems like an impractical crazy number system, but those can be the most fun, so I'm going to show you one.

The number system we most commonly use is a base 10 place-value system. A number is described by digits from 0 to 9, and the position of each digit denotes the exponent of the base, multiplied by the value of the digit. For instance, 341 = 3*10² + 4*10¹ + 1*10⁰. In binary (base 2), each digit represents a power of two, e.g. 2⁵ 2⁴ 2³ 2² 2¹ 2⁰. 2^-1 etc. With these bases, each quantity has only a single representation. For instance in base 10, 2 is equal only to 2 [1] and no other finite number [2].

This is what we consider normal and natural. However, if we choose an irrational number such as the golden ratio as the base of a number system, we get something entirely different.

In an earlier post I derived the golden ratio, showing its origin from the equation φ = 1 + 1/φ. Rewrite it as φ² = φ + 1. Since the digits in base φ represent exponent multiples of φ, all numbers take the form of φ² φ¹ φ⁰, etc. In base φ, the equation above can be directly represented as numbers, stating

1* φ² + 0*φ¹ + 0* φ^{0 =} 0*φ² + 1*φ¹ +1*φ⁰

or simply, 100_φ = 11_φ, or even more simply 100 = 11 (assuming base φ). This demonstrates that a number in base φ has non-unique finite representations! One number (100) can be represented as another (11). Multiplying or dividing the original equation by φ shows that we can apply this rule universally. Just as 100 = 11, 1000 = 110, 10 = 1.1, 0.001 = 0.00011, etc.

The next question is how many representations does each number have in base φ? Look at the case of 100 again. 100 can be represented as 11, 11 can be represented as 10.11, 10.11 can be represented as 10.1011, 10.1011 can be represented as 10.101011, ad infinitum. Not only are finite numbers in base φ representable in non-unique ways, there are an infinite number of ways to finitely represent any number in base φ!

Now that's a fun number system!

Burton MacKenZie www.burtonmackenzie.com

[1] Even for large values of 2!

[2] strictly speaking, 2 can also be represented in base 10 as an infinite series, 1.9999999....