The Golden Ratio in Base φ
To obtain the golden ratio, imagine having two amounts a and b, where a is larger than b. These are two parts of a whole, where the whole is a + b. If the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part, that is, if (a + b)/a equals a/b, then that ratio is called the golden ratio.
This is a number denoted by the Greek letter φ (phi), equal to (1 + √5)/2, and approximately 1.618.
What Does Base φ Mean?
So what does it mean to have a number in base φ? φ is not the number of digits in our system, since you can’t have φ digits. However, a positional numeral system can be defined by the specific number which is raised to a given power for each position and multiplied by the digit at that position.
In our standard base 10 system, each position represents a power of 10. The rightmost digit is multiplied by 10⁰ = 1, the next position to the left is multiplied by 10¹ = 10, then 10² = 100, and so on. For example, the number 325 in base 10 means 3 × 10² + 2 × 10¹ + 5 × 10⁰.
In base φ, each position represents a power of φ instead. Moving from right to left, positions are multiplied by φ⁰ = 1, φ¹ = φ, φ² = φ × φ, φ³ = φ × φ × φ, and so on. The digits used are typically 0 and 1, making it a binary-like system, though the base itself is irrational.
What makes base φ particularly interesting is a unique property of the golden ratio itself. Since φ satisfies the equation φ² = φ + 1, any power of φ can be expressed as a linear combination of lower powers. This means that base φ representations can be manipulated using the golden ratio’s special algebraic properties.
For instance, because φ² = φ + 1, we can replace any occurrence of φ² with φ + 1. This gives base φ arithmetic some fascinating characteristics not found in integer bases. Every positive integer can be represented uniquely in base φ using only the digits 0 and 1, with no two consecutive 1s, a property that connects to the Fibonacci sequence.
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …) is deeply connected to φ. The ratio of consecutive Fibonacci numbers approaches φ as the sequence progresses. In base φ, the nth Fibonacci number can be represented as 10…0 (a 1 followed by n−1 zeros), which provides an elegant connection between this unusual number system and one of mathematics’ most famous sequences.
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