SMRTR Science & EngineeringJun 8, 2025John D. Cook

Golden ratio base numbers

SMRTR summary

The golden ratio φ forms the basis of the "phinary" number system, expressing positive integers as sums of φ powers with 0 and 1 coefficients. Analogous to binary, it uses negative φ powers for some integers. Phinary representations are unique when consecutive 1s are prohibited. Remarkably, this system was discovered by 12-year-old George Bergman in 1957. Mathematicians have since expanded on and studied the phinary system further.

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