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Binet's Formula

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Binet formula thru zero 650x650.jpg

Fibonacci Lucas and Phi

The connection between the Fibonacci and Lucas numbers and the Golden Section is expressed by the well-known mathematical formulas, so-called Binet's formulas. (Jovanovic)[1]

Binet's Formula (see Eq. 1) is an analytic equation that provides a generalized Fibonacci sequence as real numbers between the integers of the integer-based Fibonacci Sequence.

The Fibonacci Sequence is a summation series beginning with 1 + 1, with each new value as the sum of the last two values in the series.

Binet's Formula is derived from the identity of adjacent powers of the Golden Ratio (Φ = 1.618...),

φ2 + φ = 1


φ = Φ – 1 = (√5-1)/2 = 0.6180...

The adjacent-powers identity derives Binet's Formula:

Fn = ( Φn – (–φ)n ) / √5 [2](1)

Where –1n = cos(nπ), [3] Binet's Formula (Eq. 1) is algebraically equivalent to Eq. 2:


An interesting thing about Binet's Forumula is the term (-φ)n, which is the source of a square root of a negative when used in various forms. This is the basis for imaginary numbers, also attributed to the work of Binet.

As one sequences through increasing powers of n, the result will oscillate between negative and positive solutions, e.g.

-20 = 1
-21 = -2
-22 = 4
-23 = -8
-24 = 16

However, for positive values of n there is no oscillation in sign. The oscillation for the negative values in the Fibonacci sequence crosses through zero every π radian integer spans, or at every half integer.

Jacques Philippe Marie Binet

Jacques Philippe Marie Binet was born on February 2, 1776 in Renje and died on May 12, 1856 in Paris[1]

Biographical sketch

De Moivre

Abraham de Moivre 1667 - 1754
De Moivre, Abraham De
De Moivre summary
developed formula for the normal curve
developed analytic geometry

Lucas sequence

Also a summation sequence like the Fibonacci sequence, the Lucas sequence differs in construction by the two starting seed numbers of the sequencing algorithm; the numbers '2' and '1', in that order.

See also

Binet Forms,
R. Knott on Argand Map of Binet's Forumula
Copyleft by Harry J. Smith from Geocities archives
Generalized Binet dynamics, by Clifford a Reiter[4] and Chen Ning

  1. 1.0 1.1 Binet's Formulas, © 2001-2003 Radoslav Jovanovic
  2. De Moivre(1718), Binet(1843), Lamé(1844), Vajda-58, Dunlap-69, Hoggatt-page 11, B&Q(2003)-Identity 240
  3. Negative values of powers of n are computationally difficult on most popular computer applications, failing silently in Google spreadsheet, Excell, etc.
  4. Clifford A. Reiter, Professor of mathematics, Lafayette College, Pennsylvania