Function

[Pages:3]214

MATHEMATICS MAGAZINE

Fibonacci Numbers and the Arctangent Function

KO H A Y A S H I

The King's Academy Sunnyvale, CA 94085

This note provides several geometric illustrations of three identities involving the arc-

tangent function and the reciprocals of Fibonacci numbers. The Fibonacci numbers are

+ defined by Fo = 0, F I = 1, and F, = F,_ I F,,-2, for n > 1. The following identities

link the Fibonacci numbers to the arctangent function. Only the first is evident in the literature [I, 2,3].

(&) (&) (&) arctan

= arctan

+ arctan

Identities (1)-(3) can be proven formally using Cassini's identity [I, p. 1271

and the addition formula for the tangent function. Interested readers are invited to do so.

The following six diagrams illustrate special cases of equations (1)-(3). FIGURE1, a representation of Euler's famous formula for n [4, 51, illustrates (1) for i = 1. One can see that LABD plus LDBC is equal to L A B C .

5 + Figure 1 = arctan(1) = arctan(+) arctan(;)

FIGURE2 illustrates (1) for i = 2, using the larger squares to form the arctangent of 115 and the smaller squares being used to form the arctangents of 113 and of 118.

The two diagrams in FIGURE3 illustrate (2) for the values i = 1 and i = 2. The diagrams in FIGURE4 illustrate equation (3) for the values i = 1 and i 2.

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21 5

+ Figure 2 arctan($) = arctan(:) arctan($)

Figure 3 arctan(:) = arctan(,) I

+ + Figure 4 arctan(1) = arctan($) arctan(: ); arctan({) = arctan($) arctan(&))

Acknowledgments. The author would like to thank Professor Paul Garrett for reviewing the lnathelnatics and to thank Ching-Yi Wang for his formatting of the manuscript.

REFERENCES

1. Robert hl. Young. E.rc~rr-.\ioi~i.i\z Cn/c~r/~rAriz; I?zrrrl~/cro,.j fhc. Cuntinitou.\ aizd tj2r Disc>-err.Dolciani Mathematical Expositions. MAA. Washington. DC. 1992. p. 136.

2. Marjorie Bicknell and Vernon E. Hoggatt Jr.. A Prir,~rr,forthe Fihorzucci N U I I I ~ CTh~eSF.ibonacci Association. San Jose, 1972. pp. 49-50.

3. Problem 3801. Amrr: M~ltIz~. VI~~rz4th5ij(,1938). 636. 4. Martin Gardner. I2.latlzrr,~uticciCl irc~isS. pectrum Series, MAA, Washington. DC, 1992, p. 125. 5. Edward Kitchen, Dorrie tiles and related miniatures. this M A G A L I N6E7 (1994). 128-130.

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