Binets formula for noninteger values of n

This section is optional and at an advanced level i.e. post 16 years education. Take me back to the Fibonacci Home page now or learn about square roots of negative numbers in what follows!

Well now we've tried negative values for n, why not try fractional or other non-whole values for n?

This doesn't make sense in terms of numbers in a series (there is a 2nd and a 3rd term and even perhaps a -2nd term but what can we possibly mean by a 2 5th term for instance)??

However, this could give us some interesting (mathematical) insights into the whole-number terms which are our familiar Fibonacci series.

Complex Numbers

The trouble is that in Binet's formula:

Fib(n) = { Phi» - ( phi) » }/V5 the second term (-phi)n means we have to find the n-th power of a negative number: -phi and n is not a whole number. If n was 0 5 for instance, meaning sqrt(-phi), then we are taking the square-root of a negative value which is "impossible".

Mathematicians have already extended the real-number system to cover such "imaginary" numbers. They are called complex numbers and have two parts A and B, both normal real numbers: a real part, A, and an imaginary part, B. The imaginary part is a multiple of the basic "imaginary" quantity V(-l), denoted i. So complex numbers are written as x + i y or x + y i or sometimes as x + I y or even more simply as (x,y).

Applications of Complex numbers

To me it is still surprising that such "imaginary" numbers - or numbers involving the imaginary quantity that is the square root of a negative number - have very practical applications in the real world.

For instance, electrical engineers have already found many applications for such "imaginary" or complex numbers. Whereas resistance can be described by a real number often measured in ohms, complex numbers are used for the inductance and capacitance, so they have very practical uses!

Electrical engineers tend to use j rather than i when writing complex numbers.

Mathematicians find uses for complex numbers in solving equations:

• Every equation of the form Ax+B=0 has a solution which is a fraction: namely X=-B/A if A and B are integers. These are called linear equations where A and B are, in general, any real numbers.

• Every equation of the form Ax2 + Bx + C=0 has either one or two solutions IF we allow complex numbers for x. (Here A is not zero or we just get a linear equation.)

For instance x2=2 has two solutions:

but x2=0 has just one solution namely x=0. Note that x2=-2 has two solutions too:

x=sqrt(-2)=isqrt(2) and x=-sqrt(-2)=-isqrt(2) • Every equation of the form Ax3 + Bx2 + Cx + D = 0 has at most 3 solutions again allowing x to be a complex number if necessary.

This leads to a beautiful theorem about solving equations which are sums of (real number multiples of) powers of x, called polynomials in x: If the highest power of x in a polynomial is n then there are at most n complex number solutions which make the polynomial's value zero

^^ Complex Numbers and Their Applications by F J Budden, Longman's 1968, is now out of print but is an excellent introduction to the fascinating subject of complex numbers and their applications.

Argand Diagrams

Writing (x,y) for a complex numbers suggests we might be able to plot complex numbers on a graph, the x distance being the real part of a complex number and the y height being its complex part.

Such plots are called Argand diagrams after J. R. Argand (1768-1822). We can plot an individual point such as 1 - 2i as the point (1,-2). Numbers which are real have zero as their complex part so the real number 3 is the same as the complex number 3 + 0 i and has "coordinates" (3,0). The real number -15 is the same as -15 + 0 i or (-15,0).

In general, the real number r is the complex number r + 0 i and is plotted at (r,0) on the Argand diagram.

In fact, all the real values are already in the graph along the x axis also called the real axis.

Numbers which are purely imaginary have a real-part of zero and so are of the form 0+yi always lying exactly on the y axis ( the imaginary axis).

Plotting functions on an Argand Diagram

We can plot a complex function on an Argand diagram, that is, a function whose values are complex numbers. This is where Binet's formula comes in since it will give us complex numbers as n (now a real number) varies over the real numbers.

So what happens if we plot a graph of F(n) described by Binet's formula, plotting the results on an Argand diagram?

The blue plot is for positive values of n from 0 to 6. Note how this curve crosses the x axis (representing the "real part of the complex number") at the Fibonacci numbers, 0, 1, 2, 3, 5 and 8. But there is a loop so it crosses the axis twice at x=1, and we really do get the whole Fibonacci sequence 0,1,1,2,3,5,8.. as the crossing points. The red plot is of negative values of n from -6 to 0. It also crosses the x axis at the values -8, 5, -3, 2, -1, 1 and 0

corresponding to the Fibonacci numbers F(-6), F(-5), F(-4), F(-3), F(-2), F(-l) and F(0).

FibCn^-ÈiniO, Flrgand diagram F i bCn 0<ni 6, Flrgand diagram

FibCn^-ÈiniO, Flrgand diagram F i bCn 0<ni 6, Flrgand diagram

Spirals?

• Note that the red spiral for negative values of n is NOT an equiangular or logarithmic spiral that we found in sea-shells on the Fibonacci in Nature page. This is because where the curve crosses the x axis at 1 and next at 2, so the distance from the origin has doubled, but the next crossing is not at 4 (which would mean another doubling as required for a logarithmic spiral) but at 5.

• If you adjust the width of your browser window, you should be able to see both curves side by side. Now it looks as if the two curves are made from the same 3-dimensional spiral spring-shape, a bit like a spiral bed-spring in cartoons, getting narrower towards one end. The red curve seems to be looking down the centre of the three-dimensional spring and the blue one looking at the same spring shape but from the side. Comparing the two diagrams shows even the heights of the loops are the same!

I haven't yet found an explanation for this - can you find one? [Let me know if you do!]

The plots were produced using Maple's parametric plotting provided with its built-in "plot" function:

plot([Re(f(n)),Im(f(n)),n=-6..0],color=RED, title="Fib(n),-62n20, Argand diagram"); plot([Re(f(n)),Im(f(n)),n=0..6],color=BLUE, title="Fib(n),02n26, Argand diagram");

Kurt Papke has a Web page with a Java applet to show the two Argand diagrams but animating the formula that f(n)=f(n-1)+f(n-2) for any real value n. The complex numbers f(n), f(n-1) and f(n-2) can be illustrated as vectors, and so the formula f(n)=f(n-1)+f(n-2) becomes a vector equation showing that the vector f(n-1) added to (followed by) the vector f(n-2) gives the same length-and-direction-movement as the vector f(n). Kurt has an excellent 3D version of the spiral that you can rotate on the screen (using a Java applet) AND one also for the Lucas numbers formula!

For a different complex function based on Binet's formula, see the following two articles where they both introduce the factor e*n 0 which is 1 when n is an integer:

Argand Diagrams of Extended Fibonacci and Lucas Numbers, F J Wunderlich, D E Shaw, M J

Hones Fibonacci Quarterly, vol 12 (1974), pages 233 - 234;

^^ An Extension of Fibonacci's Sequence P J deBruijn, Fibonacci Quarterly vol 12 (1974) page 251 -258;

References on Complex Numbers

Complex Numbers are included in some (UK based) Mathematics syllabuses at Advanced level (school/college examinations taken at about age 17). Here are some books relating to different Advanced level Examination Boards syllabus entries on Complex Numbers:

^ GCE A level Maths: Complex Numbers A. Nicolaides,ISBN: 1872684270, 1995.

^^ Nuffield Advanced Mathematics: Complex Numbers and Numerical Analysis June 1994, Longman, ISBN: 0582099846.

^ School Maths Project 16-19: Complex Numbers Cambridge, 1992, ISBN: 0521426529.

Fibonacci Home PageFf

The Puzzling World of the Fibonacci Numbers

^ The Mathematical Magic of the Fibonacci Numbers

The next Topic is... The Golden Section - the Number and Its Geometry

^ Fibonacci bases and other ways of representing integers

©1998-2001 Dr Ron Knott [email protected] 28 April 2001

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