Fibonacci Numbers & the Definition of φ, the Golden Ratio
by Mike Martin

This page allows you to investigate the definition of an irrational number, φ. Although we give development for it next, the defining formula for φ is:

To investigate this limit we consider the sequence whose nth term is given by

To motivate the connection between this sequence and the Golden Ratio, note that if we start with a Fibonacci sequence, Fn+1 = Fn + Fn-1 where F1 = 1 and F2 = 1, and consider a number, λ, to represent the supposed limit of the ratio of consecutive Fibonacci terms:

Then λ must satisfy   ; or, equivalently, . The positive solution to this quadratic equation is the one we're after and we may now write

So, the sequence approaches a limit, φ, as the term number grows without bound and after N terms the sequence value is within ε units of that limiting value. This tool is designed for you to investigate and visualize the relationship between N and ε, placing it in some context with the definition of the limit, φ, of this sequence. The irrational number φ is, to 20 decimal places, approximately equal to 1.6180339887498948482.

Enter the value of the parameters below, then press Evaluate.

ε =

ymin =

ymax =

nmin =

nmax =