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 n^{th} term is given by
To motivate the connection between this sequence and the Golden Ratio, note that if we start with a Fibonacci sequence, F_{n+1} = F_{n} + F_{n-1} where F_{1} = 1 and F_{2} = 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.