A Method to Calculate Exact Values of n th Power of the Irrational Number Phi

Overview

This project created an effective mathematical method that simplifies the power elevating process for the golden ratio, (Phi). When using irrational numbers, approximations are used in utilizing a finite quantity of these digits. High exponentiation make results less exact, adding a long and tedious process. Assuming (phi) is a solution to [x^2-x-1=0], then, in combination with the replacement property and multiplicative equality, you will find a relationship to the values of the powers of the number (Phi). It started with the equation [(Phi)^2-(Phi)-1=0)], cleared [(Phi)^2)] and the result was, [(Phi)^2=(Phi) +1], then (Phi) is multiplied by the complete equation [(Phi)*((Phi)^2=(Phi)+1)], resulting in, [(Phi)^3 =(Phi)^2+(Phi)], [(Phi)^2)] was replaced by [(Phi)+1], [(Phi)^3=(Phi)+1+(Phi)], sum the similar terms and resulted in [(Phi)^3=2(Phi)+1]. The procedure was repeated, and the relationship of the exponent with their coefficients and constants was noted in the form of [(Phi)^n=a*(Phi)+b)], where n represented the power of (Phi), The numbers a and b represent constants. It was found that can be used the simple algorithm of the Fibonacci Sequence to find the powers of the irrational number (Phi). A computer algorithm was later created based in the new calculation of powers of (Phi). In addition, it was found that the idea can be generalized to any irrational solution of a quadratic equation, which would help in the teaching of Algebra and Number Theory.

Competition history

• ISEF 2015 Category not listed · Entry MATH027I

Resources

• ISEF abstract page ↗

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