No Calculator: The Golden Ratio to the 12th Power — Integer & Decimal Parts Explained
Автор: Texas Little Apple
Загружено: 2025-12-25
Просмотров: 65
Описание:
What happens when you raise the Golden Ratio to a high power?
In this video, we answer a surprisingly elegant question:
What are the integer and decimal parts of 𝜙^12?
Instead of approximating the Golden Ratio numerically, we
use algebra, quadratic roots, and a powerful sequence identity
to solve the problem exactly.
You’ll see how:
The Golden Ratio and its conjugate arise from a simple quadratic equation.
Powers of these roots generate the Lucas number sequence.
A small “error term” allows us to pin down precise bounds.
We determine the integer and decimal parts without ever calculating 𝜙^12.
This is a beautiful example of how structure beats computation — perfect
for math competitions, advanced algebra students, or anyone who enjoys
elegant mathematical reasoning.
#math #goldenratio #irrationalnumbers #phi #numbertheory #algebra #mathproof #contestmath
#lucasnumbers #learnmath #mathematics #mathvideo
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