Fibonacci numbers – Singapore Maths Tuition

YouTube Video: Fibonacci Numbers and the Mysterious Golden Ratio

This is a YouTube Video, based on my earlier post on Fibonacci Numbers and the Mysterious Golden Ratio!

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The Fabulous Fibonacci Numbers

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature – from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.

With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples).

Fibonacci Numbers and the Mysterious Golden Ratio

What are Fibonacci Numbers?

Fibonacci Numbers, named after Leonardo Fibonacci, is a sequence of numbers:

$F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5$ ,

with a recurrence relation $F_n=F_{n-1}+F_{n-2}$ .

Relation to Golden Ratio

Fibonacci Numbers are linked to the mysterious Golden Ratio, $\displaystyle \phi=\frac{1+\sqrt{5}}{2}\approx 1.61803$

In fact, the ratio of successive Fibonacci numbers converges to the Golden Ratio! The first person to observe this is Johannes Kepler.

How do we prove it?

Recall the recurrence relation: $F_n=F_{n-1}+F_{n-2}$

Dividing throughout by $F_{n-1}$ , we get $\displaystyle \frac{F_n}{F_{n-1}}=1+\frac{F_{n-2}}{F_{n-1}}$

(We will first assume $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}$ exists for the time being, and prove it later)

Taking limits, we get $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}=1+\lim_{n\to\infty}\frac{F_{n-2}}{F_{n-1}}$ .

Denoting $\displaystyle\lim_{n\to\infty}\frac{F_n}{F_{n-1}}$ as $\phi$ , we get:

$\displaystyle \phi=1+\frac{1}{\phi}$

Multiplying by $\phi$ , we get $\phi^2=\phi +1$

$\phi^2-\phi-1=0$

This is a quadratic equation, solving using the quadratic equation, we get:

$\displaystyle \phi=\frac{1\pm\sqrt{1^2-4(1)(-1)}}{2}=\frac{1\pm\sqrt{5}}{2}$

Since $\phi$ is clearly positive, we have $\displaystyle \phi=\frac{1+\sqrt{5}}{2}$ which is the Golden Ratio!

For a complete proof, actually we will need to prove that $\displaystyle\frac{F_n}{F_{n-1}}$ converges. This is a bit tricky and requires some algebra.

Interested readers can refer to the excellent website at: http://pages.pacificcoast.net/~cazelais/222/fib-limit.pdf

for more details.

Interesting video on Fibonacci numbers!

Fibonacci numbers and the Golden Ratio can also be used for trading stocks.

Best Fibonacci Number Videos on Youtube

Math is logical, functional and just … awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)

Dr James Grime on the Pisano Period – a seemingly strange property of the Fibonacci Sequence.

Fibonacci Fun: Fascinating Activities With Intriguing Numbers
From “Raising Rabbits” to “Prickly Pinecones”, 24 easy-to-use, reproducible activities and projects introduce students to Fibonacci numbers and the golden ratio. Grades 4-8