Fibonacci Sequence: Examples, Golden Ratio & Nature

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  • 0:05 Definition
  • 1:03 The Golden Ratio
  • 3:27 Nature
  • 4:50 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

The Fibonacci sequence is seen all around us. Learn how the Fibonacci sequence relates to the golden ratio and explore how your body and various items, like seashells and flowers, demonstrate the sequence in the real world.

Definition

The Fibonacci sequence begins with the numbers 0 and 1. The third number in the sequence is the first two numbers added together (0 + 1 = 1). The fourth number in the sequence is the second and third numbers added together (1 + 1 = 2). Each successive number is the addition of the previous two numbers in the sequence. The sequence ends up looking like this:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on and so forth.

Looking at it, you can see that each number in the sequence is the addition or sum of the two previous numbers. For example, 34 is the addition of 21 and 13. 144 is the addition of 89 and 55. Try it out yourself and check other numbers in the sequence to see if they follow the rule.

The Golden Ratio

The golden ratio, represented by the Greek letter phi, is approximately 1.618. The golden ratio, like pi, is an irrational number that keeps going. The actual value goes like this: 1.618033988764989. . .

You might be wondering how the Fibonacci sequence relates to this number. Let us see.

Let's start by dividing pairs of numbers in the Fibonacci sequence. We will skip zero and start with the pair of ones. 1 / 1 = 1. The next pair is the one and the two. 2 / 1 = 2. In each pair, we divide the larger by the smaller number. Let's keep going and see where it takes us:

Fibonacci pair Result
2 and 3 3 / 2 = 1.5
3 and 5 5 / 3 = 1.6666. . .
5 and 8 8 / 5 = 1.6
8 and 13 13 / 8 = 1.625
13 and 21 21 / 13 = 1.6154. . .
21 and 34 34 / 21 = 1.619. . .
34 and 55 55 / 34 = 1.618. . .
55 and 89 89 / 55 = 1.618. . .
89 and 144 144 / 89 = 1.618. . .

As the numbers get larger, an interesting thing starts to happen. The result of dividing the pairs of numbers gives you the approximate value of the golden ratio, 1.618. . .

In mathematical terms, the Fibonacci sequence converges on the golden ratio. What that means is that, as the Fibonacci sequence grows, when you divide a pair of numbers from the sequence, the result will get closer and closer to the actual value of the golden ratio. Looking at the table, you can see that starting with the pair 34 and 55, the result is accurate to three decimal places. As the pairs get larger, the result will be more accurate to the decimal.

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