Fibonacci Numbers

The Man Who Brought Zero to Europe

**Leonardo of Pisa** (nicknamed Fibonacci - "son of Bonacci") was a merchant's son who traveled across North Africa. There he encountered **Hindu-Arabic numerals** (0, 1, 2... 9) and recognized their advantage over Roman numerals. In 1202 he wrote "Liber Abaci" - the book that brought zero to Europe.

These nine Indian figures - 9 8 7 6 5 4 3 2 1 - together with the sign 0 allow any number to be written.

Without Fibonacci, modern mathematics would have been impossible. Try multiplying MCMXCIX by XLVII (1999 × 47) in Roman numerals - and you'll understand why the positional system changed the world.

1, 1, 2, 3, 5, 8, 13, 21... What comes next? This simple sequence - each term the sum of the two before - appears everywhere: in sunflower spirals, nautilus shells, tree branching. Coincidence? No - mathematics of optimization.

• **Botany:** seed spirals, leaf arrangement
• **Art:** composition, proportions (sometimes)
• **Programming:** algorithms, data structures

The Fibonacci Sequence

The **Fibonacci sequence** is one of the most famous in mathematics. Each number is the sum of the two preceding ones. This simple rule produces remarkable patterns.

**Definition:** F₁ = 1, F₂ = 1 Fₙ = Fₙ₋₁ + Fₙ₋₂ (for n > 2) **Sequence:** 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

The numbers grow exponentially. Each next one is approximately 1.618 times the previous. This ratio is called the golden ratio.

Binet's Formula

There is an **explicit formula** for the nth Fibonacci number - without computing all the previous ones. It involves irrational numbers yet always yields an integer result.

**Why it works:** Binet's formula comes from solving the recurrence equation: xⁿ = xⁿ⁻¹ + xⁿ⁻² Roots of x² = x + 1: x = (1 ± √5) / 2 These are φ and ψ.

Remarkably, the formula contains √5 - an irrational number. Yet φⁿ - ψⁿ is always a multiple of √5, and the result is always an integer.

The Golden Ratio

The **golden ratio** φ ≈ 1.618 is the limit of the ratio of consecutive Fibonacci numbers. It has unique mathematical properties.

**Definition of the golden ratio:** The whole is to the larger part as the larger is to the smaller: (a + b) / a = a / b = φ This equation is x² = x + 1, whose root is φ.

The golden ratio appears in architecture, art, and nature. But not all "golden" proportions are real - many are attributed after the fact.

Fibonacci in Nature

Fibonacci numbers appear frequently in nature. But this is not magic - it is the result of simple growth rules.

**More Fibonacci in nature:** • Tree branching • Leaf arrangement (phyllotaxis) • Number of petals: 3, 5, 8, 13... • Bee ancestry tree **But not everywhere!** Many "golden proportions" in art are a myth.

Fibonacci in nature is not mysticism. It is the result of optimization. Living systems find efficient solutions, and Fibonacci numbers turn out to be one of them.

Key Ideas

• Fₙ = Fₙ₋₁ + Fₙ₋₂ - recurrence definition
• Binet's formula gives Fₙ explicitly through φ
• φ = (1+√5)/2 ≈ 1.618 - the golden ratio
• Nature uses φ for optimal packing

Related Topics

Fibonacci is related to other sequences:

• Proportions — The golden ratio is a special proportion
• Irrational Numbers — φ is irrational
• Arithmetic Progression — Another type of sequence

Вопросы для размышления

• Why does the ratio of consecutive Fibonacci numbers converge to φ?
• How does the sequence change if you start with different numbers?
• Where is the line between a genuine occurrence of φ and a fabrication?

Связанные уроки

• alg-21-dp
• calc-01-sequences