What are Fibonacci Numbers?
The Fibonacci sequence is one of the most famous sequences in all of mathematics. Each number is the sum of the two preceding ones, starting from 0 and 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597...
The sequence is named after Leonardo of Pisa, known as Fibonacci, who introduced it to European mathematics in his 1202 book Liber Abaci. He used it to model the growth of a rabbit population under idealized conditions. However, the sequence was known in Indian mathematics as early as 200 BC, described by Pingala in the context of Sanskrit poetry.
The Golden Ratio
The ratio of consecutive Fibonacci numbers converges to the golden ratio φ (phi) = (1 + √5) / 2 ≈ 1.6180339887... As you go further in the sequence, F(n+1)/F(n) approaches φ with increasing precision. The golden ratio is an irrational number with remarkable mathematical properties — it is the “most irrational” number in the sense that it is the hardest to approximate by rational numbers.
The golden ratio appears in art and architecture (the Parthenon, Leonardo da Vinci’s works), nature (the arrangement of leaves, flower petals, and seeds), and financial markets (Fibonacci retracement levels in technical analysis). The golden rectangle, whose sides are in the ratio 1:φ, is considered one of the most aesthetically pleasing shapes.
Fibonacci Numbers in Nature
Fibonacci numbers appear throughout the natural world with remarkable frequency. The number of petals on a flower is often a Fibonacci number: lilies have 3, buttercups have 5, delphiniums have 8, marigolds have 13, daisies often have 21, 34, 55, or 89 petals. Sunflower seed heads display spiral patterns where the number of clockwise and counterclockwise spirals are consecutive Fibonacci numbers (typically 34 and 55, or 55 and 89). Pinecones, pineapples, and artichokes exhibit similar Fibonacci spiral patterns.
Mathematical Properties
Fibonacci numbers possess extraordinary mathematical properties. The GCD property: gcd(F(m), F(n)) = F(gcd(m, n)). Zeckendorf’s theorem: every positive integer can be uniquely represented as a sum of non-consecutive Fibonacci numbers. Divisibility: F(n) divides F(mn) for all positive integers m and n. The sum of the first n Fibonacci numbers equals F(n+2) − 1. A number is Fibonacci if and only if 5n² + 4 or 5n² − 4 is a perfect square.