The initial number in the Fibonacci sequence is 0.

Drag the slider to the right to visualize the sequnce.

Drag the slider to the right to visualize the sequnce.

The Fibonacci numbers are a sequence of integers defined as:

F_{0} = 0

F_{1} = 1

F_{n} = F_{n − 1} + F_{n − 2}

The first two numbers are 0 and 1, and thereafter, every number is equal to the sum of the two previous numbers. This is illustrated above where the side of each square is equal to the sides of two previous squares combined. When a spiral is drawn using circular arcs across each square, it is called the Fibonacci Spiral.

The Fibonacci sequence is named after Medieval mathematician Leonardo Fibonacci, who popularized the number sequence in his book *Liber Abaci* in the early 13th century. He used the Fibonacci sequence to predict the population growth of breeding rabbits. The Fibonacci numbers appear frequently in nature, for example in the petal leaves of flowers and in the spiral shape of shells.

The Golden Ratio, φ, is defined as:

a + b | = | a | = φ ≈ 1.6180339887 |

a | b |

This means that when the ratio between **a** and **b** is the same as the ratio between **a + b** and **a**, it is the Golden Ratio. The Fibonacci sequence is heavily interconnected with the Golden Ratio, since the ratio between neighboring Fibonacci numbers gets closer and closer to the Golden Ratio, the higher the numbers get.

Here are all the Fibonacci numbers smaller than one quadrillion: