Fibonacci Numbers and the Golden Ratio

Learn the simple procedure for generating the Fibonacci sequence and see how it leads to a world of intriguing patterns. Also, get to know Fibonacci himself—a 13th-century mathematician who introduced Hindu-Arabic numerals to Europe. Finally, discover that when you divide consecutive Fibonacci numbers you get closer and closer to the golden ratio.

How do we know that some of the patterns examined in the previous lecture hold up for the entire Fibonacci sequence out to infinity? Roll up your sleeves and use different mathematical techniques to prove these conjectures. In a fun diversion, Professor Benjamin, an accomplished magician, uses a remarkable property of the Fibonacci sequence called Cassini’s identity to make a rabbit disappear.

See the professor’s favorite Fibonacci fact, which involves a concept called the greatest common divisor (GCD). Turn back the clock to the ancient Greek mathematician Euclid, who formulated an ingenious method for finding the GCD for any two integers. Then observe the Fibonacci connection emerge by working out the worst-case scenario that requires the maximum number of steps to get an answer.

Fibonacci’s real name was Leonardo of Pisa. The nickname Fibonacci (“son of Bonacci”) was bestowed in the 1800s and the Fibonacci numbers were given its name by French mathematician Édouard Lucas—a Fibonacci progression enthusiast. Lucas proposed a related sequence, now known as Lucas numbers. Compare and contrast Lucas numbers with Fibonacci numbers. Then, generate other amazing patterns.