Leonardo Pisano was an Italian mathematician, born after the dark ages in 1170 (Gies). Today he is widely known as Fibonacci, which means the son of Bonacci. His father, Guglielmo Bonacci, was a merchant, so when Guglielmo went to the port in Bugia, Algeria, Fibonacci began studying under an Arabian (Gies). During this period, the Hindu-Arabic numeral system was not commonly used or heard of. Fibonacci grew up using the Roman numeral system. His education led him to realize the simplicity of using the Hindu-Arabic numeral system for calculations. The place holder zero wasn't widely recognized, however Fibonacci recognized its value in performing calculations with the symbols for 1-9 used in India at the time. In 1202, Fibonacci published his work in a book titled Liber Abaci, which exposed all of Europe to the Hindu-Arabic numerals which influenced how we count and calculate numbers (Scott, p. 1). Today we use symbols for numbers similar to the Hindu-Arabic symbols. However, Fibonacci’s legacy didn’t come from this milestone in number theory, but rather a special recursive sequence named after him, the Fibonacci sequence.
What is a sequence? “A sequence is an ordered set of mathematical objects” (Weisstein, “Sequence”). Every sequence has a rule that the ordered set of objects follows. Then that rule is applied to the ordered object. Say that we have a sequence of {2, 4, 6, 8, 10}, we notice that each object is a multiple of two. This sequence is ordered where 2 is the first term, 4 is the second term, and so on until 10 is the fifth term. Each object is also divisible by its corresponding ordered term, where 2 is divisible by one, 4 is divisible by two, and so on until 10 is divisible by five. Therefore, the rule for our sequence is 2n where n = 1, . . ., 5.
The Fibonacci sequence is a recursive sequence, which means that “objects are defined in terms of other objects of the same type” (Weisstein, “Recursion”). A recursive sequence has a rule that each of its objects must follow, like a regular sequence, but now this rule is dependent upon the objects directly before the nth term. To start a recursive sequence, we must be given an initial value or two initial values. The Fibonacci sequence is defined as F(n) = F(n − 1) + F(n − 2) where n ≥ 3 with the initial values of F(1) = 1 and F(2) = 1 (Paul, p. 1394). “The Fibonacci sequence is the first known recursive sequence in mathematical work” (Paul, p. 1394). Below in Figure 2, a table of the first 10 terms are shown. Note that Fibonacci did not start with n = 0, however, starting with initial conditions of F(0) = 0 and F(1) = 1 which results in the same sequence and corresponding order.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ... | n |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| F(n) | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | ... | F(n-1)+F(n-2) |
Figure 2: Terms in the Fibonacci Sequence
What is the golden ratio? The golden ratio, also denoted as the Greek letter phi, φ or similarly Φ, represents the value 1.618033988749894... or (1+ √5) ⁄ 2 (Scott, p.2). The terms in the Fibonacci sequence share an interesting relationship with the golden ratio in that as n approaches infinity F(n) ⁄ F(n − 1) = (1+ √5) ⁄ 2 = φ (Kazemi, p. 3). This ratio means that F(n) is 1.618 times larger than F(n-1). The golden ratio is a concept in mathematics that has existed forever; however, we don’t know when it was first discovered because it has most likely been derived multiple times throughout history without any record of it today (Kazemi, p. 1).