Mathematics is truly one of the most important and wide ranging fields of study, and its role and influence is apparent in numerous aspects of life and civilization. Of course, math isn't always the easiest of subjects, a realization that many people will attest to. In this article, we compare two particularly challenging aspects of mathematics: Fibonacci numbers and factorials.

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Fibonacci numbers are mathematical occurrences in which numbers follow a specific sequence of integers. The first two numbers in the sequence are 0 and 1, with each subsequent number being the sum of the previous two. Some examples however omit the 0, and instead start off the sequence with two 1s.

Factorials are products of positive integers that are either less than or equal to “n”, with “n” being the factorial of a positive integer.

Fibonacci numbers always occur in sums of what are known as "shallow" diagonals in Pascal's triangle, and Lozanic's triangle. Such number sequences are more apparent in Hosoya's triangle.

Factorial operations are seen in many different mathematical fields, most commonly in combinatorics, algebra and mathematical analysis. The most basic occurrence of factorials is seen in the fact that there are “n” ways to arrange “n” objects into sequences, an occurrence that is known as the permutations of the set of objects. This occurrence was discovered as far back as the 12th century by scholars in India, although the “n!” notation commonly used today was only introduced in 1808 by Christian Kramp.

Fibonacci numbers are often used in financial market analyses, and in procedures known as Fibonacci retracements. They are also used in computer algorithms like the Fibonacci search technique and the Fibonacci heap data structure. The recursion of Fibonacci numbers has also given rise to a group of recursive graphs known as Fibonacci cubes, which are commonly utilized in interconnecting parallel and distributed systems. Fibonacci sequences are although commonly seen in biological settings, as in the case of tree branching, leaf arrangement in stems, and even in the way pineapples sprout. Other examples of Fibonacci sequences include the flowering of artichokes, the way that ferns uncurl and in the arrangement of the segments of pine cones.

Factorials are often seen in algebra in numerous instances, among them the coefficients of the binomial formula, or the averaging over permutations for symmetrization of specific operations. Factorials are also commonly seen in calculus, as in the case of the denominators of Taylor's formula terms, where they occur mainly to make up for the fact that the nth derivative of xn is n!. Other areas wherein factorials are often seen are probability theory and expression manipulation.

Fibonacci

- Are mathematical occurrences in which numbers follow a specific sequence of integers
- Always occur in sums of what are known as "shallow" diagonals in Pascal's triangle, and Lozanic's triangle

Factorial

- Products of positive integers that are either less than or equal to “n”, with “n” being the factorial of a positive integer
- Seen in many different mathematical fields, most commonly in combinatorics, algebra and mathematical analysis

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