5

---

5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.

Five is the third-smallest prime number,^[1] equal to the sum of the only consecutive positive integers to also be prime numbers (2 + 3). In integer sequences, five is also the second Fermat prime, and the third Mersenne prime exponent, as well as the fourth or fifth Fibonacci number;^[2] 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).^[3]

In geometry, the regular five-sided pentagon is the first regular polygon that does not tile the plane with copies of itself, and it is the largest face that any of the five regular three-dimensional regular Platonic solid can have, as represented in the regular dodecahedron. For curves, a conic is determined using five points in the same way that two points are needed to determine a line.^[4]

In abstract algebra and the classification of finite simple groups, five is the count of exceptional Lie groups as well as the number of Mathieu groups that are sporadic groups. Five is also, more elementarily, the number of properties that are used to distinguish between the four fundamental number systems used in mathematics, which are rooted in the real numbers.

Historically, 5 has garnered attention throughout history in part because distal extremities in humans typically contain five digits.

In the classification of number systems, the real numbers and its three subsequent Cayley–Dickson constructions of algebras over the field of the real numbers (i.e. the complex numbers, the quaternions, and the octonions) are normed division algebras that hold up to five different principal algebraic properties of interest: whether the algebras are ordered, and whether they hold commutative, associative, alternative, and power-associative multiplicative properties.^[5] Whereas the real numbers contain all five properties, the octonions are only alternative and power-associative. In comparison, the sedenions, which represent a fifth algebra in this series, is not a composition algebra unlike and , is only power-associative, and is the first algebra to contain non-trivial zero divisors as with all further algebras over larger fields.^[6] Altogether, these five algebras operate, respectively, over fields of dimension 1, 2, 4, 8, and 16.${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathbb {O} }$ ${\displaystyle \mathbb {S} }$${\displaystyle \mathbb {H} }$${\displaystyle \mathbb {O} }$

---