54 (fifty-four) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

and positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

following 53 and preceding 55. As a multiple of 2 but not of 4, 54 is an oddly

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

and a

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...

. 54 is related to the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

through

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

: the

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...

of a 54 degree angle is half of the golden ratio. Also, 54 is a regular number, and its even division of powers of 60 was useful to ancient mathematicians who used the Assyro-Babylonian mathematics system. It is also an

abundant number In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...

, since the sum of its proper divisors (66) is greater than itself.

In mathematics

Number theory

54 is an

because the sum of its proper divisors ( 66), which excludes 54 as a divisor, is greater than itself. Like all multiples of 6, 54 is equal to some of its proper divisors summed together, so it is also a semiperfect number. These proper divisors can be summed in various ways to express all positive integers smaller than 54, so 54 is a practical number as well. Additionally, as an integer for which the

arithmetic mean In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results fr ...

of all its positive

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s (including itself) is also an integer, 54 is an arithmetic number.

Trigonometry and the golden ratio

If the complementary angle of a triangle's corner is 54 degrees, the

of that angle is half the

. This is because the corresponding interior angle is equal to /5

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...

s (or 36 degrees). If that triangle is isoceles, the relationship with the golden ratio makes it a golden triangle. The golden triangle is most readily found as the spikes on a regular pentagram. If, instead, 54 is the length of a triangle's side and all the sides lengths are

rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...

, the 54 side cannot be the hypotenuse. Using the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

, there is no way to construct 54 as the sum of two other square rational numbers. Therefore, 54 is a nonhypotenuse number. However, 54 can be expressed as the

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a triangle with three rational side lengths. Therefore, it is a congruent number. One of these combinations of three rational side lengths is composed of integers: 9:12:15, which is a 3:4:5 right triangle that is a Pythagorean, a Heronian, and a Brahmagupta triangle.

Regular number used in Assyro-Babylonian mathematics

As a regular number, 54 is a divisor of many powers of 60. This is an important property in Assyro-Babylonian mathematics because that system uses a

sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...

(base-60) number system. In base 60, the reciprocal of a regular number has a finite representation. Babylonian

computers A computer is a machine that can be programmed to automatically carry out sequences of arithmetic or logical operations ('' computation''). Modern digital electronic computers can perform generic sets of operations known as ''programs'', ...

kept tables of these reciprocals to make their work more efficient. Using regular numbers simplifies multiplication and division in base 60 because dividing by can be done by multiplying by 's reciprocal when is a regular number. For instance, division by 54 can be achieved in the Assyro-Babylonian system by multiplying by 4000 because = = 4000. In base 60, 4000 can be written as 1:6:40. Because the Assyro-Babylonian system does not have a

symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...

separating the

fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...

al and integer parts of a number and does not have the concept of 0 as a number, it does not specify the power of the starting digit. Accordingly, 1/54 can also be written as 1:6:40. Therefore, the result of multiplication by 1:6:40 (4000) has the same Assyro-Babylonian representation as the result of multiplication by 1:6:40 (1/54). To convert from the former to the latter, the result's representation is interpreted as a number shifted three base-60 places to the right, reducing it by a factor of 60.

Graph theory

The second Ellingham–Horton graph was published by Mark N. Ellingham and Joseph D. Horton in 1983; it is of order 54. These graphs provided further counterexamples to the conjecture of W. T. Tutte that every

cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...

3-connected

bipartite graph In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theo ...

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

. Horton disproved the conjecture some years earlier with the Horton graph, but that was larger at 92 vertices. The smallest known counter-example is now 50 vertices.

In literature

In ''

The Hitchhiker's Guide to the Galaxy ''The Hitchhiker's Guide to the Galaxy'' is a Science fiction comedy, comedy science fiction franchise created by Douglas Adams. Originally a The Hitchhiker's Guide to the Galaxy (radio series), radio sitcom broadcast over two series on BBC ...

'' by Douglas Adams, the " Answer to the Ultimate Question of Life, the Universe, and Everything" famously was 42. Eventually, one character's unsuccessful attempt to divine the Ultimate Question elicited "What do you get if you multiply six by nine?" The mathematical answer was 54, not 42. Some readers who were trying to find a deeper meaning in the passage soon noticed the fact was true in base 13: the base-10 expression 54 can be encoded as the base-13 expression = 42. Adams said this was a coincidence.

List of basic calculations

Explanatory footnotes

Genji_chapter_symbols_groupings_of_5_elements

References

{{DEFAULTSORT:54 (Number) Integers