54 (number)

Template:About Template:Infobox number Template:Sister project 54 (fifty-four) is the natural number and positive integer following 53 and preceding 55. As a multiple of 2 but not of 4, 54 is an oddly even number and a composite number.

54 is related to the golden ratio through trigonometry: the sine 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,<ref name="abundant">Template:Cite OEIS</ref> since the sum of its proper divisors (66)<ref name="aliquot">Template:Cite OEIS</ref> is greater than itself.

In mathematics

Number theory

File:Square-sum-54.png

54 as the sum of three positive squares

54 is an abundant number<ref name="abundant"></ref> because the sum of its proper divisors (66),<ref name="aliquot"></ref> which excludes 54 as a divisor, is greater than itself. Like all multiples of 6,<ref>Template:Cite journal</ref> 54 is equal to some of its proper divisors summed together,Template:Efn so it is also a semiperfect number.<ref>Template:Cite OEIS</ref> 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.<ref>Template:Cite OEIS</ref> Additionally, as an integer for which the arithmetic mean of all its positive divisors (including itself) is also an integer, 54 is an arithmetic number.<ref>Template:Cite OEIS</ref>

Trigonometry and the golden ratio

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio.<ref>Template:Cite journal</ref><ref>Template:Cite OEIS</ref> This is because the corresponding interior angle is equal to [[Pi|Template:Pi]]/5 radians (or 36 degrees).Template:Efn = \frac{2\sqrt{5}}{4} = \frac{\phi}{2}</math>.}} 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, the 54 side cannot be the hypotenuse. Using the Pythagorean theorem, there is no way to construct 54² as the sum of two other square rational numbers. Therefore, 54 is a nonhypotenuse number.<ref>Template:Cite OEIS</ref>

However, 54 can be expressed as the area of a triangle with three rational side lengths.Template:Efn Therefore, it is a congruent number.<ref>Template:Cite OEIS</ref> 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.Template:Efn This is an important property in Assyro-Babylonian mathematics because that system uses a sexagesimal (base-60) number system. In base 60, the reciprocal of a regular number has a finite representation. Babylonian computers kept tables of these reciprocals to make their work more efficient. Using regular numbers simplifies multiplication and division in base 60 because dividing Template:Mvar by Template:Mvar can be done by multiplying Template:Mvar by Template:Mvar's reciprocal when Template:Mvar is a regular number.<ref>Template:Citation.</ref><ref>Template:Citation</ref>

For instance, division by 54 can be achieved in the Assyro-Babylonian system by multiplying by 4000 because Template:Nowrap = Template:Nowrap = 4000. In base 60, 4000 can be written as 1:6:40.Template:Efn Because the Assyro-Babylonian system does not have a symbol separating the fractional and integer parts of a number<ref name="BabFrac">Template:Cite journal</ref> and does not have the concept of 0 as a number,<ref>Template:Cite journal</ref> it does not specify the power of the starting digit. Accordingly, 1/54 can also be written as 1:6:40.Template:Efn<ref name="BabFrac"/> 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 60³.Template:Efn

Graph theory

File:Ellingham-Horton 54-graph.svg

The Ellingham–Horton 54-graph

The second Ellingham–Horton graph was published by Mark N. Ellingham and Joseph D. Horton in 1983; it is of order 54.<ref>Template:Citation.</ref> These graphs provided further counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.<ref>Template:Citation.</ref> Horton disproved the conjecture some years earlier with the Horton graph, but that was larger at 92 vertices.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The smallest known counter-example is now 50 vertices.<ref>Template:Citation.</ref>

In literature

In The Hitchhiker's Guide to the Galaxy by Douglas Adams, the "Answer to the Ultimate Question of Life, the Universe, and Everything" famously was 42.<ref>Template:Cite book</ref> Eventually, one character's unsuccessful attempt to divine the Ultimate Question elicited "What do you get if you multiply six by nine?"<ref name="Restaurant">Template:Cite book</ref> 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 54₁₀ can be encoded as the base-13 expression Template:Nowrap = 42₁₃.<ref name="scripts">Template:Cite book</ref> Adams said this was a coincidence.<ref>Template:Cite news</ref>

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
54 × x	54	108	162	216	270	324	378	432	486	540	594	648	702	756	810

Division	1	2	3	4	5	6	7	8	9	10
54 ÷ x	54	27	18	13.5	10.8	9	7.Template:Overline	6.75	6	5.4
x ÷ 54	0.0Template:Overline	0.Template:Overline	0.0Template:Overline	0.Template:Overline	0.0Template:Overline	0.Template:Overline	0.1Template:Overline	0.Template:Overline	0.1Template:Overline	0.Template:Overline

Exponentiation	1	2	3
54x	54	2916	157464
x54	1	18014398509481984	58149737003040059690390169
<math>\sqrt[x]{54}</math>	54	7.34846...Template:Efn	3.77976...

Explanatory footnotes

File:Genji chapter symbols groupings of 5 elements.svg

Genji-mon, the traditional symbols that represent the fifty-four chapters of The Tale of Genji