54 (number)

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.

54 is an abundant number^[1] because the sum of its proper divisors (66),^[2] is greater than itself. Like all multiples of 6,^[3] 54 is equal to some of its proper divisors summed together,^[a] so it is also a semiperfect number.^[4] 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.^[5] 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.^[6]

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio.^[7][8] This is because the corresponding interior angle is equal to π/5 radians (or 36 degrees).^[b] 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.

As a regular number, 54 is a divisor of many powers of 60.^[c] 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 a by b can be done by multiplying a by b's reciprocal when b is a regular number.^[9][10]

For instance, division by 54 can be achieved in the Assyro-Babylonian system by multiplying by 4000 because 60³ ÷ 54 = 60³ × (1/54) = 4000. In base 60, 4000 can be written as 1:6:40.^[d] Because the Assyro-Babylonian system does not have a symbol separating the fractional and integer parts of a number^[11] and does not have the concept of 0 as a number,^[12] it does not specify the power of the starting digit. Accordingly, 1/54 can also be written as 1:6:40.^[e][11] 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.^[f]

The second Ellingham–Horton graph was published by Mark N. Ellingham and Joseph D. Horton in 1983; it is of order 54.^[13] These graphs provided further counterexamples to the conjecture of W. T. Tutte that every cubic 3-connected bipartite graph is Hamiltonian.^[14] Horton disproved the conjecture some years earlier with the Horton graph, but that was larger at 92 vertices.^[15] The smallest known counter-example is now 50 vertices.^[16]

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.^[17] Eventually, one character's unsuccessful attempt to divine the Ultimate Question elicited "What do you get if you multiply six by nine?"^[18] 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 6₁₃ × 9₁₃ = 42₁₃.^[19] Adams said this was a coincidence.^[20]

Exponentiation	1	2	3
54x	54	2916	157464
x54	1	18014398509481984	58149737003040059690390169
${\displaystyle {\sqrt[{x}]{54}}}$	54	7.34846...[g]	3.77976...