54 (number)

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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 an abundant number^[1] because the sum of its proper divisors (66),^[2] which excludes 54 as a divisor, 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]

54 can be constructed mathematically in a variety of ways. It is the smallest number that can be expressed as the sum of three positive squares in more than two different ways.^[b][7] It can also be expressed as twice the third power of 3, so it is a Leyland number.^[c][8] Additionally, fifty-four objects can be positioned to construct the vertices of a polygon with nineteen sides, so 54 is an enneadecagonal number and the first 19-gonal number after 19 itself.^[d][9]

If the complementary angle of a triangle's corner is 54 degrees, the sine of that angle is half the golden ratio.^[10][11] This is because the corresponding interior angle is equal to π/5 radians (or 36 degrees).^[e] 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.^[12]

However, 54 can be expressed as the area of a triangle with three rational side lengths.^[f] Therefore, it is a congruent number.^[13] 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.

As a regular number, 54 is a divisor of many powers of 60,^[g] which is an important property in Assyro-Babylonian mathematics because it uses a sexagesimal (base 60) number system. In base 60, the reciprocal of a regular number has a finite representation, and that simplified multiplication and division.^[14] Babylonian calculators kept tables of these reciprocals in order to make their work more efficient.^[15] For instance, 60³ ÷ 54 = 4000. In base 60, division by 54 can be accomplished by multiplying by 4000 and shifting three places. In base 60, 4000 can be written as 1:6:40.^[h] Thus, 1/54 can be written as 1×60^-1 + 6×60^-2 + 40×60^-3, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/60³, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three base-60 places.

Exponentiation	1	2	3	4	5
54x	54	2916	157464	8503056	459165024
x54	1	18014398509481984	58149737003040059690390169	324518553658426726783156020576256	55511151231257827021181583404541015625
${\displaystyle {\sqrt[{x}]{54}}}$	54	7.34846...[i]	3.77976...	2.71080...	2.22064...

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 42₁₃.^[19] Adams said this was a coincidence.^[20]