54 (number)

In mathematics

Number theory

54 is an abundant number 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 of all its positive divisors (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 sine of that angle is half the golden ratio. This is because the corresponding interior angle is equal to /5 radians (or 36 degrees). = \frac{2\sqrt{5}}{4} = \frac{\phi}{2}.}} 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 54 as the sum of two other square rational numbers. Therefore, 54 is a nonhypotenuse number.

However, 54 can be expressed as the area 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 (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.

For instance, division by 54 can be achieved in the Assyro-Babylonian system by multiplying by 4000 because 60 ÷ 54 = 60 × (1/54) = 4000. In base 60, 4000 can be written as 1:6:40. Because the Assyro-Babylonian system does not have a symbol separating the fractional 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 3-connected bipartite graph is 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 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 6 × 9 = 42. Adams said this was a coincidence.

List of basic calculations

Explanatory footnotes

References

References

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