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.

In mathematics

Number theory

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]

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.^[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.

Regular number used in Assyro-Babylonian mathematics

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]

Graph theory

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 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.^[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]

List of basic calculations

Explanatory footnotes

1. ^ 54 can be expressed as: 9 + 18 + 27 = 54.
2. ^ There are various ways to prove this, but the algebraic method will eventually show that \sin{54} = \cos{\frac{\pi}{5}} = \frac{2\sqrt{5}}{4} = \frac{\phi}{2}.
3. ^ 60³ and its multiples are divisible by 54.
4. ^ 1:6:40 = 1×60² + 6×60¹ + 40×60⁰ = 4000. This is the number written in Babylonian numerals: 𒐕𒐚𒐏.
5. ^ 1:6:40 = 1×60^-1 + 6×60^-2 + 40×60^-3 = 1/54. This is the number written in Babylonian numerals: 𒐕𒐚𒐏.
6. ^ For example, 6534 ÷ 54 = 121. The Assyro-Babylonian method is to calculate 6534 × 4000 = 26136000. This result can be written in Babylonian numerals as 𒐖𒐕 (2:1), meaning 2×60⁴ + 1×60³. To complete the division by 54, one must divide by 60³. Shifting the numeral three base-60 digits to the right divides the number by 60³, so 𒐖𒐕 (2:1) is already the answer: 2×60¹ + 1×60⁰ = 121.
7. ^ Because 54 is a multiple of 2 but not a square number, its square root is irrational.^[21]

References

1. ^
2. ^
3. ^ Zachariou, Andreas & Zachariou, Eleni (1972). "Perfect, Semiperfect and Ore Numbers". Bull. Soc. Math. Grèce. 13: 12–22. Nouvelle Série. MR 360455. Zbl 0266.10012.
4. ^
5. ^
6. ^
7. ^ Khan, Sameen Ahmed (2020-10-11). "Trigonometric Ratios Using Geometric Methods". Advances in Mathematics: Scientific Journal. 9 (10): 8698. doi:10.37418/amsj.9.10.94. ISSN 1857-8365
8. ^
9. ^ Aaboe, Asger (1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)", Journal of Cuneiform Studies. 19 (3): 79–86, The American Schools of Oriental Research, doi:10.2307/1359089. JSTOR 1359089. MR 0191779. S2CID 164195082.
10. ^ Sachs, A. J. (1947), "Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers", Journal of Cuneiform Studies. 1 (3): 219–240, The American Schools of Oriental Research, doi:10.2307/1359434. JSTOR 1359434. MR 0022180. S2CID 163783242
11. ^ Cajori, Florian (1922). "Sexagesimal Fractions Among the Babylonians". The American Mathematical Monthly. 29 (1): 8–10. doi:10.2307/2972914. ISSN 0002-9890. JSTOR 2972914
12. ^ Boyer, Carl B. (1944). "Zero: The Symbol, the Concept, the Number". National Mathematics Magazine. 18 (8): 323–330. doi:10.2307/3030083. ISSN 1539-5588. JSTOR 3030083
13. ^ Ellingham, M. N. & Horton, J. D. (1983), "Non-Hamiltonian 3-connected cubic bipartite graphs", Journal of Combinatorial Theory, Series B. 34 (3): 350–353, doi:10.1016/0095-8956(83)90046-1.
14. ^ Tutte, W. T. (1971), "On the 2-factors of bicubic graphs", Discrete Mathematics. 1 (2): 203–208, doi:10.1016/0012-365X(71)90027-6.
15. ^ "Horton Graphs". Retrieved 2025-04-11.
16. ^ Georges, J. P. (1989), "Non-hamiltonian bicubic graphs", Journal of Combinatorial Theory, Series B. 46 (1): 121–124, doi:10.1016/0095-8956(89)90012-9.
17. ^ Adams, Douglas (1979). The Hitchhiker's Guide to the Galaxy. p. 179-80.
18. ^ Adams, Douglas (1980). The Restaurant at the End of the Universe. p. 181-84.
19. ^ Adams, Douglas (1985). The Original Hitchhiker Radio Scripts. Perkins, Geoffrey (ed.). London: Pan Books. p. 128. ISBN 0-330-29288-9.
20. ^ Diaz, Jesus. "Today Is 101010: The Ultimate Answer to the Ultimate Question". io9. Archived 26 May 2017 at the Wayback Machine. Retrieved 8 May 2017.
21. ^ Jackson, Terence (2011-07-01). "95.42 Irrational square roots of natural numbers — a geometrical approach". The Mathematical Gazette. 95 (533): 327–330. doi:10.1017/S0025557200003193. ISSN 0025-5572. S2CID 123995083