81

81, or eighty-one, is the natural number that follows 80 and precedes 82. It is an odd composite number with a simple prime factorization of 3^4.

In mathematics, 81 is notable for being both a perfect square and a perfect fourth power: 9^2 and 3^4. Its positive divisors are 1, 3, 9, 27, and 81. The sum of all divisors is 121, so the sum of proper divisors is 40, making 81 a deficient number. Its representations in other bases include 1010001 in binary, 121 in octal, 51 in hexadecimal, 10000 in base 3, and 100 in base 9. In Roman numerals it is written as LXXXI. The digit sum in base 10 is 8 + 1 = 9, so 81 is divisible by 9 and has a digital root of 9. As a 3-smooth number, its prime factors are exclusively powers of 3.

In science, 81 is the atomic number of thallium, a post-transition metal with the symbol Tl. Thallium was discovered in 1861 by William Crookes and has been used in various applications, including electronics and historically in certain luminescent materials, though it is highly toxic.

Historically, the year 81 CE falls in the early Roman Empire, during the Flavian dynasty. In that year, Domitian began his reign as emperor after the death of Titus, marking a transition in Roman leadership and policy.

Beyond mathematics and science, 81 appears in cultural and numeral systems as a benchmark for base conversions and as a reference point in arithmetic and number theory.

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