In the English language, the numbers 90 and 19 are often confused, as they sound very similar. When carefully enunciated, they differ in which syllable is stressed: 19 /naɪnˈtiːn/ vs 90 /ˈnaɪnti/. However, in dates such as 1999, and when contrasting numbers in the teens and when counting, such as 17, 18, 19, the stress shifts to the first syllable: 19 /ˈnaɪntiːn/.
In mathematics
Ninety is a
pronic number as it is the
product of
9 and
10,[1] and along with
12 and
56, one of only a few pronic numbers whose digits in
decimal are also successive. 90 is divisible by the sum of its
base-ten digits, which makes it the thirty-second
Harshad number.[2]
Properties of the number
90 is the only number to have an aliquot sum of
144 = 122.
Only three numbers have a set of
divisors that generate a sum equal to 90, they are
40,
58 and
89.[3]
90 is the tenth and largest number to hold an
Euler totient value of
24;[7] no number has a totient that is 90, which makes it the eleventh
nontotient (with
50 the fifth).[8]
The twelfth
triangular number78[9] is the only number to have an
aliquot sum equal to 90, aside from the
square of the twenty-fourth prime,
892 (which is
centered octagonal).[10][11] 90 is equal to the fifth sum of non-triangular numbers, respectively between the fifth and sixth triangular numbers,
15 and
21 (equivalently
16 +
17 ... +
20).[12] It is also twice
45, which is the ninth triangular number, and the second-smallest sum of twelve integers, from two through thirteen .
90 can be expressed as the sum of distinct non-zero
squares in six ways, more than any smaller number (see image):[13]
The square of eleven is the ninetieth indexed
composite number,[14] where the sum of integers is
65, which in-turn represents the composite index of 90.[14] In the
fractional part of the
decimal expansion of the reciprocal of
11 in
base-10, "" repeats periodically (when leading zeroes are moved to the end).[15]
The members of the first
prime sextuplet (
7,
11,
13, 17,
19,
23) generate a
sum equal to 90, and the difference between respective members of the first and second prime sextuplets is also 90, where the second prime sextuplet is (
97,
101,
103,
107,
109,
113).[18][19] The last member of the second prime sextuplet, 113, is the 30th
prime number. Since prime sextuplets are formed from prime members of lower order
prime k-tuples, 90 is also a record maximal gap between various smaller pairs of prime k-tuples (which include
quintuplets,
quadruplets, and
triplets).[a]
This Witting configuration when reflected under the
finite space splits into points and planes, alongside lines.[23]
Whereas the rhombic enneacontahedron is the
zonohedrification of the regular dodecahedron,[24] a
honeycomb of Witting polytopes holds vertices
isomorphic to the lattice, whose symmetries can be traced back to the regular icosahedron via the
icosian ring.[25]
Cutting an annulus
The maximal number of pieces that can be obtained by cutting an
annulus with twelve cuts is 90 (and equivalently, the number of 12-dimensional
polyominoes that are prime).[26]
Other fields
In science
The latitude in degrees of the
North and the
South geographical poles.
The atomic number of
thorium, an
actinide. As an atomic weight, 90 identifies an
isotope of
strontium, a by-product of nuclear reactions including fallout. It contaminates
milk.
^90 is the record gap between the first pair of
prime quintuplets of the form (p, p+2, p+6, p+8, p+12) (
A201073), while 90 is a record between the second and third prime quintuplets that have the form (p, p+4, p+6, p+10, p+12) (
A201062). Regarding
prime quadruplets, 90 is the gap record between the second and third set of quadruplets (
A113404).
Prime triplets of the form (p, p+4, p+6) have a third record maximal gap of 90 between the second and ninth triplets (
A201596), and while there is no record gap of 90 for prime triplets of the form (p, p+2, p+6), the first and third record gaps are of
6 and 60 (
A201598), which are also unitary perfect numbers like 90 (
A002827).