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Look up
ninety in Wiktionary, the free dictionary.
Natural number
Cardinal ninety Ordinal 90th (ninetieth) Factorization 2 × 32 × 5 Divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 Greek numeral Ϟ´ Roman numeral XC Binary 10110102 Ternary 101003 Senary 2306 Octal 1328 Duodecimal 76 12 Hexadecimal 5A16 Armenian Ղ Hebrew צ / ץ Babylonian numeral 𒐕𒌍 Egyptian hieroglyph 𓎎
90 (ninety ) is the natural number following 89 and preceding 91 .
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 [ edit ]
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 [ edit ]
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 number 78 [9] is the only number to have an aliquot sum equal to 90, aside from the square of the twenty-fourth prime, 89 2 (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 non-zero integers, from two through thirteen
{
2
,
3
,
.
.
.
,
13
}
{\displaystyle \{2,3,...,13\}}
.
90 can be expressed as the sum of distinct non-zero squares in six ways, more than any smaller number (see image):[13]
(
9
2
+
3
2
)
,
(
8
2
+
5
2
+
1
2
)
,
(
7
2
+
5
2
+
4
2
)
,
(
8
2
+
4
2
+
3
2
+
1
2
)
,
(
7
2
+
6
2
+
2
2
+
1
2
)
,
(
6
2
+
5
2
+
4
2
+
3
2
+
2
2
)
.
{\displaystyle (9^{2}+3^{2}),(8^{2}+5^{2}+1^{2}),(7^{2}+5^{2}+4^{2}),(8^{2}+4^{2}+3^{2}+1^{2}),(7^{2}+6^{2}+2^{2}+1^{2}),(6^{2}+5^{2}+4^{2}+3^{2}+2^{2}).}
90 as the sum of distinct nonzero squares
The square of eleven
11
2
=
121
{\displaystyle 11^{2}=121}
is the ninetieth indexed composite number ,[14] where the sum of integers
{
2
,
3
,
.
.
.
,
11
}
{\displaystyle \{2,3,...,11\}}
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 , "
90
{\displaystyle 90}
" repeats periodically (when leading zeroes are moved to the end).[15]
The eighteenth Stirling number of the second kind
S
(
n
,
k
)
{\displaystyle S(n,k)}
is 90, from a
n
{\displaystyle n}
of
6
{\displaystyle 6}
and a
k
{\displaystyle k}
of
3
{\displaystyle 3}
, as the number of ways of dividing a set of six objects into three non-empty subsets .[16] 90 is also the sixteenth Perrin number from a sum of 39 and 51 , whose difference is 12 .[17]
Prime sextuplets [ edit ]
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]
Unitary perfect number [ edit ]
90 is the third unitary perfect number (after 6 and 60 ), since it is the sum of its unitary divisors excluding itself,[20] and because it is equal to the sum of a subset of its divisors, it is also the twenty-first semiperfect number .[21]
Right angle [ edit ]
A right angle measures ninety degrees .
An angle measuring 90 degrees is called a right angle .[22] In normal space , the interior angles of a rectangle measure 90 degrees each, while in a right triangle , the angle opposing the hypotenuse measures 90 degrees, with the other two angles adding up to 90 for a total of 180 degrees.
Icosahedral symmetry [ edit ]
The Witting polytope , with ninety van Oss polytopes
The rhombic enneacontahedron is a zonohedron with a total of 90 rhombic faces : 60 broad rhombi akin to those in the rhombic dodecahedron with diagonals in
1
:
2
{\displaystyle 1:{\sqrt {2}}}
ratio, and another 30 slim rhombi with diagonals in
1
:
φ
2
{\displaystyle 1:\varphi ^{2}}
golden ratio . The obtuse angle of the broad rhombic faces is also the dihedral angle of a regular icosahedron , with the obtuse angle in the faces of golden rhombi equal to the dihedral angle of a regular octahedron and the tetrahedral vertex-center-vertex angle, which is also the angle between Plateau borders :
109.471
{\displaystyle 109.471}
°. It is the dual polyhedron to the rectified truncated icosahedron , a near-miss Johnson solid . On the other hand, the final stellation of the icosahedron has 90 edges. It also has 92 vertices like the rhombic enneacontahedron, when interpreted as a simple polyhedron . Meanwhile, the truncated dodecahedron and truncated icosahedron both have 90 edges . A further four uniform star polyhedra (U 37 , U 55 , U 58 , U 66 ) and four uniform compound polyhedra (UC 32 , UC 34 , UC 36 , UC 55 ) contain 90 edges or vertices .
Witting polytope [ edit ]
The self-dual Witting polytope contains ninety van Oss polytopes such that sections by the common plane of two non-orthogonal hyperplanes of symmetry passing through the center yield complex
3
{
4
}
3
{\displaystyle _{3}\{4\}_{3}}
Möbius–Kantor polygons .[23] The root vectors of simple Lie group E 8 are represented by the vertex arrangement of the
4
21
{\displaystyle 4_{21}}
polytope , which shares 240 vertices with the Witting polytope in four-dimensional complex space . By Coxeter , the incidence matrix configuration of the Witting polytope can be represented as:
[
40
9
12
4
90
4
12
9
40
]
{\displaystyle \left[{\begin{smallmatrix}40&9&12\\4&90&4\\12&9&40\end{smallmatrix}}\right]}
or
[
40
12
12
2
240
2
12
12
40
]
.
{\displaystyle \left[{\begin{smallmatrix}40&12&12\\2&240&2\\12&12&40\end{smallmatrix}}\right].}
This Witting configuration when reflected under the finite space
PG
(
3
,
2
2
)
{\displaystyle \operatorname {PG} {(3,2^{2})}}
splits into
85
=
45
+
40
{\displaystyle 85=45+40}
points and planes, alongside
27
+
90
+
240
=
357
{\displaystyle 27+90+240=357}
lines.[23]
Whereas the rhombic enneacontahedron is the zonohedrification of the regular dodecahedron,[24] a honeycomb of Witting polytopes holds vertices isomorphic to the
E
8
{\displaystyle \mathrm {E} _{8}}
lattice , whose symmetries can be traced back to the regular icosahedron via the icosian ring .[25]
Cutting an annulus [ edit ]
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 [ edit ]
In science [ edit ]
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 .
In sports [ edit ]
References [ edit ]
^ 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 ).
^ "Sloane's A005349 : Niven (or Harshad) numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-29 .
^ Sloane, N. J. A. (ed.). "Sequence A000203 (...the sum of the divisors of n.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-30 .
^ Sloane, N. J. A. (ed.). "Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-23 .
^ Sloane, N. J. A. (ed.). "Sequence A002093 (Highly abundant numbers)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-23 .
^ Sloane, N. J. A. (ed.). "Sequence A071395 (Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-23 .
^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n ): count numbers less than or equal to n and prime to n.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-01-16 .
^ "Sloane's A005277 : Nontotients" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-29 .
^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-11-01 .
^ Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-30 .
^ Sloane, N. J. A. (ed.). "Sequence A016754 (Centered octagonal numbers.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-02 .
^ Sloane, N. J. A. (ed.). "Sequence A006002 (...also: Sum of the nontriangular numbers between successive triangular numbers.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A033461 (Number of partitions of n into distinct squares.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ a b Sloane, N. J. A. (ed.). "Sequence A02808 (The composite numbers.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A060283 (Periodic part of decimal expansion of reciprocal of n-th prime (leading 0's moved to end).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^ "Sloane's A008277 :Triangle of Stirling numbers of the second kind" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2021-12-24 .
^ "Sloane's A001608 : Perrin sequence" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-29 .
^ Sloane, N. J. A. (ed.). "Sequence A022008 (Initial member of prime sextuples (p, p+4, p+6, p+10, p+12, p+16).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-11 .
^ Sloane, N. J. A. (ed.). "Sequence A200503 (Record (maximal) gaps between prime sextuplets (p, p+4, p+6, p+10, p+12, p+16).)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-06-23 .
^ "Sloane's A002827 : Unitary perfect numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-29 .
^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-29 .
^ Friedman, Erich (n.d.). "What's Special About This Number?" . www.stetson.edu . Archived from the original on February 23, 2018. Retrieved February 27, 2023 .
^ a b Coxeter, Harold Scott MacDonald (1974). Regular Complex Polytopes (1st ed.). Cambridge University Press. p. 133. ISBN 978-0-52-1201254 .
^ Hart, George W. "Zonohedrification" . Virtual Polyhedra (The Encyclopedia of Polyhedra) . Retrieved 2023-06-23 .
^ Baez, John C. (2018). "From the Icosahedron to E8 ". London Math. Soc. Newsletter . 476 . London, UK: London Mathematical Society : 18–23. arXiv :1712.06436 . Bibcode :2017arXiv171206436B . MR 3792329 . S2CID 119151549 . Zbl 1476.51020 .
^ Sloane, N. J. A. (ed.). "Sequence A000096 (a(n ) equal to n*(n+3)/2.)" . The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
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e
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1,000,000
10,000,000
100,000,000
1,000,000,000
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