# 90 (number)

90 (ninety) is the natural number preceded by 89 and followed by 91.

 ← 89 90 91 →
Cardinalninety
Ordinal90th
(ninetieth)
Factorization2 × 32 × 5
Divisors1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Greek numeralϞ´
Roman numeralXC
Binary10110102
Ternary101003
Senary2306
Octal1328
Duodecimal7612

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

90 is a pronic number, as it is the product of 9 and 10.[1] It is nontotient,[2] and divisible by the sum of its base 10 digits, which makes it a Harshad number.[3]

• 90 is a Stirling number of the second kind ${\displaystyle S(n,k)}$  from a ${\displaystyle n}$  of ${\displaystyle 6}$  and a ${\displaystyle k}$  of ${\displaystyle 3}$ , as it is the number of ways of dividing a set of six objects into three non empty subsets.[6]
• 90 is the fifth sum of non-triangular numbers, respectively between the fifth and sixth triangular numbers, 15 and 21 (equivalently 16 + 17 ... + 20).[9]

In normal space, the interior angles of a rectangle measure 90 degrees each. Also, 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.[10] Thus, an angle measuring 90 degrees is called a right angle.[11]

The truncated dodecahedron and truncated icosahedron both have 90 edges. A further four uniform star polyhedra (U37, U55, U58, U66) and four uniform compound polyhedra (UC32, UC34, UC36, UC55) contain 90 edges or vertices.

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 ${\displaystyle 1:{\sqrt {2}}}$  ratio, and another 30 slim rhombi with diagonals in ${\displaystyle 1:\varphi ^{2}}$  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: ${\displaystyle 109.471}$ °. The rhombic enneacontahedron is the zonohedrification of the regular dodecahedron, and 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.

The 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 Möbius–Kantor polygons.[12] The root vectors of simple Lie group E8 are represented by the vertex arrangement of the 421 polytope, which is shared by the Witting polytope in four-dimensional complex space.

## In science

Ninety is:

• 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.
• the latitude in degrees of the North and the South geographical poles.

## In other fields

Interstate 90 is a freeway that runs from Washington to Massachusetts.

## References

1. ^ "Sloane's A002378 : Oblong (or promic, pronic, or heteromecic) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
2. ^ "Sloane's A005277 : Nontotients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
3. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
4. ^ "Sloane's A002827 : Unitary perfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
5. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
6. ^ "Sloane's A008277 :Triangle of Stirling numbers of the second kind". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-12-24.
7. ^ "Sloane's A001608 : Perrin sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
8. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-11-01.
9. ^
10. ^ http://www.sparknotes.com/testprep/books/newsat/chapter20section4.rhtml