36 (thirty-six) is the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

following 35 and preceding 37.

In mathematics

36 is both the

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

six 6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. In mathematics A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. A hexagon a ...

, and the eighth

triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

or the sum of the first eight non-zero positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s, which makes 36 the first non-trivial

square triangular number In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a square number, in other words, the sum of all integers from 1 to n has a square root that is an integer. There are inf ...

. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

is also a triangular number. 36 is also the eighth

refactorable number A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n with \tau(n)=\sigma_0(n)=\prod_^(e_i+1) for n=\prod_^np_i^. The first few refact ...

, as it has exactly nine positive divisors, and 9 is one of them; in fact, it is the smallest positive integer with at least nine

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

s, which leads 36 to be the 7th

highly composite number A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(' ...

. It is the sum of the fourth pair of twin-primes ( 17 + 19), and the 18th

Harshad number In mathematics, a harshad number (or Niven number) in a given radix, number base is an integer that is divisible by the digit sum, sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers ...

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

, as it is divisible by the sum of its digits (9). It is the smallest number

n

with exactly eight solutions ( 37, 57, 63, 74, 76, 108,

114 114 may refer to: *114 (number) *AD 114 *114 BC *114 (1st London) Army Engineer Regiment, Royal Engineers, an English military unit *114 (Antrim Artillery) Field Squadron, Royal Engineers, a Northern Irish military unit *114 (MBTA bus) *114 (New Je ...

, 126) to the

Euler totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In oth ...

\phi(x)=n

. Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is also the eighth

semiperfect number In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. ...

. This number is the sum of the cubes of the first three positive

s and also the product of the squares of the first three positive integers. 36 is the number of degrees in the

interior angle In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point withi ...

of each tip of a regular

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...

. The thirty-six officers problem is a

mathematical puzzle Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that sati ...

with no solution. The number of possible outcomes (not summed) in the roll of two distinct

dice A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...

. 36 is the largest numeric base that some computer systems support because it exhausts the numerals, 0–9, and the letters, A-Z. See

Base 36 A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base is a semiprime, though it is unique as the product o ...

. The

truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangle (geometry), triangular), 36 edges, and 24 vertices. If the truncated cube has unit edge length, its dual triak ...

and the

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

are

Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

s with 36 edges. The number of

domino tiling In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by domino (mathematics), dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a matching (graph theory), ...

s of a 4×4

checkerboard A checkerboard (American English) or chequerboard (British English) is a game board of check (pattern), checkered pattern on which checkers (also known as English draughts) is played. Most commonly, it consists of 64 squares (8×8) of alternating ...

is 36. Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an

Erdős–Woods number In number theory, a positive integer is said to be an Erdős–Woods number if it has the following property: there exists a positive integer such that in the sequence of consecutive integers, each of the elements has a non-trivial common fac ...

. The sum of the integers from 1 to 36 is

666 666 may refer to: * 666 (number) * 666 BC, a year * AD 666, a year * The number of the beast, a reference in the Book of Revelation in the New Testament Places * 666 Desdemona, a minor planet in the asteroid belt * List of highways numbered 6 ...

(see

number of the beast The number of the beast (, ) is associated with the The Beast (Revelation), Beast of Revelation in chapter 13, verse 18 of the Book of Revelation. In most manuscripts of the New Testament and in English translations of the Bible, the number of ...

). 36 is also a Tridecagonal number. The

cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...

of an angle of 36 degrees is half the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

. This is equivalent to cos(/5) and sin( 54). The point of the

golden triangle Golden Triangle may refer to: Places Asia * Golden Triangle (Southeast Asia), named for its opium production * Golden Triangle (Yangtze), China, named for its rapid economic development * Golden Triangle (India), comprising the popular tourist sp ...

is 36 degrees.

In religion

* Jewish tradition holds that the number 36 has had special significance since the beginning of time: According to the

Midrash ''Midrash'' (;"midrash"
. ''Random House Webster's Unabridged Dictionary''. ; or ''midrashot' ...

, the light created by God on the first day of creation shone for exactly 36 hours; it was replaced by the light of the

Sun The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...

that was created on the Fourth Day. The

Torah The Torah ( , "Instruction", "Teaching" or "Law") is the compilation of the first five books of the Hebrew Bible, namely the books of Genesis, Exodus, Leviticus, Numbers and Deuteronomy. The Torah is also known as the Pentateuch () ...

commands 36 times to love, respect and protect the stranger. Furthermore, in every generation there are 36 righteous people (the ''"Lamed Vav Tzadikim"'') in whose merit the world continues to exist. In the modern celebration of

Hanukkah Hanukkah (, ; ''Ḥănukkā'' ) is a Jewish holidays, Jewish festival commemorating the recovery of Jerusalem and subsequent rededication of the Second Temple at the beginning of the Maccabean Revolt against the Seleucid Empire in the 2nd ce ...

, 36 candles are kindled in the menorah over the eight days of that holiday (not including the ''shamash'' candle).

In culture

* In French-speaking countries, 36 is often used as a placeholder number.

References

{{Integers, zero Integers