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111 (one hundred ndeleven) is the natural number following 110 and preceding 112.


In mathematics

111 is the fourth non-trivial nonagonal number, and the seventh perfect totient number. 111 is furthermore the ninth number such that its Euler totient \varphi(n) of 72 is equal to the totient value of its sum-of-divisors: :\varphi(111) = \varphi(\sigma(111)). Two other of its multiples ( 333 and 555) also have the same property (with totients of 216 and 288, respectively).


Magic squares

The smallest
magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
using only 1 and
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s has a magic constant of 111: Also, a six-by-six
magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...
using the numbers 1 to 36 also has a magic constant of 111: (The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111). On the other hand, 111 lies between 110 and 112, which are the two smallest edge-lengths of
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 ...
s that are tiled in the interior by smaller squares of distinct edge-lengths (see,
squaring the square Squaring the square is the problem of tessellation, tiling an integral square using only other integral squares. (An integral square is a square (geometry), square whose sides have integer length.) The name was coined in a humorous analogy with sq ...
).


Properties in certain radices

111 is R_ or the second repunit in
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 ...
, a
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
like 11, 111, or 1111 that consists of repeated units, or ones. 111 equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are repdigits of the form 3n \times 37. As a repunit, it also follows that 111 is a
palindromic number A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ...
. All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in
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 ...
, but is prime in base two, where 1112 = 710. It is also prime in many other bases up to 128 (3, 5, 6, ..., 119) . In base 10, it is furthermore a strobogrammatic number, as well as a Harshad number. In base 18, the number 111 is 73 (= 34310) which is the only base where 111 is a perfect power.


Nelson

In cricket, the number 111 is sometimes called "a Nelson" after Admiral Nelson, who allegedly only had "One Eye, One Arm, One Leg" near the end of his life. This is in fact inaccurate—Nelson never lost a leg. Alternate meanings include "One Eye, One Arm, One Ambition" and "One Eye, One Arm, One Arsehole". Particularly in
cricket Cricket is a Bat-and-ball games, bat-and-ball game played between two Sports team, teams of eleven players on a cricket field, field, at the centre of which is a cricket pitch, pitch with a wicket at each end, each comprising two Bail (cr ...
, multiples of 111 are called a double Nelson (222), triple Nelson (333), quadruple Nelson (444; also known as a salamander) and so on. A score of 111 is considered by some to be unlucky. To combat the supposed bad luck, some watching lift their feet off the ground. Since an umpire cannot sit down and raise his feet, the international umpire David Shepherd had a whole retinue of peculiar mannerisms if the score was ever a Nelson multiple. He would hop, shuffle, or jiggle, particularly if the number of wickets also matched—111/1, 222/2 etc.


In other fields

111 is also: *The
emergency telephone number An emergency telephone number is a number that allows a caller to contact local emergency services for assistance. The emergency number differs from country to country; it is typically a three-digit number so that it can be easily remembered and ...
in
New Zealand New Zealand () is an island country in the southwestern Pacific Ocean. It consists of two main landmasses—the North Island () and the South Island ()—and List of islands of New Zealand, over 600 smaller islands. It is the List of isla ...
; see 111 (emergency telephone number). * NHS 111, a medical helpline in England and Scotland.


Notes


References


Further reading

Wells, D. '' The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 134


External links

{{DEFAULTSORT:111 (Number) Horatio Nelson Integers