# Beyond Numbers

Numbers can be spooky, and when empowered with math and coincidence can be hypnotic and mind bogglingly influential and symbolic, or just weird and wonderful. Play on.

# 11 (number)

11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name.

## Name

Eleven derives from the Old Englishęndleofon which is first attested in Bede‘s late 9th-century Ecclesiastical History of the English People.[2][3] It has cognates in every Germanic language (for example, German elf), whose Proto-Germanicancestor has been reconstructed as *ainalifa-,[4] from the prefix *aina- (adjectival “one“) and suffix *-lifa- of uncertain meaning.[3] It is sometimes compared with the Lithuanianvienúolika, although -lika is used as the suffix for all numbers from 11 to 19 (analogous to “-teen”).[3]

The Old English form has closer cognates in Old Frisian, Saxon, and Norse, whose ancestor has been reconstructed as *ainlifun. This has formerly been considered derived from Proto-Germanic *tehun (“ten“);[3][5] it is now sometimes connected with *leikʷ- or *leip- (“left; remaining”), with the implicit meaning that “one is left” after having already counted to ten.[3]

## In languages

While, as mentioned above, 11 has its own name in Germanic languages such as English, German, or Swedish, and some Latin based languages such as Spanish, Portuguese, and French, it is the first compound number in many other languages: Italian ùndici , Chinese 十一 shí yī, Korean 열한 yeol han.

## In mathematics

11 is a prime number. It is the smallest two-digit prime number in the decimalbase.

The next prime is 13, with which it comprises a twin prime.

If a number is divisible by 11, reversing its digits will result in another multiple of 11. As long as no two adjacent digits of a number added together exceed 9, then multiplying the number by 11, reversing the digits of the product, and dividing that new number by 11, will yield a number that is the reverse of the original number. (For example: 142,312 × 11 = 1,565,432 → 2,345,651 ÷ 11 = 213,241.)

Multiples of 11 by one-digit numbers all have matching double digits: 00 (=0), 11, 22, 33, 44, etc.

An 11-sided polygon is called a hendecagon or undecagon.

There are 11 regular and semiregular convex uniform tilings in the second dimension, and 11 planigons that correspond to these 11 regular and semiregular tilings.

In base 10, there is a simple test to determine if an integer is divisible by 11: take every digit of the number located in odd position and add them up, then take the remaining digits and add them up. If the difference between the two sums is a multiple of 11, including 0, then the number is divisible by 11.[6] For instance, if the number is 65,637 then (6 + 6 + 7) – (5 + 3) = 19 – 8 = 11, so 65,637 is divisible by 11. This technique also works with groups of digits rather than individual digits, so long as the number of digits in each group is odd, although not all groups have to have the same number of digits. For instance, if one uses three digits in each group, one gets from 65,637 the calculation (065) – 637 = -572, which is divisible by 11.

Another test for divisibility is to separate a number into groups of two consecutive digits (adding a leading zero if there is an odd number of digits), and then add up the numbers so formed; if the result is divisible by 11, the number is divisible by 11. For instance, if the number is 65,637, 06 + 56 + 37 = 99, which is divisible by 11, so 65,637 is divisible by eleven. This also works by adding a trailing zero instead of a leading one: 65 + 63 + 70 = 198, which is divisible by 11. This also works with larger groups of digits, providing that each group has an even number of digits (not all groups have to have the same number of digits).

An easy way of multiplying numbers by 11 in base 10 is: If the number has:

• 1 digit – Replicate the digit (so 2 × 11 becomes 22).
• 2 digits – Add the 2 digits together and place the result in the middle (so 47 × 11 becomes 4 (11) 7 or 4 (10+1) 7 or (4+1) 1 7 or 517).
• 3 digits – Keep the first digit in its place for the result’s first digit, add the first and second digits together to form the result’s second digit, add the second and third digits together to form the result’s third digit, and keep the third digit as the result’s fourth digit. For any resulting numbers greater than 9, carry the 1 to the left. Example 1: 123 × 11 becomes 1 (1+2) (2+3) 3 or 1353. Example 2: 481 × 11 becomes 4 (4+8) (8+1) 1 or 4 (10+2) 9 1 or (4+1) 2 9 1 or 5291.
• 4 or more digits – Follow the same pattern as for 3 digits.

In base 13 and higher bases (such as hexadecimal), 11 is represented as B, where ten is A. In duodecimal, however, 11 is sometimes represented as E and ten as T or X.

There are 11 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the 3-variable Helmholtz equation can be solved using the separation of variables technique.

11 of the thirty-five hexominoes can be folded to form cubes. 11 of the sixty-six octiamonds can be folded to form octahedra.

11 is the fourth Sophie Germain prime,[7] the third safe prime,[8] the fourth Lucas prime,[9] the first repunit prime,[10] the second good prime,[11] and the second unique prime.[12] Although it is necessary for n to be prime for 2n − 1 to be a Mersenne prime, the converse is not true: 211 − 1 = 2047 which is 23 × 89.

11 raised to the nth power is the nth row of Pascal’s Triangle. (This works for any base, but the number eleven must be changed to the number represented as 11 in that base; for example, in duodecimal this must be done using thirteen.)

11 is a Heegner number, meaning that the ring of integers of the fieldQ(−11) has the property of unique factorization.

One consequence of this is that there exists at most one point on the elliptic curve x3 = y2 + 11 that has positive-integer coordinates. In this case, this unique point is (15, 58).

Cardinal eleven Ordinal 11th (eleventh) FactorizationprimePrime 5th Divisors 1, 11 Greek numeral ΙΑ´ Roman numeral XI Greekprefixhendeca-/hendeka-Latinprefixundeca-Binary 10112Ternary 1023Quaternary 234Quinary 215Senary 156Octal 138Duodecimal B12Hexadecimal B16Vigesimal B20Base 36 B36Hebrew numeral יא

## In other fields

Sector 11[18] in the North American Industry Classification System is the code for Agriculture, Forestry, Fishing and Hunting industries.

Being only one hour before 12:00, the eleventh hour means the last possible moment to take care of something, and often implies a situation of urgent danger or emergency (see Doomsday clock).

In Astrology,Aquarius is the 11th astrological sign of the Zodiac.[citation needed]

In Basque,hamaika (“eleven”) has the double meaning of “infinite“, probably from amaigabe, “endless”, as in Hamaika aldiz etortzeko esan dizut! (“I told you infinite/eleven times to come!”).

American Airlines flight 11, a BostonLos Angeles flight which crashed into the North Tower of the World Trade Center in New York City, New York after being hijacked by terrorists on September 11, 2001.

The number 11 bus is a low-cost way of sightseeing in London.

In the game of blackjack, an Ace can be counted as either one or 11, whichever is more advantageous for the player.

11 is the number of the French department Aude.

Three filmsBen-Hur (1959), Titanic (1997), and The Lord of the Rings: The Return of the King (2003) – have each won 11 Academy Awards, including Best Picture of their respective years.

In the anime seriesCode Geass, Japan is known as Area 11 of the Brittanian Empire.

Eleven is the name of a character in the popular 2016 Netflix original series Stranger Things, Presented by Millie Bobby Brown.

## In religion

### Christianity

After Judas Iscariot was disgraced, the remaining apostles of Jesus were sometimes described as “the Eleven” (Mark 16:11; Luke 24:9 and 24:33); this occurred even after Matthias was added to bring the number to twelve, as in Acts 2:14:[14]Peter stood up with the eleven (New International Version). The New Living Translationsays Peter stepped forward with the eleven other apostles, making clear that the number of apostles was now twelve.

Saint Ursula is said to have been martyred in the third or fourth century in Cologne with a number of companions, whose reported number “varies from five to eleven”.[15]A legend that Ursula died with eleven thousand virgin companions[16] has been thought to appear from misreading XI. M. V. (Latin abbreviation for “Eleven martyr virgins”) as “Eleven thousand virgins”.

### Babylonian

In the Enûma Eliš the goddess Tiamat creates eleven monsters to take revenge for the death of her husband, Apsû.

## In music

• The interval of an octave and a fourth is an 11th. A complete 11th chord has almost every note of a diatonic scale.
• The number of thumb keys on a bassoon, not counting the whisper key. (A few bassoons have a 12th thumb key.)
• In the mockumentary This Is Spinal Tap, Spinal Tap‘s amplifiers go up to eleven.
• In Igor Stravinsky’s The Rite of Spring, there are 11 consecutive repetitions of the same chord.
• In Tool’s song “Jimmy” and in Negativland‘s song “Time Zones”, the number 11 is heard numerous times in the lyrics.
• “Eleven pipers piping” is the gift on the 11th day of Christmas in the carol “The Twelve Days of Christmas.”
• In Green Grow the Rushes, O, Eleven is for “the eleven who went to heaven.”
• The Eleven” is a song by The Grateful Dead.
• In “Time Enough For Rocking When We’re Old” by The Magnetic Fields, a lyric references “when our pheromones go up to eleven.”
• Eleven Records is the record label of Jason Webley, and many of Webley’s works feature the number 11.[17]

## In numeral systems

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## 7 (seven)

7 is the natural number following 6 and preceding 8. It is a prime number, and is often considered lucky in Western culture, and is often seen as highly symbolic.

## Evolution of the glyph

In the beginning, various Hindus wrote 7 more or less in one stroke as a curve that looks like an uppercaseJ vertically inverted. The western Ghubar Arabs’ main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the character more rectilinear. The eastern Arabs developed the character from a 6 lookalike into an uppercase V lookalike. Both modern Arab forms influenced the European form, a two-stroke character consisting of a horizontal upper line joined at its right to a line going down to the bottom left corner, a line that is slightly curved in some font variants. As is the case with the European glyph, the Cham and Khmer glyph for 7 also evolved to look like their glyph for 1, though in a different way, so they were also concerned with making their 7 more different. For the Khmer this often involved adding a horizontal line above the glyph.[1] This is analogous to the horizontal stroke through the middle that is sometimes used in handwriting in the Western world but which is almost never used in computer fonts. This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph for one in writings that use a long upstroke in the glyph for 1. In some Greek dialects of early 12th century the longer line diagonal was drawn in a rather semicircular transverse line.

On the seven-segment displays of pocket calculators and digital watches, 7 is the number with the most common glyph variation (1, 6 and 9 also have variant glyphs). Most calculators use three line segments, but on Sharp, Casio, and a few other brands of calculators, 7 is written with four line segments because, in Japan, Korea and Taiwan 7 is written as ① in the illustration to the right.

While the shape of the 7 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

Most people in Continental Europe,[2] and some in Britain and Ireland as well as Latin America, write 7 with a line in the middle (“7“), sometimes with the top line crooked. The line through the middle is useful to clearly differentiate the character from the number one, as the two can appear similar when written in certain styles of handwriting. This glyph is used in official handwriting rules for primary school in Russia, Ukraine, Bulgaria, Poland, other Slavic countries,[3]France, Italy, Belgium, Finland,[4] Romania, Germany, Greece,[5] and Hungary.[6][failed verification]

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## Why a simple number has such psychological and visceral appeal.

With thanks to Guy Winch – Posted Jun 27, 2015

Even people who are not “numbers people” may have a soft spot for one number in particular: 7. For various reasons, many of us find it fascinating, even magical.

Consider: Bestselling books feature it in their titles, (e.g., The Seven Habits of Highly Effective People), articles use it to intrigue readers (e.g., Seven Characteristics of Emotionally Strong People), and the mere sight of three sevens standing side-by-side can make us shriek with excitement—especially if we happen to be standing by a slot machine with a plastic bucket of quarters.

What is it about the number that causes such a visceral response? Why didn’t any slot-machine manufacturers make their jackpot sequence three ones in a row or even three fives? After all, one and five are perfectly good numbers. The fact is that, historically, the number 7 has played a significant role in society, culture, religion, and even psychology.

### 7 reasons we are so drawn to 7:

1. It has been significant since ancient times. It was prominent in many ancient cultures. Most famous of all were the seven wonders of the world which comprised a bucket list for every world traveler — the Great Pyramid of Giza, the Hanging Gardens of Babylon, the Temple of Artemis at Ephesus, the Statue of Zeus at Olympia, the Mausoleum at Halicarnassus, the Colossus of Rhodes, and the Lighthouse of Alexandria.

2. It has had significance in almost every major religion. In the Old Testament the world was created in six days and God rested on the seventh, creating the basis of the seven-day-week we use to this day. In the New Testament the number seven symbolizes the unity of the four corners of the Earth with the Holy Trinity. The number seven is also featured in the Book of Revelation (seven churches, seven angels, seven seals, seven trumpets, and seven stars). The Koran speaks of seven heavens and Muslim pilgrims walk around the Kaaba in Mecca (Islam’s most sacred site) seven times. In Hinduism there are seven higher worlds and seven underworlds, and in Buddhism the newborn Buddha rises and takes seven steps.

3. It is associated with luck and magical properties. Seven not only represents the jackpot on slot machines, but it s also the basis for many myths and folklore. For example, various parts of the world had beliefs about the seventh son of a seventh son, legends that endowed him with magical powers of both the good and evil variety (e.g., the child was believed to have healing powers according to Irish folklore).

4. It matches our memory capacity. In 1956, George Miller of Harvard University wrote what is today considered one of the classic papers in psychology in which he demonstrated that most people can retain roughly seven items of information in their short-term memory. That is why phone numbers in the U.S. and many other countries tend to have seven digits (area code not withstanding) — as it is the most digits most people are likely to recall (although cell phones have done away with the need to recall anyone’s phone number, even our own).

5. It fits our attention spans. Because of our mental capacity favoring seven items, seven is also a good fit for our attentions spans, as long as the information is presented in seven groups. While bloggers and authors tend to take advantage of this fact, some seem to take more advantage the others. For example, the book: 7 Pages to Success: The Ultimate Short-Attention-Span Guide to Unlock Your Full Potential! is only 7 pages long, which is fine, but it costs \$7 (!), which leads me to believe the only real potential being unlocked is that of the author’s bank account.

6. It is a prime number. Prime numbers are those that can only be divided by themselves and by the number 1, and have long been considered special to mathematicians and non-mathematicians alike. Seven is considered by some to be the most ‘prime’ number within the first 10 numbers as you cannot multiply it within the group, making it a kind of optimal-prime (not to be confused with Optimus Prime, who is a Transformer).

7. It is the most popular number. The mathematician Alex Bellos asked 44,000 people to name their favorite number and over 4,000 of them named the number 7, far more than any other number. Indeed, ask people to name a number from one to ten and many will name the number seven — as many magicians know.

It is this combination of cultural, historical, religious, numerical, and psychological factors that contribute to the allure of the number 7 and the sway it has over our decisions. For example, my book, Emotional First Aid: Healing Rejection, Guilt, Rejection, and Other Everyday Hurts, was initially supposed to have eight chapters (one for each psychological ‘wound’) but after consulting with my editor, I decided to reduce the number of chapters (and wounds) to seven (loss, rumination, loneliness, and low self-esteem, in addition to the three in the subtitle) as I believed focusing on seven psychological wounds would be easier for readers to absorb and to recall rather than eight. Was I correct? Check out the book and let me know.