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The Number Zero

The number Zero isn’t really a number, it’s a placeholder. Without Zero, 1205 would look like 125.

0_for_Post

Zero


The idea of nothingness and emptiness has inspired and puzzled mathematicians, physicists, and even philosophers. What does empty space mean? If the space is empty, does it have any physical meaning or purpose?

From the mathematical point of view, the concept of zero has eluded humans for a very long time. In his book, The Nothing That Is, author Robert Kaplan writes, “Zero’s path through time and thought has been as full of intrigue, disguise and mistaken identity as were the careers of the travellers who first brought it to the West.” But our own familiarity with zero makes it difficult to imagine a time when the concept of zero did not exist. When the last pancake is devoured and the plate is empty, there are zero pancakes left. This simple example illustrates the connection between counting and zero.

Counting is a universal human activity. Many ancient cultures, such as the Sumerians, Indians, Chinese, Egyptians, Romans, and Greeks, developed different symbols and rules for counting. But the concept of zero did not appear in number systems for a long time; and even then, the Roman number system had no symbol for zero. Sometime between the sixth and third centuries b.c.e., zero made its appearance in the Sumerian number system as a slanted double wedge.

To appreciate the significance of zero in counting, compare the decimal and Roman number system. In the decimal system, all numbers are composed of ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. After counting to nine, the digits are repeated in different sequences so that any number can be written with just ten digits. Also, the position of the number indicates the value of the number. For example, in 407, 4 stands for four hundreds, 0 stands for no tens, and 7 stands for seven.

The Roman number system consists of the following few basic symbols: I for 1, V for 5, and X for 10. Here are some examples of numbers written with Roman numerals.

IV = 4     XV = 15

VIII = 8     XX = 20

XIII = 13     XXX = 30

Without a symbol for zero, it becomes very awkward to write large numbers. For 50, instead of writing five Xs, the Roman system has a new symbol, L.

Performing a simple addition, such as 33 + 22, in both number systems further shows the efficiency of the decimal system. In the decimal number system, the two numbers are aligned right on top of each other and the corresponding digits are added.

In the Roman number system, the same problem is expressed as XXXIII + XXII, and the answer is expressed as LV. Placing the two Roman numbers on top of each other does not give the digits LV, and therefore when adding, it is easier to find the sum with the decimal system.

Properties of Zero

All real numbers, except 0, are either positive (x > 0) or negative (x < 0). But 0 is neither positive nor negative. Zero has many unique and curious properties, listed below.

Additive Identity: Adding 0 to any number x equals x. That is, x + 0 = x. Zero is called the additive identity.

Multiplication property: Multiplying any number b by 0 gives 0. That is, b × 0 = 0. Therefore, the square of 0 is equal to zero (02 = 0).

Exponent property: Any number other than zero raised to the power 0 equals 1. That is, b 0 = 1.

Division property: A number cannot be divided by 0. Consider the problem 12/0 = x. This means that 0 × x must be equal to 12. No value of x will make 0 × x = 12. Therefore, division by 0 is undefined.

Undefined Division

Because division by 0 is undefined, many functions in which the denominator becomes 0 are not defined at certain points in their domain sets. For instance, is not defined at x = 0; is not defined at x = 1; is not defined at either x = 1 or x = −1.

Even though the function is not defined at 0, it is possible to see the behavior of the function around 0. Points can be chosen close to 0; for instance, x equal to 0.001, 0.0001, and 0.00001. The function values at these points are f (0.001) 1/0.001 1,000; f (0.0001) = 10,000; and f (0.00001) = 100,000.

As x becomes smaller and approaches 0, the function values become larger. In fact, the function grows without bound; that is, the function values has no upper ceiling, or limit, at x = 0. In mathematics, this behavior is described by saying that as x approaches 0, the function approaches infinity.

Approaching Zero

Consider a sequence of numbers which in decimal notation is expressed as 1, 0.5, 0.33, 0.25, 0.2, 0.16, 0.14, and so on. Each number in the sequence is called a term. As n becomes larger, becomes increasingly smaller. When n = 10,000 is 0.0001.

The sequence approaches 0, but its terms never equals 0. However, the terms of the sequence can be as close to 0 as wanted. For instance, it is possible for the terms of the sequence to get close enough to 0 so that the difference between the two is less than a billionth, or 10−6. If one takes , then the sequence terms will be smaller than 10−6. (Encyclopedia.com)

see also Division by Zero; Limit.

For More Information visit:

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0
−1 0 1 2 3 4 5 6 7 8 9
Cardinal 0, zero, “oh” (/ˈ/), nought, naught, nil
Ordinal zeroth, noughth
Divisors all other numbers
Binary 02
Ternary 03
Quaternary 04
Quinary 05
Senary 06
Octal 08
Duodecimal 012
Hexadecimal 016
Vigesimal 020
Base 36 036
Arabic ٠,0
Urdu ۰
Bengali
Devanāgarī ० (shunya)
Chinese 零, 〇
Japanese 零, 〇
Khmer
Thai

The Number 9

The last number is Nine.

9_for_PostNine in medieval angelology there were traditionally nine orders to the celestial hierarchy; cats proverbially have nine lives; nine days or nights is the period during which a novelty is supposed to attract attention.
Nine Days’ Queen a name for Lady Jane Grey (1537–54), Queen of England for nine days following the death of Edward VI. She was quickly deposed by forces loyal to Edward’s (Catholic) sister Mary, who had popular support, and executed the following year.
nine days’ wonder a person who or thing which is briefly famous.
nine-eleven (written 9/11, in American usage) refers to the terrorist action of 11 September 2001, in which hijacked passenger planes were flown into the World Trade Center and the Pentagon, and a fourth hijacked plane crashed near Pittsburgh.
nine men’s morris a game, also known as merrill, played on a board between two players, each with an equal number of pebbles, discs of wood or metal, pegs, or pins.
nine tailors make a man proverbial saying, early 17th century; the literal meaning is that a gentleman must select his attire from various sources. It is now also associated with bell-ringing: tailors = tellers = strokes, the number of strokes on the passing bell indicating the sex of the deceased.
nine times out of ten on nearly every occasion.
nine-to-five used in reference to typical office hours, often to express an idea of predictable routine.
Nine Worthies nine famous personages of ancient Jewish and classical and medieval Christian history and legend ( Joshua, David, and Judas Maccabaeus; Hector, Alexander, and Julius Caesar; and King Arthur, Charlemagne, and Godfrey of Bouillon).
to the nines to a great or elaborate extent (now often in dressed up to the nines). The origin is unclear: it could refer to the number just below the highest point in a scale of 1 to 10, or to the nine Muses. It has also been suggested that it could represent a misdivision of an unrecorded medieval phrase ‘to the eyne [eyes]’. The later association with dress may have been influenced by the 19th-century sartorial reputation of the 99th Wiltshire Regiment. A baseball team has nine players. (Encyclopedia.com)

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
9 \times x 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 450 900 9000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
9 \div x 9 4.5 3 2.25 1.8 1.5 1.285714 1.125 1 0.9 0.81 0.75 0.692307 0.6428571 0.6
x \div 9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 1.1 1.2 1.3 1.4 1.5 1.6
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
9 ^ x\, 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481 2541865828329
x ^ 9\, 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 2357947691 5159780352 10604499373
Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100
110 120 130 140 150 200 250 500 1000 10000 100000 1000000
x_{9} \ 1 5 119 169 229 279 339 449 559 669 779 889 1109 1219
1329 1439 1549 1659 1769 2429 3079 6159 13319 146419 1621519 17836619
9
−1 0 1 2 3 4 5 6 7 8 9
Cardinal nine
Ordinal 9th
(ninth)
Numeral system nonary
Factorization 32
Roman numeral IX
Unicode symbol(s) Ⅸ, ⅸ
Greek prefix ennea-
Latin prefix nona-
Binary 10012
Ternary 1003
Quaternary 214
Quinary 145
Senary 136
Octal 118
Duodecimal 912
Hexadecimal 916
Vigesimal 920
Base 36 936
Amharic
Arabicl ٩
Urdu ۹
Armenian numeral Թ
Bengali
Chinese/Japanese
/Korean numeral
九 (jiu)
玖 (formal writing)
Devanāgarī ९ (nau)
Greek numeral θ´
Hebrew numeral ט (Tet)
Tamil numerals
Khmer
Telugu numeral
Thai numeral

The Number 8

The next number is Eight.

8_for_PostEight the word is recorded from Old English (in form ehta, eahta) and is of Germanic origin; it comes from an Indo-European root shared by Latin octo and Greek oktō.
The Eight is the name given to a group of American realist painters who exhibited together in 1908, united by a concern to involve painting with the realities of contemporary, especially urban, life.
behind the eight ball (chiefly in North American usage) at a disadvantage; in a variety of the game of pool, the black ball is numbered eight.
have one over the eight have one drink too many, probably from the assumption that the average person can drink eight pints of beer without getting drunk. (Encyclopedia.com)

8
−1 0 1 2 3 4 5 6 7 8 9
Cardinal eight
Ordinal 8th
(eighth)
Numeral system octal
Factorization 23
Divisors 1, 3, 4, 8
Roman numeral VIII
Roman numeral (unicode) Ⅷ, ⅷ
Greek prefix octa-/oct-
Latin prefix octo-/oct-
Binary 10002
Ternary 223
Quaternary 204
Quinary 135
Senary 126
Octal 108
Duodecimal 812
Hexadecimal 816
Vigesimal 820
Base 36 836
Greek η (or Η)
Arabic ٨,8
Urdu ۸
Amharic
Bengali
Chinese numeral 八,捌
Devanāgarī
Kannada
Telugu
Tamil
Hebrew ח (Het)
Hebrew שמונה (shmoneh)
Khmer
Korean
Thai

 

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The Number Seven

The next number is Seven.

7_for_PostSeven the number seven is often used symbolically, denoting completion or perfection, especially in echoes of biblical phraseology.
Seven against Thebes in Greek legend, the name given to the expedition against Thebes led by Polynices, son of Oedipus, against his brother Eteocles. When both young men were killed, their uncle Creon decreed that Polynices was not to be buried because he had attacked his own city; his niece Antigone defied the order.
Seven Churches of Asia the seven churches addressed by John in Revelation, the Churches of Ephesus, Smyrna, Pergamos, Thyatira, Sardis, Philadelphia, and Laodicea.
seven corporal works of mercy, as enumerated in medieval theology, are taken from Matthew 25:35–37 and Tobit 12:12; they are feeding the hungry, giving the thirsty to drink, sheltering the stranger, clothing the naked, visiting the sick, comforting the prisoner, and (from Tobit) burying the dead.
seven deadly sins in Christian tradition, the sins of pride, covetousness, lust, anger, gluttony, envy, and sloth. They are listed (with minor variation) by the monk John Cassian (d.435), St Gregory the Great, and St Thomas Aquinas.
Seven Dwarfs in the story of Snow White, the seven dwarfs living in a hut in the forest who shelter the fugitive princess from her stepmother; in the Walt Disney cartoon film (1937), they are named Happy, Sleepy, Doc, Bashful, Sneezy, Grumpy, and Dopey.
seven gifts of the Holy Spirit wisdom, understanding, counsel, fortitude, knowledge, piety, and fear of the Lord. The list is taken from Isaiah 11:2; six as in the AV ‘And the spirit of the Lord shall rest upon him, the spirit of wisdom and understanding, the spirit of counsel and might, the spirit of knowledge and of the fear of the Lord,’ with piety (pietas) added from the Vulgate text.
seven heavens recognized in later Jewish and in Muslim belief; the highest is the abode of God and the most exalted angels. The division may have been of Babylonian origin, and founded on astronomical theories.
Seven Hills of Rome the seven hills on which the ancient city of Rome was built: Aventine, Caelian, Capitoline, Esquiline, Quirinal, Viminal, and Palatine. Rome is informally known as the City of the Seven Hills.
Seven Joys of Mary special occasions for joy on the part of the Virgin Mary, as traditionally enumerated; the Annunciation, Visitation, Nativity, Epiphany, Finding in the Temple, Resurrection, and Ascension. The medieval church reckoned five (although lists differ); an early 14th century poem gives the Annunciation, Nativity, Epiphany, Resurrection, and the Assumption of the Virgin, with later Roman Catholic writers adding the Visitation and the Finding as the second and fifth respectively, and making the seventh the Ascension.
Seven Last Words the last seven utterances of Christ on the Cross: ‘Father, forgive them, for they know not what they do’ (Luke 23:34); ‘Woman, behold thy son!’ (John 19:26); ‘Behold thy mother!’ (John 19:27); ‘Eli, Eli, lama sabachthani? [My God, my God, why has thou forsaken me?]’ (Matthew 27:46); ‘I thirst’ (John 19:28); ‘It is finished’ (John 19:30); ‘Father, into thy hands I commend my spirit’ (Luke 23:46, a quotation from Psalm 31:5). Also known as the Seven Words.
seven-league boots the boots which in the fairy story of Hop o’ my Thumb enabled their wearer to cover seven leagues at each step.
seven liberal arts in the Middle Ages, the quadrivium and the trivium, a course of seven subjects of study introduced in the 6th century and regarded as essential grounding for more advanced studies: they are arithmetic, geometry, astronomy, music, grammar, rhetoric, and logic. Also known as the seven sciences.
seven sacraments the sacraments as enumerated in Christian belief, a list thought to have been formulated first by Peter Lombard in the 12th century: Baptism, Confirmation, the Eucharist, Penance, Extreme Unction, Holy Orders, and Matrimony. Since the Reformation, Protestant usage has generally recognized two sacraments, Baptism and the Eucharist or Lord’s Supper.
Seven Sages seven wise Greeks of the 6th century bc, to each of whom a moral saying is attributed. The seven, named in a traditional list found in Plato, are Bias, Chilon, Cleobulus, Periander, Pittacus, Solon, and Thales.
seven samurai the eponymous heroes of a Japanese film (1954), depicting a group of warriors who come together to protect a village against marauding bandits (it was remade in 1960 as an American film, The Magnificent Seven). In extended usage, the term may be used for a number of individuals who decide to act together when conventional systems, and protection, have failed.
seven seas all the oceans of the world (conventionally listed as the Arctic, Antarctic, North Pacific, South Pacific, North Atlantic, South Atlantic, and Indian Oceans).
Seven Sisters the star cluster of the Pleiades, traditionally believed to represent the seven daughters of the Titan Atlas and the Oceanid Pleione. In the late 20th century, the seven international oil companies noted for their dominant influence on the production and marketing of petroleum, Exxon, Mobil, Gulf, Standard Oil of California, Texaco, British Petroleum, and Royal Dutch Shell, became known as the Seven Sisters.
Seven Sleepers, in early Christian legend, seven noble Christian youths of Ephesus who fell asleep in a cave while fleeing from the Decian persecution and awoke 187 years later. The legend was translated from the Syriac by Gregory of Tours (6th century) and is mentioned in other sources, including the Koran.
Seven Sorrows of Mary seven particular griefs of the Virgin Mary, as enumerated in medieval theology: the prophecy of Simeon; the flight into Egypt; the three-day loss of the child Jesus in Jerusalem; the meeting with Jesus on the way to Calvary; the Crucifixion; the taking down from the Cross; and the entombment of Jesus. The Servite order, founded in the 13th century, was devoted to meditation on the sorrows of the Virgin, and the enumeration developed from this.
seven spiritual works of mercy as enumerated in Christian belief: conversion of the sinner, instruction of the ignorant, counselling the doubtful, comforting the sorrowful, patient endurance of wrong, forgiveness of injuries, and prayer for the living and the dead.
seven stars a former name for the Pleiades and the Great Bear.
Seven Wonders of the World the seven most spectacular man-made structures of the ancient world. Traditionally they comprise (1) the pyramids of Egypt, especially those at Giza; (2) the Hanging Gardens of Babylon; (3) the Mausoleum of Halicarnassus; (4) the temple of Artemis at Ephesus in Asia Minor, rebuilt in 356 bc; (5) the Colossus of Rhodes; (6) the huge ivory and gold statue of Zeus at Olympia in the Peloponnese, made by Phidias c.430 bc; (7) the Pharos of Alexandria (or in some lists, the walls of Babylon). The earliest extant list of these dates from the 2nd century bc.
seven year itch originally (in literal use, recorded from the late 19th century) a condition lasting for or recurring after seven years; now, a supposed tendency to infidelity after seven years of marriage; in modern usage, the term was reinforced by Billy Wilder’s film The Seven Year Itch (1955), starring Marilyn Monroe.
Seven Years War a war (1756–63) which ranged Britain, Prussia, and Hanover against Austria, France, Russia, Saxony, Sweden, and Spain. Its main issues were the struggle between Britain and France for supremacy overseas, and that between Prussia and Austria for the domination of Germany. The war was ended by the Treaties of Paris and Hubertusburg in 1763, leaving Britain the supreme European naval and colonial power and Prussia in an appreciably stronger position than before in central Europe. (Encyclopedia.com)

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
7 × x 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 350 700 7000
Division 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15
7 ÷ x 7 3.5 2.3 1.75 1.4 1.16 1 0.875 0.7 0.7
0.63 0.583 0.538461 0.5 0.46
x ÷ 7 0.142857 0.285714 0.428571 0.571428 0.714285 0.857142 1 1.142857 1.285714 1.428571
1.571428 1.714285 1.857142 2 2.142857
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
7x 7 49 343 2401 16807 117649 823543 5764801 40353607 282475249 1977326743 13841287201 96889010407
x7 1 128 2187 16384 78125 279936 823543 2097152 4782969 10000000 19487171 35831808 62748517
Radix 1 5 10 15 20 25 30 40 50 60 70 80 90 100
110 120 130 140 150 200 250 500 1000 10000 100000 1000000
x7 1 5 137 217 267 347 427 557 1017 1147 1307 1437 1567 2027
2157 2317 2447 2607 3037 4047 5057 13137 26267 411047 5643557 113333117

 

7
−1 0 1 2 3 4 5 6 7 8 9
Cardinal seven
Ordinal 7th
(seventh)
Numeral system septenary
Factorization prime
Prime 4th
Divisors 1, 7
Roman numeral VII
Roman numeral (unicode) Ⅶ, ⅶ
Greek prefix hepta-/hept-
Latin prefix septua-
Binary 1112
Ternary 213
Quaternary 134
Quinary 125
Senary 116
Octal 78
Duodecimal 712
Hexadecimal 716
Vigesimal 720
Base 36 736
Greek numeral Z, ζ
Amharic
Arabic ٧,7
Persian ٧ – هفت
Urdu ۷
Bengali
Chinese numeral 七(qi)
Devanāgarī ७ (sat)
Telugu
Tamil
Hebrew ז (Zayin)
Khmer
Thai
Saraiki ٧

The Number Six

The next number is Six.

scan0001six Six Counties the counties of Northern Ireland, Antrim, Armagh, Down, Londonderry, Tyrone, and Fermanagh, which by the Treaty of 1920 were constituted as a separate province.
Six Day War a war, 5–10 June 1967, in which Israel occupied Sinai, the Old City of Jerusalem, the West Bank, and the Golan Heights and defeated an Egyptian, Jordanian, and Syrian alliance. Arab name June War.
six hours’ sleep for a man, seven for a woman, and eight for a fool proverbial saying, early 17th century, implying that the more sleep a person needs, the less vigorous and effective they are likely to be. (In the late 20th century, it was widely reported that Margaret Thatcher when British Prime Minister habitually slept for only four hours a night.)
six of one and half a dozen of the other a traditional saying, mid 19th century; meaning that there is little or nothing to choose between two sides. (Encyclopedia.com)

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
6 \times x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 300 600 6000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 \div x 6 3 2 1.5 1.2 1 0.\overline{85714}\overline{2} 0.75 0.\overline{6} 0.6 0.\overline{5}\overline{4} 0.5 0.\overline{46153}\overline{8} 0.\overline{42857}\overline{1} 0.4
x \div 6 0.1\overline{6} 0.\overline{3} 0.5 0.\overline{6} 0.8\overline{3} 1 1.1\overline{6} 1.\overline{3} 1.5 1.\overline{6} 1.8\overline{3} 2 2.1\overline{6} 2.\overline{3} 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
6 ^ x\, 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
x ^ 6\, 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 4826809

 

6
−1 0 1 2 3 4 5 6 7 8 9
Cardinal six
Ordinal 6th
(sixth)
Numeral system senary
Factorization 2 × 3
Divisors 1, 2, 3, 6
Roman numeral VI
Roman numeral (unicode) Ⅵ, ⅵ, ↅ
Greek prefix hexa-/hex-
Latin prefix sexa-/sex-
Binary 1102
Ternary 203
Quaternary 124
Quinary 115
Senary 106
Octal 68
Duodecimal 612
Hexadecimal 616
Vigesimal 620
Base 36 636
Greek στ (or ΣΤ or ς)
Arabic ٦
Persian ۶
Urdu ۶
Amharic
Bengali
Chinese numeral 六,陆
Devanāgarī
Hebrew ו (Vav)
Khmer
Thai
Telugu
Tamil
Saraiki ٦

The Number Five

The next number is Five.

scan0013Five five-and-dime in North America, a shop selling a wide variety of inexpensive household and personal goods (originally, a shop where all the articles were priced at five or ten cents).
five-finger exercise an easy task (literally, an exercise on the piano for all the fingers on both hands).
five Ks in Sikh belief, five signs, kangha (comb), kara (steel bangle), kesh (uncut hair, covered by a turban, and beard), kirpan (short sword) and kuccha (short trousers, originally for riding) which show allegiance to the Khalsa.
Five Members the members of the Long Parliament, Pym, Hampden, Haselrig, Holles, and Strode, whose arrest was unsuccessfully attempted by Charles I on 4 January 1642 in the House of Commons; having been warned in advance, they had escaped. The event contributed materially to the final break between king and Parliament.
five o’clock shadow a dark appearance on a man’s chin and face caused by the slight growth of beard that has occurred since he shaved in the morning.
Five Pillars of Islam the five duties expected of every Muslim—profession of the faith in a prescribed form, observance of ritual prayer, giving alms to the poor, fasting during the month of Ramadan, and performing a pilgrimage to Mecca.
five senses the faculties of sight, smell, hearing, taste, and touch. (Encyclopedia.com)

5
−1 0 1 2 3 4 5 6 7 8 9
Cardinal five
Ordinal 5th
(fifth)
Numeral system quinary
Factorization prime
Divisors 1, 5
Roman numeral V
Roman numeral (unicode) Ⅴ, ⅴ
Greek prefix penta-/pent-
Latin prefix quinque-/quinqu-/quint-
Binary 1012
Ternary 123
Quaternary 114
Quinary 105
Senary 56
Octal 58
Duodecimal 512
Hexadecimal 516
Vigesimal 520
Base 36 536
Greek ε (or Ε)
Arabic ٥,5
Persian ۵
Urdu ۵
Ge’ez
Bengali
Kannada
Punjabi
Chinese numeral 五,伍
Korean 다섯,오
Devanāgarī ५ (panch)
Hebrew ה (Hey)
Khmer
Telugu
Malayalam
Tamil

The Number 4

The next number is Four.

scan0012Four four-colour problem a mathematical problem to prove that any plane map can be coloured with only four colours so that no two same-coloured regions have a common boundary.
four eyes see more than two two people are more observant than one alone (compare two heads are better than one). The saying is recorded from the late 16th century.
four freedoms the four essential human freedoms as proclaimed in a speech to Congress by Franklin D. Roosevelt in 1941: freedom of speech and expression, freedom of worship, freedom from want, and freedom from fear.
Four Horsemen of the Apocalypse War, Famine, Death, and Pestilence; they are traditionally identified with the riders of the white, red, black, and pale horses seen in Revelation.
four last things in Christian belief, death, judgement, heaven, and hell, as studied in eschatology.
four-letter word any of several short words referring to sexual or excretory functions, regarded as coarse or offensive; the term is recorded from the 1920s.
four noble truths the four central beliefs containing the essence of Buddhist teaching; they are that human life is characterized by frustration and suffering, that the cause of this is desire and greed, that desire must therefore be got rid of, and that following the eightfold path is the way to achieve this. (Encyclopedia.com)

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
4 \times x 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 200 400 4000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4 \div x 4 2 1.\overline{3} 1 0.8 0.\overline{6} 0.\overline{571428} 0.5 0.\overline{4} 0.4 0.\overline{36} 0.\overline{3} 0.\overline{307692} 0.\overline{285714} 0.2\overline{6} 0.25
x \div 4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
4 ^ x\, 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864
x ^ 4\, 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561