An element p of a commutative integral domain R is called prime if it is nonzero and whenever p|ab in R then either p|a or p|b.

Here we write x|y for x divides y.

For example, prime numbers are prime elements of the ring of integers.

Lemma A prime element of R is irreducible

Proof Suppose that p is prime but that it is not irreducible. Then there exists non-units x,y in R such that p=xy. Clearly then p|xy. Since p is prime it divides one of x or y. Without loss of generality, it divides x. Thus pa=x, for some a in R. Thus we have p=pay or p(1-ay)=0. Since R is an integral domain it follows that y must be a unit. This contradicts the choice of y and proves the lemma.

in photography, a prime is short for a prime lens, that's to say a lens whose focal length is fixed.
The opposite of a zoom (read that node for a comparison of zooms vs. primes).

To paint with a primer, a kind of paint formulated to help stick another coat of paint (top coat) onto the surface which you are painting. Presumably, the chemisty which makes for ideal top coat paint is not the same as the chemistry for ideal primer paint.

In bicycle racing, a prime (pronounced "preem") is a special prize for the winner of the next lap, usually in the middle of the race somewhere. This is used mainly in criterium races, which involve the riders going around a short (less than a few miles) course. If the races are long (20-50 or more laps), the winning strategy may be to hide in the pack, waiting for the sprint finish. To discourage such boring race tactics, organizers put in prime laps. They are usually (but not always) randomly selected, non-previously announced laps in which a bell is rung. The winner of the next lap is given the prime prize. Racers are usually told there will be a prime lap, but not told which lap they will be.

Sometimes, the prizes for the prime are collected from the spectators. Sometimes, spectators give out money in order to have the prime when they want. Frequently, primes are given out when the race organizer is noticing that the race is looking boring.

Racers get really angry and tired when a diabolical race organizer sets up three primes consecutively near the end of the race. This wrecks any sneaky race tactics, and insures that the really, really strong will win the race.

In backgammon, a prime is a block of six contiguous points that have two or more of one's pieces on them, thereby blocking any of the opposing player's pieces behind the prime from crossing them and escaping. Primes are effective because in backgammon one moves pieces a die at a time, ie. four then three, not seven. Since no die can have more than six on it, no piece can be moved more than six points at once and a prime of six is thus a total block until it's dismantled.

A prime on all six points of your inner board (the quarter of the board where your opponent's pieces start) is an inside prime; one with any of its points outside one's inner board is an outside prime, and one of less than six contiguous points is a partial prime. If you can put together an inside prime you will often win the game by a large margin, because if you then hit one or more of your opponent's blots he will be completely unable to reenter the board with that piece and must wait until you break the prime, which you ordinarily do only when bearing off at the end of the game. Thus, one of the strategies of backgammon: the priming game.

A 2005 film written and directed by Ben Younger, starring Uma Thurman and Bryan Greenberg. It's basically a chick flick, but a decent one, at least. It's about a 37-year-old woman who's recently divorced and starts dating a 23-year-old guy. The twist, though, is that he is her therapist's son. So once his mother finally figures this out, there's some complications (obviously), and the couple hit some rough patches. Eventually (Major spoiler ahead) they break up, never to see each other again. I liked that about this film, it was a much better ending than those found in most of the films of this type. (spoilers end here)

Of course, there's gay friends of the woman, as every film apparently needs these days. I can see how these things happen, deep inside Hollywood: "Hmm... this movie's not funny enough yet. Any ideas?", "Well sir, how about we add some flaming homosexuals?". There's also the guy's wacky friend who hits girls with pies, and the stereotypically jewish grandparents ("Did you have something to eat? Would you like a sandwich?"). It's mostly pretty predictable, but rather well executed. The couple's relationship is warm and believable, the writing is generally quite good, and (best of all) they don't over-sell the point.

Perhaps it's just because it had Uma Thurman in it, but I genuinely liked this chick flick (a rare occurence).

Director & writer: Ben Younger
Producer(s): Jennifer Todd & Suzanne Todd
Music by: Ryan Shore

Cast:
Rafi Gardet: Uma Thurman
David Bloomberg: Bryan Greenberg
Lisa Metzger: Meryl Streep
Morris: Jon Abrahams

"Marked or distinguished by a mark (′) called a prime mark."
--Webster 1913

The prime mark looks very much like an apostrophe, but it is a separate symbol, leaning a bit towards the right, as in the the French accent aigu (but doesn't lean quite so far). The Unicode character is (U+2032), and the HTML is (&prime;) or (&#8242;). To make things a bit more confusing, outside of the USA the prime mark is sometimes called the 'dash', but is still written the same: x′ may be referred to as 'x prime' or 'x dash'.

The most common use for the prime mark is to denote feet and inches (6′6″) or latitude and longitude (1°24′12.2″N 2°10′26.5″E). They are also used for arc minutes and seconds. As you can see, these use both the prime and the double prime; there are additionally triple and quadruple prime marks.

Prime marks are also common in math, where they denote a number of different things. Originally, prime was used in the sense of 'first', so the prime mark marked the first item in a series. This also leads to another difference in terminology, with some people refering to x″ as 'x second', and others as 'x double prime'.

In calculus, ƒ′ is the first derivative, ƒ″ is the second derivative, and so on.

When x is a simple variable, x′, x″, and so on may be defined as new variables of the same type, e.g., refering to integers q and q′. Physics uses a variant of this where the prime mark is used to denote the state after an event; e.g., v before impact and v′ after impact. These uses are also commonly found in philosophy and logic.

In set theory, A′ is the complement of the set A. However, the notation Ac is also used, and seems to be more popular.

In probability theory, Pr(A′) is the complementary event, i.e., the negation of an event: Pr(A′) = 1 − Pr(A). This can also be denoted with ¬A, Ac, or an A with a line over it.

It may also be used to denote a transformation, a transpose, or a number of other odds and ends as needed.

In quantum physics, prime refers to an upper state of the quantum number, while double prime refers to the lower state.

In organic chemistry, molecular biology, and genetics, the prime mark is used in a number of different roles to denote specific structures; if you see an apostrophe in a molecular notation, it is probably actually a prime mark. The most familiar usage is probably the 5′ ('five prime end') and 3′ ('three prime end') notation in refering to the DNA molecule, which is a specific case of identifying the position of carbon on a ring of deoxyribose or ribose. However, this is specific to genetics, and the prime mark is in no way specific to identifying carbon in other branches of chemistry.

As you may have gathered, the prime mark is a useful, and small, mark that can be used easily to denote that something interesting is going on. The usages listed above are some of the most common, but are by no means exhaustive. It is generally assumed that if you are seeing it, you are part of the ingroup and already know what it means. This can be frustrating to people who are entering new fields, and have seen it used in a very specific -- and useful -- context that no longer applies. The range of uses also means it has a range of names; not just prime and dash, but also second and inch and complement.

It does not help that the prime mark is largely indistinguishable from the apostrophe, and worse, does not appear on most keyboards. Most primes and double primes you see are actually single and double quotation marks. If you would like to use the proper symbols, you can use the HTML or Unicode below, but be warned that many people's browsers may not support them all:

```′      Prime:           U+2032, HTML &#8242; or &prime; (lower case p)
″      Double Prime:    U+2033, HTML &#8243; or &Prime; (upper case P)
‴      Triple Prime:    U+2034, HTML &#8244; or &tprime;
⁗      Quadruple Prime: U+2057, HTML &#8279; or &qprime;
```

Prime (?), a. [F., fr. L. primus first, a superl. corresponding to the compar. prior former. See Prior, a., Foremost, Former, and cf. Prim, a., Primary, Prince.]

1.

First in order of time; original; primeval; primitive; primary. "Prime forests." Tennyson.

She was not the prime cause, but I myself.
Milton.

⇒ In this sense the word is nearly superseded by primitive, except in the phrase prime cost.

2.

First in rank, degree, dignity, authority, or importance; as, prime minister. "Prime virtues." Dryden.

3.

First in excellence; of highest quality; as, prime wheat; a prime quality of cloth.

4.

Early; blooming; being in the first stage. [Poetic]

His starry helm, unbuckled, showed him prime
In manhood where youth ended.
Milton.

5.

Lecherous; lustful; lewd. [Obs.] Shak.

6.

Marked or distinguished by a mark (′) called a prime mark.

Prime and ultimate ratio. (Math.). See Ultimate. --
Prime conductor. (Elec.) See under Conductor. --
Prime factor (Arith.), a factor which is a prime number. --
Prime figure (Geom.), a figure which can not be divided into any other figure more simple than itself, as a triangle, a pyramid, etc. --
Prime meridian (Astron.), the meridian from which longitude is reckoned, as the meridian of Greenwich or Washington. --
Prime minister, the responsible head of a ministry or executive government; applied particularly to that of England. --
Prime mover. (Mech.)
(a) A natural agency applied by man to the production of power. Especially: Muscular force; the weight and motion of fluids, as water and air; heat obtained by chemical combination, and applied to produce changes in the volume and pressure of steam, air, or other fluids; and electricity, obtained by chemical action, and applied to produce alternation of magnetic force.
(b) An engine, or machine, the object of which is to receive and modify force and motion as supplied by some natural source, and apply them to drive other machines; as a water wheel, a water-pressure engine, a steam engine, a hot-air engine, etc.
(c) Fig.: The original or the most effective force in any undertaking or work; as, Clarkson was the prime mover in English antislavery agitation. --
Prime number (Arith.), a number which is exactly divisible by no number except itself or unity, as 5, 7, 11. --
Prime vertical (Astron.), the vertical circle which passes through the east and west points of the horizon. --
Prime-vertical dial, a dial in which the shadow is projected on the plane of the prime vertical. --
Prime-vertical transit instrument, a transit instrument the telescope of which revolves in the plane of the prime vertical, -- used for observing the transit of stars over this circle.

Prime (?), n.

1.

The first part; the earliest stage; the beginning or opening, as of the day, the year, etc.; hence, the dawn; the spring. Chaucer.

In the very prime of the world.
Hooker.

Hope waits upon the flowery prime.
Waller.

2.

The spring of life; youth; hence, full health, strength, or beauty; perfection. "Cut off in their prime." Eustace. "The prime of youth." Dryden.

3.

That which is first in quantity; the most excellent portion; the best part.

Give him always of the prime.
Swift.

4. [F. prime, LL. prima (sc. hora). See Prime, a.]

The morning; specifically (R. C. Ch.), the first canonical hour, succeeding to lauds.

Early and late it rung, at evening and at prime.
Spenser.

⇒ Originally, prime denoted the first quarter of the artificial day, reckoned from 6 a. m. to 6 p. m. Afterwards, it denoted the end of the first quarter, that is, 9 a. m. Specifically, it denoted the first canonical hour, as now. Chaucer uses it in all these senses, and also in the sense of def. 1, above.

They sleep till that it was pryme large.
Chaucer.

5. (Fencing)

The first of the chief guards.

6. (Chem.)

Any number expressing the combining weight or equivalent of any particular element; -- so called because these numbers were respectively reduced to their lowest relative terms on the fixed standard of hydrogen as 1. [Obs. or Archaic]

7. (Arith.)

A prime number. See under Prime, a.

8.

An inch, as composed of twelve seconds in the duodecimal system; -- denoted by [′]. See 2d Inch, n., 1.

Prime of the moon, the new moon at its first appearance.

Prime, v. t. [imp. & p. p. Primed (?); p. pr. & vb. n. Priming.] [From Prime, a.]

1.

To apply priming to, as a musket or a cannon; to apply a primer to, as a metallic cartridge.

2.

To lay the first color, coating, or preparation upon (a surface), as in painting; as, to prime a canvas, a wall.

3.

To prepare; to make ready; to instruct beforehand; to post; to coach; as, to prime a witness; the boys are primed for mischief. [Colloq.] Thackeray.

4.

To trim or prune, as trees. [Obs. or Prov. Eng.]

5. (Math.)

To mark with a prime mark.

To prime a pump, to charge a pump with water, in order to put it in working condition.

Prime, v. i.

1.

To be renewed, or as at first. [Obs.]

Night's bashful empress, though she often wane,
As oft repeats her darkness, primes again.
Quarles.

2.

To serve as priming for the charge of a gun.

3.

To work so that foaming occurs from too violent ebullition, which causes water to become mixed with, and be carried along with, the steam that is formed; -- said of a steam boiler.

Prime, a. (Math.)

(a)

Divisible by no number except itself or unity; as, 7 is a prime number.

(b)

Having no common factor; -- used with to; as, 12 is prime to 25.

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