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"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 (′) or (′). 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 ′ or ′ (lower case p)
″ Double Prime: U+2033, HTML ″ or ″ (upper case P)
‴ Triple Prime: U+2034, HTML ‴ or ‴
⁗ Quadruple Prime: U+2057, HTML ⁗ or ⁗