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Write half a dozen or a half dozen, not a half a dozen. Do not write the half a or one half a.
Incorrect:    the past half a century, one-half a million.
            Correct:         the past half century, one-half million, a half million men, half a million men,
                                      a million and a half Americans, a half dozen valleys.

Do not hyphenate a half; hyphenate one-half.

Hyphenate compound adjectives before a noun: a half-serious remark. Do not hyphenate such compounds when they follow a noun unless hyphenated in Webster's: He was only half serious. Do not hyphenate adverbs: half jokingly.

Hyphenate fractions: a half-millionth of an ounce; one half-millionth of an ounce. Incorrect: one-half millionth of an ounce.

Hyphenate measurements that include the unit of measurement before a noun: a half-mile trek, an eight-and-a-half-inch snake, but eight and a half inches of snake; a million-and-a-half-pelt shipment, a million and a half pelts; a four-and-a-half-million-dollar program, four and a half million dollars; a half million residents; two and a half hours' walk.

Hyphenate with verb forms: half-see the castle, a half-seen castle, the castle could be half-seen; we half-walked, half-crawled.

Miscellaneous: half brother, half Hawaiian, half a dollar, one half-dollar (the coin), halfway, halfhearted. Examples: he threw down a half-dollar; halfway there; halfhearted efforts.

Use an apostrophe in such expressions as half an hour's work, half a dollar's worth of gum, a half month's wages.

When half precedes a noun, of may be dropped: half an orange, half a mile, though half of an orange and half of a mile are not incorrect. When half is followed by a pronoun, of is retained: half of them, half of it.

Verb agreement: Half may be singular or plural depending on the intention of the writer. Generally the object of the preposition of following half governs the verb: Half the world's population lives there; half of them are illiterate.