Half-width, hankaku (半角), refers to a character which is half as wide as it is high. Characters of equal height and width are called full-width, zenkaku (全角).
Half-width katakana, hankaku katakana (半角片仮名), is a type of katakana on a computer where each letter is tall and thin, compared to full width katakana, zenkaku katakana (全角片仮名), where each character is about the same height and width. Originally, before there was any kanji encoding for Japanese, half-width katakana was the only way to output Japanese text on a computer. Nowadays, the need for this encoding has reduced, but half-width katakana is still used, for example, on cash register receipts and for television and film subtitles to save space. It is also used in the Japanese banking system, with account names all being in half-width katakana.
In half-width katakana, there are no forms of characters with combined dakuten, like ば, or handakuten, like ぱ. Instead, the dakuten or handakuten are represented by a separate character which has the same width as one katakana character. Thus, for example, "Nippon" (Japan) in full-width katakana is ニッポン, but in half-width katakana it is ﾆｯﾎﾟﾝ, with the handakuten as a separate ﾟ symbol. (See also What are the names of the Japanese non-kana, non-kanji symbols?)
Half-width katakana was originally encoded by the JIS X 0201 encoding standard. (See Encodings of Japanese.) The common "Shift JIS" encoding of Japanese is an attempt to enable both half-width katakana and full-width characters together. (See Encodings of Japanese.)
Half-width, or hankaku, is also used in a computer context to describe the ASCII character set, which is referred to as hankaku eisūji (半角英数字) in Japanese, in contrast to zenkaku eisūji (全角英数字), the "full width" versions of ASCII characters which occupy the same space as a kanji, as in ＡＢＣ１２３＄ ％＆. See What is "wide ASCII"? for more details on this.
There is no half-width version of the hiragana characters.
To input half-width katakana, see How is Japanese input on a computer? For more about the uses of katakana, see What is katakana used for?
In Unicode, half-width katakana is represented from position
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