The traditional terms "long" and "short" for teaching English vowels were borrowed from Latin grammar back in the days when Latin was the only language for which formal grammatical discourse was available. These terms don't represent the linguistic facts of English. That is, some English vowels do have a longer duration than others, but the primary feature which distinguishes English vowels is not length, as it appears to have been in Latin, but "quality", the sound imposed by the variable shape of the oral cavity.
The representation of English vowels in IPA and similar systems of notation does represent the linguistic facts:
"long a", "long i" and "long o" are diphthongs—they consist of a glide from one sound to another—so they are represented with two glyphs each, /eɪ/, /aɪ/, /oʊ/ (or /əʊ/).
"long e" and "long u", however, are monophthongs—they consist of a single more or less homongeneous sound—so they are represented with single glyphs, /i/ and /u/.
Note that diphthongs are in fact, by default, longer than monophthongs, and that "long e" and "long u" are, in fact, usually pronounced with a longer duration (and consequently are often represented with a colon, or a glyph similar to a colon, to represent the increased length: /iː/, /uː/. That, no doubt, is why early English grammarians adopted the terms "long" and "short".
"Short a, e, i, o, u" are also monophthongs, but they have different qualities than the long versions, and are represented with different glyphs: /æ/, /ɛ/ or /e/, /ɪ/, /ɒ/, /ʊ/ or /ʌ/ (depending on which sound your teacher names a "short u").
There are also a whole bunch of vowels which don't have long/short distinctions, or even names, in traditional grammar, because they didn't occur in Latin (or at least weren't discussed by Latin grammarians). Some are diphthongs: "ow" /aʊ/, "oy" /ɔɪ/, and the vowels preceding /r/ in non-rhotic dialects like BrE, /eə/ as in "fair", /ʊə/ as in "cure", /ɪə/ as in "fear". Others are monophthongs: "aw" /ɔː/, "ah" /ɑː/, and the destressed central vowel /ə/.