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I read a book the author wrote that /æ/ sound like the words cat, rat and so on means /ɑ/+/e/ sound. Is it real?

I wonder this sound is a single vowel or a dipthong.

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/æ/ is a monophthong. You can also find a list of English diphthongs in Wikipedia.

Technically, æ is the near-open front unrounded vowel (or near-low front unrounded vowel). Perhaps in your book, the author might write that the /æ/ sound, the vowel of words such as, cat and rat is a combination of /ɑ/+/e/ sound. I believe that the book just tries to help the reader to imagine the position of the mouth, assuming that the reader already knows how to produce both the /ɑ/ and /e/ sounds.

This is fine because phonetically /æ/ sound is something somewhere between /ɑ/ and /e/ sounds. (I personally prefer to say that it's more like somewhere between the /a/ and /ɛ/.) See their positions in the vowel chart below.

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However, the result of the combination is a single vowel, not two. It is a monophthong, not diphthong.

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    Just to round out this admirable answer, the <æ> ligature is employed because that's the character Old English scribes used to express a sound which did not exist in Latin and therefore had no character in the Roman alphabet they used; and the sound didn't exist in any modern European language, either, when the IPA was developed. Some very early texts employ the digraph <ae>, and some texts ligate <e> with a form like <ɑ> instead of <a>. Both presumably represent the scribes' sense that, as you say, the sound /æ/ lies between the sounds represented by <e> and <a> (or <ɑ>) – StoneyB on hiatus Feb 4 '14 at 1:05
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/æ/ is also a ordinary letter in the Danish alphabet. Unless other countries, Denmark uses the letters(vowels) /æ/,/ø/ and /å/ in their alphabet. /æ/ doesn't exactly sounds like an /a/+/e/, but I guess its the closest you can get to it.

If you want to hear how its exactly pronounced, I suggest you go to Google translate, translate from "Danish" to "English", type in /æ/, and click the "listen" icon. (OBS!) There's a Danish and English listen Icon, make sure you click the "Danish" listen Icon.

Hope it helps!

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    The fact it is a letter in Danish does not mean it represents the same sound. ("i" is a letter in English, but [i] sounds more like that same letter in Dutch.) – oerkelens Feb 2 '14 at 21:29
  • You need to distinguish between the letter ‹æ› and the sound [æ]. As it happens, Danish words written with ‹æ› aren't necessarily pronounced with the sound [æ]. (We can't use slashes here because those are only defined within the context of the phonemic system of a single language, and English and Danish are different languages.) – snailplane♦ Sep 27 '15 at 23:39

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