A very strange question is why do we pronounce give as [ɡɪv] instead of [gʌɪv] like in hive, dive, wife, life, knife?
Or should it actually be [jʌɪv] by the grammar standards?
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As with many words in the English language, pronunciation is rooted in the Old English spelling of the word, rather than how the spelling of the word might have evolved to its present day form.
I'll preface this by saying that this is just my layman's point of view, but looking at the historical spelling of the word 'give', and one of your sample words, 'hive' yields a far clearer explanation than looking at their current day spellings.
Our current day pronunciation of the word 'give' stems from the Old English form of the word 'giefan', as shown.
giefan (Old English) > given (Middle English) > give
. (interestingly, the past tense form of the word - 'geaf' - also seems closer in pronunciation, than 'gave').
The word 'hive' on the other hand, although having similar spelling in modern times, evolved from a very different pronunciation in Old English, hence the difference in pronunciation.
hȳf (Old English) > hyfe (Middle English > hive
It may be more useful to ask the converse question: Why do we write [ɡɪv] as “give” when it's contrary to the Magic E rule? Blame Norman scribes who introduced the rule that English words can't end in the letter V. (The word “rev”, a 20th century addition to the language, is exempt from this rule, as are proper names [mostly of Slavic origin] like Gorbachev or Romanov.) So the final silent E is added even though it does not turn the [ɪ] into [aɪ].