Indeed. And that’s why you will hear (and often see) [니가] for ⟨네가⟩. 내가 remains the same.
Historically, ⟨ㅐ⟩ has been closer to the a sound in the American English pronunciation of cat ([æ]) and the e ([ɛ]) sound in bet. ⟨ㅔ⟩ has been closer to the e ([e]) in Spanish. In term of vowel quality, the [e] sound is closer to [i]. That is why American English—which does not have a standalone [e] resort to “pulling” the [ɛ] closer to [i] using a diphthong or replacing the [e] outright with [i]:
Spanish ⟨pedro⟩ → English “paydro”
Japanese /sa.ke/ → English “sahkee”
French ⟨mêlée⟩ → English “maylay”
In Korean loanwords from American English—strangely enough—words from English that have /æ/ will be represented with ⟨ㅐ⟩ as in 프로그램 (program), 밴드 (band), and 팬 (fanatic). But /ɛ/ will be represented by ⟨ㅔ⟩ as in 펜 (pen) and 레몬 (lemon). I suppose it’s done to maintain the contrast between /æ/ and /ɛ/ in writing, but does a native Korean speaker actually internalize the difference between the phonemes and think of them as different sounds with the same pronunciation?
My wife is a native Korean speaker and claims to hear the difference, but I really cannot hear it and I'm typically pretty good at such things.
I'm convinced she knows from context which word it is and can thus distinguishes them after the fact, but I really am not sure. I'm pretty much resigned yo just using the same sound for both and memorizing the spelling of words.
She may actually hear the difference if her Korean acquaintances are generally old. Many aged Koreans pronounce them differently as the distinction has been blurred very recently. I can articulate them, but I normally don't and you can never tell the difference from what I say.
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Why does it say "Let's" when it is just a basic ending? Shouldn't the Korean sentence read "내일 제 친구의 집에서 놀아자" (or something similar) if they wanted us to translate it like this?
The basic ending is also used with the propositive mood. 놀자 is in 해라체, the lowest speech level.
Or this sentence can mean Tomorrow, I will hang out at my friend's house. My Korean husband told me.
Does anyone know if the [ㅔ] sound is similar to the [e] sound in Vietnamese? And does the [ㅐ] sound like the Vietnamese [ê]?
The Korean language prescriptivists would tell you that there a difference between ㅐ and ㅔ. In practical standard Korean, they sound alike with the difference only in spelling. However, for more conservative pronunciation, ㅐ is ê and ㅔ is e.