Throughout the Tanakh, I have come across אֲנִי (ʾani) and אָנֹכִי (ʾanokhi) many times, which both mean "I". What is the difference between the two words in Tanakh?
Notice that the 10 Commandments begin "Anokhi Hashem Elokekha", while Shema ends "Ani, Hashem Elokeikhem." And how Yaakov says "וַיֹּאמֶר יַעֲקֹב אֶל אָבִיו, "אָנֹכִי עֵשָׂו בְּכֹרֶךָ עָשִׂיתִי כַּאֲשֶׁר דִּבַּרְתָּ אֵלָי (Bereishis 27:19) While the real Esav says וַיֹּאמֶר לוֹ יִצְחָק אָבִיו "מִי אָתָּה?" וַיֹּאמֶר "אֲנִי בִּנְךָ בְכֹרְךָ עֵשָׂו." (Ibid. v. 32)
Therefore commentators were motivated to look at the difference. But this is in terms of connotation and derashah, not translation.
Rav SR Hirsch (Shemos 2:20, on the first commandment) says that "ani" refers to the person in contrast to other people, whereas "anokhi" refers to the person in relation to the listener. The Malbim (on Bereishis 27) and similarly Shadal write that "ani" is the normal word for "I", whereas "anokhi" is used to emphasize "and not others".
So in general, "ani" would mean that the emphasis of the phrase in on the who, whereas "anokhi" would emphasize what it was that I did or that happened to me.
I think all of these are consistent with a derashah R' Yochanan makes on Shabbos 105a on the "Anokhi" at the start of the Commandments. He says Hashem used "Anokhi" as an acronym for "אנא נפשי כתיבת יהבית -- I Myself [lit: My Soul] made this writing." Generalize that to what you think "Anokhi" connotes across the board, and you have the kinds of ideas we see above.
(In the Zohar, it says that Hashem's "Ani" refers to the Lower Throne [Malkhus as it is manifest within creation], and His use of "Anokhi" refers to the Upper Throne [comprised of Binah as connected to Malkhus]. I have no idea what that means.)
After lots of searching, I finally found a paper on this topic:
"The Two Forms of First Person Singular Pronoun in Biblical Hebrew: Redundancy or Expressive Contrast?" by E. J. Revell, Journal of Semitic Studies 40 (1995), pp. 199–207.
The crux of Revell's argument is that "אני is typically used by status-marked human speakers, אנכי by others."
He notes also that אני is also used often when someone is volunteering for, requesting or claiming a position, expressing emotion or care, etc. אנכי is used when describing a "shameful or distasteful" quality, solidarity with addressee, etc. G-d uses אנכי where "speech concerns the addressee on a personal level" or in statements describing background context. אני is used by G-d "in clauses which do not refer to the addressee".
אֲנִי is the basic word "I". It is just about always followed by an adjective or noun. It is used the vast majority of times both in Tanakh and later writings.
אָנֹכִי is a more nuanced version of אֲנִי. It also means "I", but it's a more robust, stronger version. It's main use is for emphasis, a bold I as it were.
Let me give an example. "Ani Hashem" is found multiple times in Tanakh. It simply means, "I am G-d" as is often a final clause to a sentence. However, "Anokhi Hashem" found in the Asereth HaDibroth is more emphatic: "It is I who is G-d" - in fact, the commentators understand it as the commandment to believe in G-d. For such a function, one would use the stronger word "Anokhi".
Not really a complete answer, but a path to an understanding of the difference between the two words. The Gur Aryeh on Rashi Bereishit 29:17 explains a grammatical difference between the word Anochi and the word Ani.
The word Anochi can be read with and without a pause afterwards without changing the pronunciation of the word.
Ani, on the other hand is vowelled differently (and therefore pronounced differently) when it is connected to the following word and when there is a break afterwards.
Therefore, one of the differences between Anochi and Ani is that Ani is always unambiguously understood (depending on the Nikkud), while Anochi is more ambiguous.
The Arvei Nachal in Toldos writes that אנכי is used to mean 'me and no one else.'
As quoted above from Maseches Shabbos page 105 ,the Gemara referring to the posuk אנכי ד' breaks down the word ANOCHI to imply-אנא נפשי כתבית יהבית literally saying according to some,'I MY SOUL I WROTE AND SHARED as such that the word takes on a new meaning -As opposed to ANI which connotes a sence of athority ,the word ANOCHI implies his giving over of himself TO YOU .
See how amazingly the midrash comments on this posuk in מדרש רבה שמות פכ'ט אות ז' אנכי.. מלך בשר ודם בונה פלטין שמא יכול להזיז אותו ממקומו אני איני כך שנאמר אני עשיתי ואני אשא ואני אסבל ואמלט meaning I HASHEM ARRANGE SOMETHING ONE WAY AND YET I LET MY WORK BE ALTERED by others which BS'D is the concept of Anochi 'ones self dedication to a person or a cause in a humble way rather than I am the one in charge
.Moreover we find this exact thing by YOSEF HATZADIK who said to Yaakov' אנכי אעשה כדבריך' meaning i am totally dedicated to fulfill your request regarding your burial in ארץ ישראל.
Now Hashem has opened my eyes to the amazing accuracy our Torah was written in .In Parashas Toldos when describing the event of Yaakov receiving the Berochos instead of Aisov ,when quoting Yaakov saying to Yitzchak" I am your son" it says אנכי עשיו בכוריך .However when quoting Aisov it says אני בנך בכורך עשיו telling us the contrast of how Yaakov says humbly" I am Aisov and I am at your service" .Whereas Aisov is quoted saying "I am your son and it is me who is in charge of your care ".
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