Maths of the Exodus

BySam Brummer

Pesach

שמות י״ב:מ׳

(מ) וּמוֹשַׁב֙ בְּנֵ֣י יִשְׂרָאֵ֔ל אֲשֶׁ֥ר יָשְׁב֖וּ בְּמִצְרָ֑יִם שְׁלֹשִׁ֣ים שָׁנָ֔ה וְאַרְבַּ֥ע מֵא֖וֹת שָׁנָֽה׃

Exodus 12:40

(40) The length of time that the Israelites lived in Egypt was four hundred and thirty years;

350 Years?

בראשית מ״ו:י״א

(יא) וּבְנֵ֖י לֵוִ֑י גֵּרְשׁ֕וֹן קְהָ֖ת וּמְרָרִֽי׃

Genesis 46:11

(11) Levi’s sons: Gershon, Kohath, and Merari.

שמות ו׳:י״ח

(יח) וּבְנֵ֣י קְהָ֔ת עַמְרָ֣ם וְיִצְהָ֔ר וְחֶבְר֖וֹן וְעֻזִּיאֵ֑ל וּשְׁנֵי֙ חַיֵּ֣י קְהָ֔ת שָׁלֹ֧שׁ וּשְׁלֹשִׁ֛ים וּמְאַ֖ת שָׁנָֽה׃

Exodus 6:18

(18) The sons of Kohath: Amram, Izhar, Hebron, and Uzziel; and the span of Kohath’s life was 133 years.

שמות ו׳:כ׳

(כ) וַיִּקַּ֨ח עַמְרָ֜ם אֶת־יוֹכֶ֤בֶד דֹּֽדָתוֹ֙ ל֣וֹ לְאִשָּׁ֔ה וַתֵּ֣לֶד ל֔וֹ אֶֽת־אַהֲרֹ֖ן וְאֶת־מֹשֶׁ֑ה וּשְׁנֵי֙ חַיֵּ֣י עַמְרָ֔ם שֶׁ֧בַע וּשְׁלֹשִׁ֛ים וּמְאַ֖ת שָׁנָֽה׃

Exodus 6:20

(20) Amram took into his [household] as wife his father’s sister Jochebed, and she bore him Aaron and Moses; and the span of Amram’s life was 137 years.

שמות ז׳:ז׳

(ז) וּמֹשֶׁה֙ בֶּן־שְׁמֹנִ֣ים שָׁנָ֔ה וְאַֽהֲרֹ֔ן בֶּן־שָׁלֹ֥שׁ וּשְׁמֹנִ֖ים שָׁנָ֑ה בְּדַבְּרָ֖ם אֶל־פַּרְעֹֽה׃ {פ}

Exodus 7:7

(7) Moses was eighty years old and Aaron eighty-three, when they made their demand on Pharaoh.

400 Years?

בראשית ט״ו:י״ג

(יג) וַיֹּ֣אמֶר לְאַבְרָ֗ם יָדֹ֨עַ תֵּדַ֜ע כִּי־גֵ֣ר ׀ יִהְיֶ֣ה זַרְעֲךָ֗ בְּאֶ֙רֶץ֙ לֹ֣א לָהֶ֔ם וַעֲבָד֖וּם וְעִנּ֣וּ אֹתָ֑ם אַרְבַּ֥ע מֵא֖וֹת שָׁנָֽה׃

Genesis 15:13

(13) And [God] said to Abram, “Know well that your offspring shall be strangers in a land not theirs, and they shall be enslaved and oppressed four hundred years;

בראשית כ״א:ה׳

(ה) וְאַבְרָהָ֖ם בֶּן־מְאַ֣ת שָׁנָ֑ה בְּהִוָּ֣לֶד ל֔וֹ אֵ֖ת יִצְחָ֥ק בְּנֽוֹ׃

Genesis 21:5

(5) Now Abraham was a hundred years old when his son Isaac was born to him.

בראשית כ״ה:כ״ו

(כו) וְאַֽחֲרֵי־כֵ֞ן יָצָ֣א אָחִ֗יו וְיָד֤וֹ אֹחֶ֙זֶת֙ בַּעֲקֵ֣ב עֵשָׂ֔ו וַיִּקְרָ֥א שְׁמ֖וֹ יַעֲקֹ֑ב וְיִצְחָ֛ק בֶּן־שִׁשִּׁ֥ים שָׁנָ֖ה בְּלֶ֥דֶת אֹתָֽם׃

Genesis 25:26

(26) Then his brother emerged, holding on to the heel of Esau; so they named him Jacob. Isaac was sixty years old when they were born.

בראשית מ״ז:ט׳

(ט) וַיֹּ֤אמֶר יַעֲקֹב֙ אֶל־פַּרְעֹ֔ה יְמֵי֙ שְׁנֵ֣י מְגוּרַ֔י שְׁלֹשִׁ֥ים וּמְאַ֖ת שָׁנָ֑ה מְעַ֣ט וְרָעִ֗ים הָיוּ֙ יְמֵי֙ שְׁנֵ֣י חַיַּ֔י וְלֹ֣א הִשִּׂ֗יגוּ אֶת־יְמֵי֙ שְׁנֵי֙ חַיֵּ֣י אֲבֹתַ֔י בִּימֵ֖י מְגוּרֵיהֶֽם׃

Genesis 47:9

(9) And Jacob answered Pharaoh, “The years of my sojourn [on earth] are one hundred and thirty. Few and hard have been the years of my life, nor do they come up to the life spans of my ancestors during their sojourns.”

Resolution or 430 Years?

מגילה ט׳ א:ט״ז

״וּמוֹשַׁב בְּנֵי יִשְׂרָאֵל אֲשֶׁר יָשְׁבוּ בְּמִצְרָיִם וּבִשְׁאָר אֲרָצוֹת אַרְבַּע מֵאוֹת שָׁנָה״.

Megillah 9a:16

Instead of: “And the residence of the children of Israel, who resided in Egypt, was four hundred and thirty years” (Exodus 12:40), which when read literally is imprecise, for they did not dwell in Egypt that long, they wrote: And the residence of the children of Israel, who resided in Egypt and in other lands, was four hundred years.

בראשית י״ב:ד׳

(ד) וַיֵּ֣לֶךְ אַבְרָ֗ם כַּאֲשֶׁ֨ר דִּבֶּ֤ר אֵלָיו֙ יְהֹוָ֔ה וַיֵּ֥לֶךְ אִתּ֖וֹ ל֑וֹט וְאַבְרָ֗ם בֶּן־חָמֵ֤שׁ שָׁנִים֙ וְשִׁבְעִ֣ים שָׁנָ֔ה בְּצֵאת֖וֹ מֵחָרָֽן׃

Genesis 12:4

(4) Abram went forth as יהוה had commanded him, and Lot went with him. Abram was seventy-five years old when he left Haran.

בראשית י״א:ל״א

(לא) וַיִּקַּ֨ח תֶּ֜רַח אֶת־אַבְרָ֣ם בְּנ֗וֹ וְאֶת־ל֤וֹט בֶּן־הָרָן֙ בֶּן־בְּנ֔וֹ וְאֵת֙ שָׂרַ֣י כַּלָּת֔וֹ אֵ֖שֶׁת אַבְרָ֣ם בְּנ֑וֹ וַיֵּצְא֨וּ אִתָּ֜ם מֵא֣וּר כַּשְׂדִּ֗ים לָלֶ֙כֶת֙ אַ֣רְצָה כְּנַ֔עַן וַיָּבֹ֥אוּ עַד־חָרָ֖ן וַיֵּ֥שְׁבוּ שָֽׁם׃

Genesis 11:31

(31) Terah took his son Abram, his grandson Lot the son of Haran, and his daughter-in-law Sarai, the wife of his son Abram, and they set out together from Ur of the Chaldeans for the land of Canaan; but when they had come as far as Haran, they settled there.

Egypt is not an independent experience, but is the condition of being away from the land of Canaan, or of lacking an enduring footing while yet in it. Therefore, the Torah properly begins the count of the sojourn in Egypt not from the time of Yaakov's arrival, nor even of Yitzchak's birth, but rather from the time of Avraham leaving Ur forever.

Mathematics of Exodus

במדבר א׳:מ״ה-מ״ו

(מה) וַיִּֽהְי֛וּ כׇּל־פְּקוּדֵ֥י בְנֵֽי־יִשְׂרָאֵ֖ל לְבֵ֣ית אֲבֹתָ֑ם מִבֶּ֨ן עֶשְׂרִ֤ים שָׁנָה֙ וָמַ֔עְלָה כׇּל־יֹצֵ֥א צָבָ֖א בְּיִשְׂרָאֵֽל׃ (מו) וַיִּֽהְיוּ֙ כׇּל־הַפְּקֻדִ֔ים שֵׁשׁ־מֵא֥וֹת אֶ֖לֶף וּשְׁלֹ֣שֶׁת אֲלָפִ֑ים וַחֲמֵ֥שׁ מֵא֖וֹת וַחֲמִשִּֽׁים׃

Numbers 1:45-46

(45) All the Israelite males, aged twenty years and over, enrolled by ancestral houses, all those in Israel who were able to bear arms— (46) all who were enrolled came to 603,550.

View of the Shadal:
Torah may leave out some generations. It mentions those important to the narrative, but should not be read as a historical database

בראשית מ״א:נ׳

(נ) וּלְיוֹסֵ֤ף יֻלַּד֙ שְׁנֵ֣י בָנִ֔ים בְּטֶ֥רֶם תָּב֖וֹא שְׁנַ֣ת הָרָעָ֑ב אֲשֶׁ֤ר יָֽלְדָה־לּוֹ֙ אָֽסְנַ֔ת בַּת־פּ֥וֹטִי פֶ֖רַע כֹּהֵ֥ן אֽוֹן׃

Genesis 41:50

(50) Before the years of famine came, Joseph became the father of two sons, whom Asenath daughter of Poti-phera, priest of On, bore to him.

בראשית מ״ח:ה׳-ו׳

(ה) וְעַתָּ֡ה שְׁנֵֽי־בָנֶ֩יךָ֩ הַנּוֹלָדִ֨ים לְךָ֜ בְּאֶ֣רֶץ מִצְרַ֗יִם עַד־בֹּאִ֥י אֵלֶ֛יךָ מִצְרַ֖יְמָה לִי־הֵ֑ם אֶפְרַ֙יִם֙ וּמְנַשֶּׁ֔ה כִּרְאוּבֵ֥ן וְשִׁמְע֖וֹן יִֽהְיוּ־לִֽי׃ (ו) וּמוֹלַדְתְּךָ֛ אֲשֶׁר־הוֹלַ֥דְתָּ אַחֲרֵיהֶ֖ם לְךָ֣ יִהְי֑וּ עַ֣ל שֵׁ֧ם אֲחֵיהֶ֛ם יִקָּרְא֖וּ בְּנַחֲלָתָֽם׃

Genesis 48:5-6

(5) Now, your two sons, who were born to you in the land of Egypt before I came to you in Egypt, shall be mine; Ephraim and Manasseh shall be mine no less than Reuben and Simeon. (6) And whoever is born to you, whom you bear after them, will be yours; they shall be called after the name of their brethren in their inheritance.

Is it possible that the thirty-nine year old Yosef had no more children? Or Yehuda, aged forty-three? And why should we assume that Peretz, who was a whole generation younger than his father, stopped having children after Chetzron and Chamol, who went down with him to Egypt? But if he did indeed bear more children, why are the families of these other children not listed in the census of those who inherit the Land, in the fortieth year? We are forced to conclude, then, that those born in Egypt are subsumed under the households of their brethren who went down to Egypt, and that they join in their inheritance.

מדרש תנחומא, שמות ה׳

(א) ‎וַיָּמָת יוֹסֵף וְכָל אֶחָיו, וְאַף עַל פִּי כֵן וּבְנֵי יִשְׂרָאֵל פָּרוּ וַיִּשְׁרְצוּ. רַבִּי יַנַּאי אוֹמֵר: כָּל אַחַת וְאַחַת יוֹלֶדֶת שִׁשָּׁה בְּכֶרֶס אֶחָת.

Midrash Tanchuma, Shemot 5

(1) And Joseph died, and all his brethren (Exod. 1:6). Nevertheless, the children of Israel were fruitful and increased abundantly (ibid., v. 7). R. Yannai declared: Each woman bore six children at one time

דברי הימים א ז׳:כ׳-כ״ז

(כ) וּבְנֵ֥י אֶפְרַ֖יִם שׁוּתָ֑לַח וּבֶ֤רֶד בְּנוֹ֙ וְתַ֣חַת בְּנ֔וֹ וְאֶלְעָדָ֥ה בְנ֖וֹ וְתַ֥חַת בְּנֽוֹ׃ (כא) וְזָבָ֥ד בְּנ֛וֹ וְשׁוּתֶ֥לַח בְּנ֖וֹ וְעֵ֣זֶר וְאֶלְעָ֑ד וַהֲרָג֗וּם אַנְשֵׁי־גַת֙ הַנּוֹלָדִ֣ים בָּאָ֔רֶץ כִּ֣י יָֽרְד֔וּ לָקַ֖חַת אֶת־מִקְנֵיהֶֽם׃ (כג) וַיָּבֹא֙ אֶל־אִשְׁתּ֔וֹ וַתַּ֖הַר וַתֵּ֣לֶד בֵּ֑ן וַיִּקְרָ֤א אֶת־שְׁמוֹ֙ בְּרִיעָ֔ה כִּ֥י בְרָעָ֖ה הָיְתָ֥ה בְּבֵיתֽוֹ׃ (כה) וְרֶ֣פַח בְּנ֗וֹ וְרֶ֧שֶׁף וְתֶ֛לַח בְּנ֖וֹ וְתַ֥חַן בְּנֽוֹ׃ (כו) לַעְדָּ֥ן בְּנ֛וֹ עַמִּיה֥וּד בְּנ֖וֹ אֱלִישָׁמָ֥ע בְּנֽוֹ׃ (כז) נ֥וֹן בְּנ֖וֹ יְהוֹשֻׁ֥עַ בְּנֽוֹ׃

I Chronicles 7:20-27

(20) The sons of Ephraim: Shuthelah, his son Bered, his son Tahath, his son Eleadah, his son Tahath, (21) his son Zabad, his son Shuthelah, also Ezer and Elead. The men of Gath, born in the land, killed them because they had gone down to take their cattle. (23) He cohabited with his wife, who conceived and bore a son; and she named him Beriah, because it occurred when there was misfortune in his house. (25) His son Rephah, his son Resheph, his son Telah, his son Tahan, (26) his son Ladan, his son Ammihud, his son Elishama, (27) his son Non, his son Joshua.

Based on the assumption that the generation that went down to Egypt numbered sixty males who were still of child-bearing age, the next generation would have numbered 180 males; the third generation - 540, the fourth - 1620, the fifth - 4860, the sixth - 14,580, the seventh - 43,760, the eighth - 131,280, and the ninth - some four hundred thousand.
Assuming that it was the seventh, eighth and ninth generations that left Egypt, after about 190 years there were six hundred thousand, and after 210 years - all were aged 20.