Torah and Mathematics: A Mathematical Analysis of Rebuke and Blessing in Parshat Ha'azinu

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In a related article1 on parshat Ha’azinu, we discussed how in the two final parshahs of the Pentateuch, Ha’azinu and Vezot Habrachah, Moses’ unites his rebuke of the Jewish people together with his blessings to them. Let us now analyze the properties of this unification from a mathematical perspective.

Through our analysis we will see how ideas uncovered in our study of the Torah can be used to discover beautiful relationships between numbers. Of course, there is nothing mysterious about these relationships and they could be discovered by any mathematician playing around with numbers. Still, the fact that they are found while studying Torah gives them greater spiritual significance.

  • The Hebrew word for “rebuke” is תּוֹכֵחָה , whose gematria is 439.
  • The Hebrew word for “blessing” is בְּרָכָה , whose gematria is 227.

Thus, the sum of the two words is 666, which is also the triangle of 36 (the sum of integers from 1 to 36), which we write as 666 = 36.
In passing let us note that 36 is itself a triangular number:

36 = 8

And, it is also a square number:

36 = 62.

Elsewhere, we have dealt with the question of when a triangular number can also be a square number, but 36 is one of these special numbers.

Since in these two words there are 9 letters we can draw them as a square like this:


Note that the left-to-right diagonal contains the letters כ ה ר , which are the same letters as those that appear in the bottom row. The sum of these letters is 225 = 152 or the value of the first two letters of Havayah—which constitute the holy Name that corresponds to thesefirah of wisdom2יה = 15, squared.

Apart from the letters in the left-to-right diagonal (or the bottom row), the sum of the rest of the letters in the square equal 441, the gematria of "truth" (אֶמֶת ). But, 441 is equal to 212, or the value of the first three letters of Havayahיהו , squared!

Thus, we have found that 666 is the sum of the two squares 441 and 225, or 212 ┴ 152. But, now note that the roots of the two squares—21 and 15—sum to 36; while, 666 is the triangle of 36, as noted earlier. Moreover, both 21 and 15 are themselves triangular numbers: 21 = 6, and 15 = 5.

Let us write our finding out in mathematical notation:
(5)2 ┴ (6)2 = (5 ┴ 6)

Since 5 and 6 are consecutive integers, we can generalize and write:

(k)2 ┴ ([k ┴ 1])2 = (k ┴ [k ┴ 1])

Does this equation hold true for all k? Knowing the algebraic equation for computing the triangle of k, we can use simple algebra to prove that indeed, both sides of this equation are equivalent. So, this equation does indeed hold true for all k.

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