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Various numbers play a significant role in Jewish texts or practice. Some such numbers were used as mnemonics to help remember concepts, while other numbers were considered to have intrinsic significance or allusive meaning. Numbers such as 7, 10, 12, and 40 were known for recurring in symbolic contexts.
Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a b is rational: [28] [29] Consider √ 2 √ 2; if this is rational, then take a = b = √ 2. Otherwise, take a to be the irrational number √ 2 √ 2 and b = √ 2. Then a b = (√ 2 √ 2) √ 2 = √ 2 √ 2 · √ 2 = √ 2 2 = 2 ...
11 is a prime number, and a super-prime. 11 forms a twin prime with 13, [6] and sexy pair with 5 and 17. The first prime exponent that does not yield a Mersenne prime is 11. 11 is part of a pair of Brown numbers. Only three such pairs of numbers are known. [citation needed] Rows in Pascal's triangle can be seen as representation of powers of 11 ...
The following 1953 proof by Dov Jarden has been widely used as an example of a non-constructive proof since at least 1970: [4] [5] CURIOSA 339. A Simple Proof That a Power of an Irrational Number to an Irrational Exponent May Be Rational. is either rational or irrational. If it is rational, our statement is proved.
Names of God in Judaism have further prominence, though infinite meaning turns the whole Torah into a Divine name. As the Hebrew name of things is the channel of their lifeforce, parallel to the sephirot, so concepts such as "holiness" and "mitzvot" embody ontological Divine immanence, as God can be known in manifestation as well as transcendence.
This shows that any irrational number has irrationality measure at least 2. The Thue–Siegel–Roth theorem says that, for algebraic irrational numbers, the exponent of 2 in the corollary to Dirichlet’s approximation theorem is the best we can do: such numbers cannot be approximated by any exponent greater than 2.
For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x 2 − 2 = 0. The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2 − x − 1 = 0.
Examples include the seven days of creation and so seven days that make up a week, and the seven lamps on the Temple Menorah. One variation on the use of seven is the use of the number six in numerology, used as a final hallmark in a series leading to a seven (e.g. mankind is created on the sixth day in Genesis, out of the seven days of creation).