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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 ...
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written in mathematics as 2 {\displaystyle {\sqrt {2}}} or 2 1 / 2 {\displaystyle 2^{1/2}} .
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.
Hence, the set of real numbers consists of non-overlapping sets of rational, algebraic irrational, and transcendental real numbers. [3] 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 proof by Pythagoras about 500 BCE has had a profound effect on mathematics. It shows that the square root of 2 cannot be expressed as the ratio of two integers. The proof bifurcated "the numbers" into two non-overlapping collections—the rational numbers and the irrational numbers.
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
Season to date, The Irrational is averaging 6.3 million total viewers and a 0.5 demo rating, ranking No. 2 and tying for third among all entertainment programs that have aired on NBC this fall.
The following famous example of a nonconstructive proof shows that there exist two irrational numbers a and b such that is a rational number. This proof uses that is irrational (an easy proof is known since Euclid), but not that is irrational (this is true, but the proof is not elementary).