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The square root of 2 (approximately 1.4142) is the positive real number that, ... Figure 2. Tom Apostol's geometric proof of the irrationality of √ 2.
Irrationality, by infinite reciprocal subtraction, can be easily seen in the golden ratio of the regular pentagon. [ 26 ] Some scholars in the early 20th century credited Hippasus with the discovery of the irrationality of 2 {\displaystyle {\sqrt {2}}} , the square root of 2 .
The square root of the Gelfond–Schneider constant is the transcendental number = 1.632 526 919 438 152 844 77.... This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence.
The square root of 2 was likely the first number proved irrational. [27] The golden ratio is another famous quadratic irrational number. The square roots of all natural numbers that are not perfect squares are irrational and a proof may be found in quadratic irrationals.
Rational approximations to the Square root of 2. In mathematics , an irrationality measure of a real number x {\displaystyle x} is a measure of how "closely" it can be approximated by rationals .
The square root of 2 was the first such number to be proved irrational. Theodorus of Cyrene proved the irrationality of the square roots of non-square natural numbers up to 17, but stopped there, probably because the algebra he used could not be applied to the square root of numbers greater than 17. Euclid's Elements Book 10 is dedicated to ...
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. There is a famous passage in Plato's Theaetetus in which it is stated that Theodorus (Plato's teacher) proved the irrationality of
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.