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The square root of 2 (approximately 1.4142) is the positive real number that, ... Square root of 2 is irrational, a collection of proofs; Haran, Brady (27 Jan 2012).
Among irrational numbers are the ratio π of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. [1] In fact, all square roots of natural numbers, other than of perfect squares, are irrational. [2] Like all real numbers, irrational numbers can be expressed in positional notation, notably ...
2.1 Irrational numbers. 3 References. ... [2] [3] Little is known ... , the square root of 2. Plato in his Theaetetus, [27] describes how ...
The proof that the square root of 2 (√ 2) is irrational (i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the ancient Greeks, and is perhaps the earliest known example of a proof by infinite descent.
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.
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.
A real number that is not rational is called irrational. [5] Irrational numbers include the square root of 2 ( ), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. [1]
The concept of irrationality was implicitly accepted by early Indian mathematicians such as Manava (c. 750–690 BC), who was aware that the square roots of certain numbers, such as 2 and 61, could not be exactly determined. [7] Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational.