Search results
Results From The WOW.Com Content Network
Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. [1]
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.
An illustration of the irrationality of the square root of 2. Suppose m and n are integers. Then a similar triangle can be constructed with legs m-n and hypotenuse 2n-m, leading to infinite descent. Date: 25 July 2007: Source: Own work: Author: Smjg: Other versions: Image:Irrationality of sqrt2.png (previous, uncoloured version)
English: geometric proof for the irrationality of the square root of 2: If the isosceles right triangle ABC had integer side lengths (AB = q = BC and AC = p = q √ 2), so had the isosceles right triangle A'B'C. Since q < p < 2q, the side lengths of A'B'C are strictly smaller than those of ABC, i.e. p-q < q and 2q-p < p.
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 φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x 2 − ...
The irrationality base or Sondow irrationality measure is obtained by setting (,) =. [ 1 ] [ 6 ] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding β ( x ) = 1 {\displaystyle \beta (x)=1} for all other real numbers:
The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.. An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients.