irrational number


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irrational number

any real number that cannot be expressed as the ratio of two integers, such as π

Irrational Number

 

a number that is not rational (that is, not an integer or fraction). Real irrational numbers can be represented by an infinite non repeating decimal; for example, √2 = 1.41 …, π = 3.14 …. The existence of irrational ratios (for example, the irrationality of the ratio of the diagonal of a square to its side) was known in antiquity. The irrationality of the number π was established by the German mathematician J. Lambert (1766). However, a rigorous theory of irrational numbers was constructed only in the second half of the 19th century. Irrational numbers are divided into nonrational algebraic numbers and transcendental numbers.

irrational number

[i′rash·ən·əl ′nəm·bər]
(mathematics)
A number which is not the quotient of two integers.

irrational number

(mathematics)
A real number which is not a rational number, i.e. it is not the ratio of two integers.

The decimal expansion of an irrational is infinite but does not end in an infinite repeating sequence of digits.

Examples of irrational numbers are pi, e and the square root of two.
References in periodicals archive ?
My favourite (and almost certainly false) story regarding Pythagoras is that of the student who attracted his ire by proving the existence of irrational numbers. Enraged - as, allegedly, one of his core beliefs was that all numbers could be represented as a perfect ratio - Pythagoras drowned his follower to keep the story from spreading.
Considering the existence of two complementary fractals on the sets of rational and irrational numbers accordingly [17], the probability that small variations (fluctuations) lead to coincidences between irrational and rational numbers of small quotients is minimum.
For a given continuous F : T [right arrow] T, Theorem 4 and its corollary shed no light on how to determine which irrational numbers can be realized by rotational subsystems of (T,F).
(1) Regardless of the fact that the determined bit of the test data is 0 or 1, it has a little effect on compression ratio in the process of finding the irrational number. (2) Do not-care bit can accelerate the search speed of irrational number, reduce the value of irrational number used to store the test data, and improve the compression ratio.
Niven: Irrational Numbers, MAA, John Wiley & Sons, Inc., 1956.
The irrationality of [pi] was not proved until 1761 by Johann Lambert (1728-77), then in 1882 Ferdinand Lindemann (1852-1939) proved that [pi] was a non-algebraic irrational number, a transcendental number (one which is not a solution of an algebraic equation, of any degree, with rational coefficients).
It is usually simply stated the real numbers are comprised of rational and irrational numbers. It is very common to start from such a generality.
The digits of the irrational number pi (3.1415926535897932384626 ...) not only go on forever but behave statistically like a sequence of random numbers uniformly distributed on 0,1, ...,9; no matter how long it may be, any finite subsequence of digits will appear an infinite number of times.
For older rivers, which will have had a chance to develop lengthy, meandering courses, the ratio often approaches 3.14, similar to the value of pi, the irrational number that links the radius of a circle to its circumference and area.
In contrast, Pi is like s, what mathematicians call an "irrational number," that is, 3.14 if rounded off, but with endlessly unfolding decimal places if carried out.
In terms closer to everyday language, it asks the following question: can one speak of the existence of an irrational number, for instance, a number familiar to each one of us, known as the "square root of 2"?
The infinite binary sequences generated here by an irrational number still have MP.