Reciprocals of Irrational Numbers
Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers often can prove useful for reasons linked to the irrational number in question. Examples of this are the reciprocal of e which is special because no other positive number can produce a lower number when put to the power of itself, and the golden ratio's reciprocal which, being roughly 0.6180339887, is exactly one less than the golden ratio and in turn illustrates the uniqueness of the number.
There are an infinite number of irrational reciprocal pairs that differ by an integer (giving the curious effect that the pairs share their infinite mantissa). These pairs can be found by simplifying n+√(n2+1) for any integer n, and taking the reciprocal.
Read more about this topic: Multiplicative Inverse
Famous quotes containing the words irrational and/or numbers:
“It is not to be forgotten that what we call rational grounds for our beliefs are often extremely irrational attempts to justify our instincts.”
—Thomas Henry Huxley (1825–95)
“The barriers of conventionality have been raised so high, and so strangely cemented by long existence, that the only hope of overthrowing them exists in the union of numbers linked together by common opinion and effort ... the united watchword of thousands would strike at the foundation of the false system and annihilate it.”
—Mme. Ellen Louise Demorest 1824–1898, U.S. women’s magazine editor and woman’s club movement pioneer. Demorest’s Illustrated Monthly and Mirror of Fashions, p. 203 (January 1870)