# Algebraic Numbers

### Some articles on numbers, algebraic numbers, number, algebraic:

Lindemann–Weierstrass Theorem
... theorem is a result that is very useful in establishing the transcendence of numbers ... αn are algebraic numbers which are linearly independent over the rational numbers Q, then eα1.. ... αn are distinct algebraic numbers, then the exponentials eα1.. ...
Cantor's First Uncountability Proof - Is Cantor's Proof of The Existence of Transcendentals Constructive or Non-constructive?
... claim that Cantor's proof of the existence of transcendental numbers is constructive—that is, it provides a method of constructing a transcendental number ... Cantor's proof is not "constructive," and so does not yield a tangible transcendental number ... If we set up a definite listing of all algebraic numbers … and then apply the diagonal procedure …, we get a perfectly definite transcendental number (it could be computed to any number of decimal ...
Strong Four Exponentials Conjecture
... deals with the vector space over the algebraic numbers generated by 1 and all logarithms of non-zero algebraic numbers, denoted here as L∗ ... So L∗ is the set of all complex numbers of the form for some n ≥ 0, where all the βi and αi are algebraic and every branch of the logarithm is considered ... Let x1, x2, and y1, y2 be two pairs of complex numbers with each pair being linearly independent over the algebraic numbers, then at least one of the four numbers xi ...
Transcendental Number - Properties
... The set of transcendental numbers is uncountably infinite ... are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable ... diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable so the set of all transcendental numbers must also be uncountable ...
Irrational Number - Transcendental and Algebraic Irrationals
... Almost all irrational numbers are transcendental and all real transcendental numbers are irrational (there are also complex transcendental numbers) the article on transcendental numbers ... Another way to construct irrational numbers is as irrational algebraic numbers, i.e ... Because the algebraic numbers form a field, many irrational numbers can be constructed by combining transcendental and algebraic numbers ...

### Famous quotes containing the words numbers and/or algebraic:

Think of the earth as a living organism that is being attacked by billions of bacteria whose numbers double every forty years. Either the host dies, or the virus dies, or both die.
Gore Vidal (b. 1925)

I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
Henry David Thoreau (1817–1862)