Algebraic Numbers

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

Strong Four Exponentials Conjecture
... The statement of this 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 ... 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 yj for 1 ...
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 ... Another way to construct irrational numbers is as irrational algebraic numbers, i.e ... Suppose you know that there exists some real number x with p(x) = 0 (for instance if n is odd and an is non-zero, then because of the intermediate value theorem) ...
Cantor's First Uncountability Proof - Is Cantor's Proof of The Existence of Transcendentals Constructive or Non-constructive?
... Some mathematicians claim that Cantor's proof of the existence of transcendental numbers is constructive—that is, it provides a method of constructing a ... 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 ...
Lindemann–Weierstrass Theorem
... 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.. ...
Transcendental Number - Properties
... The set of transcendental numbers is uncountably infinite ... and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable ... Cantor's 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 ...

Famous quotes containing the words numbers and/or algebraic:

    Our religion vulgarly stands on numbers of believers. Whenever the appeal is made—no matter how indirectly—to numbers, proclamation is then and there made, that religion is not. He that finds God a sweet, enveloping presence, who shall dare to come in?
    Ralph Waldo Emerson (1803–1882)

    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)