Dedekind Cut - Construction of The Real Numbers

Construction of The Real Numbers

See also: Construction of the real numbers#Construction by Dedekind cuts

A typical Dedekind cut of the rational numbers is given by

This cut represents the irrational number √2 in Dedekind's construction. To establish this truly, one must show that this really is a cut and that it is the square root of two. However, neither claim is immediate. Showing that it is a cut requires showing that for any positive rational with, there is a rational with and The choice works. Then we have a cut and it has a square no larger than 2, but to show equality requires showing that if is any rational number less than 2, then there is positive in with .

Note that the equality b2 = 2 cannot hold since √2 is not rational.

Read more about this topic:  Dedekind Cut

Famous quotes containing the words construction of the, construction of, construction, real and/or numbers:

    “No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
    —T.S. (Thomas Stearns)

    “There is, I think, no point in the philosophy of progressive education which is sounder than its emphasis upon the importance of the participation of the learner in the formation of the purposes which direct his activities in the learning process, just as there is no defect in traditional education greater than its failure to secure the active cooperation of the pupil in construction of the purposes involved in his studying.”
    —John Dewey (1859–1952)

    “The construction of life is at present in the power of facts far more than convictions.”
    —Walter Benjamin (1892–1940)

    “Shakespeare carries us to such a lofty strain of intelligent activity, as to suggest a wealth which beggars his own; and we then feel that the splendid works which he has created, and which in other hours we extol as a sort of self-existent poetry, take no stronger hold of real nature than the shadow of a passing traveller on the rock. The inspiration which uttered itself in Hamlet and Lear could utter things as good from day to day, for ever.”
    —Ralph Waldo Emerson (1803–1882)

    “The land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it.”
    —Bible: Hebrew Numbers 35:33.