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:

    “When the leaders choose to make themselves bidders at an auction of popularity, their talents, in the construction of the state, will be of no service. They will become flatterers instead of legislators; the instruments, not the guides, of the people.”
    —Edmund Burke (1729–1797)

    “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)

    “There’s no art
    To find the mind’s construction in the face.”
    —William Shakespeare (1564–1616)

    “Society is a masked ball, where every one hides his real character, and reveals it by hiding.”
    —Ralph Waldo Emerson (1803–1882)

    “Out of the darkness where Philomela sat,
    Her fairy numbers issued. What then ailed me?
    My ears are called capacious but they failed me,
    Her classics registered a little flat!
    I rose, and venomously spat.”
    —John Crowe Ransom (1888–1974)