Construction of The Real Numbers
See also: Construction of the real numbers#Construction by Dedekind cutsA 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:
“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)
“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 real American type can never be a ballet dancer. The legs are too long, the body too supple and the spirit too free for this school of affected grace and toe walking.”
—Isadora Duncan (1878–1927)
“All ye poets of the age,
All ye witlings of the stage,
Learn your jingles to reform,
Crop your numbers to conform.
Let your little verses flow
Gently, sweetly, row by row;
Let the verse the subject fit,
Little subject, little wit.
Namby-Pamby is your guide,
Albion’s joy, Hibernia’s pride.”
—Henry Carey (1693?–1743)