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:
“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)
“The construction of life is at present in the power of facts far more than convictions.”
—Walter Benjamin (1892–1940)
“Striving toward a goal puts a more pleasing construction on our advance toward death.”
—Mason Cooley (b. 1927)
“There is no real teacher who in practise does not believe in the existence of the soul, or in a magic that acts on it through speech.”
—Allan Bloom (1930–1992)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)