Constructions of the real numbers
... real numbers in a constructive way by defining what a real number is and then deducing further properties, opposed to the synthetic approach which only describes the properties of the set of real numbers and postulates the existence of an algebraic system satisfying these properties. Hence, it is re ...
... real numbers in a constructive way by defining what a real number is and then deducing further properties, opposed to the synthetic approach which only describes the properties of the set of real numbers and postulates the existence of an algebraic system satisfying these properties. Hence, it is re ...
Elementary Number Theory
... • For all integers m and n, if m+n is even then m and n are both even or m and n are both odd • The product of any non-zero rational number and any irrational number is irrational • The sum of any two positive irrational numbers is irrational • If a, b, and c are integers and a2+b2=c2, must at least ...
... • For all integers m and n, if m+n is even then m and n are both even or m and n are both odd • The product of any non-zero rational number and any irrational number is irrational • The sum of any two positive irrational numbers is irrational • If a, b, and c are integers and a2+b2=c2, must at least ...
13-integers-and
... divisor less than or equal to the square root of n • Strong inductive proof using the following logic: – Since n is composite, it has a factor a such that 1 n*n > n)
– Thus, n has a diviso ...
... divisor less than or equal to the square root of n • Strong inductive proof using the following logic: – Since n is composite, it has a factor a such that 1
A Basis Theorem for Perfect Sets
... beginning of the next step, but these variables do not achieve a limit over the course of the recursion. When we speak of the least Gk , we are referring to the Gk with minimal index. Initially X and Y are not defined anywhere, Gx = Gy = G0 , and t = 0. In the recursion step, we are given that X and ...
... beginning of the next step, but these variables do not achieve a limit over the course of the recursion. When we speak of the least Gk , we are referring to the Gk with minimal index. Initially X and Y are not defined anywhere, Gx = Gy = G0 , and t = 0. In the recursion step, we are given that X and ...
Full text
... It is again, of course, the second row number that fails to divide the coefficient in the column. For a r r a y s of the type we are studying this behavior is typical. The column entries for C = 35 in the Pascal a r r a y a r e 12, 715, 3432, 3003, 560, 17, with row numbers 12, 13, 14, 15, 16, 17. H ...
... It is again, of course, the second row number that fails to divide the coefficient in the column. For a r r a y s of the type we are studying this behavior is typical. The column entries for C = 35 in the Pascal a r r a y a r e 12, 715, 3432, 3003, 560, 17, with row numbers 12, 13, 14, 15, 16, 17. H ...