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(a) ILe Su (b) The suт of two irrational numbers.

(a) ILe Su (b) The suт of two irrational numbers.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...
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Part b please
perfectly acceptable as long as we make our assumptions explicit. However we need to be aware that, so far, the existence and properties of the real numbers is an assumption that has not been logically derived. Any time we make an assumption we need to be prepared to either abandon it completely if we find that it leads to nonsensical results, or to re-examine the assumption in the light of these results to see if we can find another assumption that subsumes the first and explains the (apparently) nonsensical results. Additional Problems Problem 11. Determine if each of the follouing is always rational or always irrational. Justify your answers. (a) The sum of two rational numbers. (b) The sum of two irrational numbers. (c) The sum of a rational and an irrational number. Problem 12. Is it possible to have two rational numbers, a and b, such that a is irrational? If so, display an erample of such a and b. If not, prove that it is not possible.
Transcribed Image Text:perfectly acceptable as long as we make our assumptions explicit. However we need to be aware that, so far, the existence and properties of the real numbers is an assumption that has not been logically derived. Any time we make an assumption we need to be prepared to either abandon it completely if we find that it leads to nonsensical results, or to re-examine the assumption in the light of these results to see if we can find another assumption that subsumes the first and explains the (apparently) nonsensical results. Additional Problems Problem 11. Determine if each of the follouing is always rational or always irrational. Justify your answers. (a) The sum of two rational numbers. (b) The sum of two irrational numbers. (c) The sum of a rational and an irrational number. Problem 12. Is it possible to have two rational numbers, a and b, such that a is irrational? If so, display an erample of such a and b. If not, prove that it is not possible.
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