MATH 23a FALL 2001 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Homework Assignment 1 Due September 21 2001 1 Read the Appendix and Section 1 1 of Edwards 2 Verify that the operations addition and multiplication are well defined for rational numbers as defined by equivalence classes of ordered pairs of integers 3 Considering the real numbers as defined by equivalence classes of Cauchy sequences of rational numbers name the equivalence class that acts as the multiplicative identity and verify that it does 4 Considering the real numbers as defined by equivalence classes of Cauchy sequences of rational numbers prove the existence of multiplicative inverses for elements other than the additive identity 5 Show that the field Z 2Z is not an ordered field That is show that there is no possible choice for a set P of positive elements such that axioms P1 P3 hold 6 Show that additive inverses in fields are unique That is show that given a F there exists a unique element b F such that a b 0 7 Use definition of limits to show that the sequence of real numbers the 1 converges to 0 sin n n 1 8 Use the definition of Cauchy sequence to show that the sequence of 1 is a Cauchy sequence rational numbers n2 n 1 9 Let x be a real number Show that there is an integer k such that x may be represented in the form x X ai 10 i ak 10 k ak 1 101 k a 1 101 a0 a1 10 1 a2 10 2 i k where ai 0 1 2 9 for every i 1 Show that this representation is unique except in the case where there exists some n N such that one such representation has ai 0 for all i n and one such representation has ai 9 for all i n 1 10 In the following exercise we begin a construction of a different completion of the rational numbers Q The eventual result is the 2 adic field Q2 instead of the usual real numbers R For any non zero x Q write x 2n ab where ab is in lowest terms that is a and b have a greatest common divisor of 1 and 2 divides neither a nor b Define a new absolute value 2 on Q as follows x 2 2 n We also define 0 2 0 a Show that this absolute value satisfies the usual rules i x 2 0 for all non zero x Q ii xy 2 x 2 y 2 for all x y Q iii x y 2 x 2 y 2 for all x y Q b Show that not only is the Triangle Inequality valid for this absolute value but the following Ultra metric Inequality is also valid iv x y 2 max x 2 y 2 for all x y Q 2