Assignment 1 (pdf)

Make sure to write complete proofs. Try to avoid skipping steps. Write clear sentences.

1. A "multiplicative inverse" for a number \(x\) is a number \(y\) such that \(xy = 1\).
   1. Show that every non-zero rational number has a multiplicative inverse (that is also a rational number). Don't just write \(1/x\), the point of this question is to show that "\(1/x\)" exists. Assume that the rational number is written in the form that rational numbers are, and use that to explicitly write down the inverse.
   2. If \(z\) is a complex number \(a + bi\), then we define the conjugate \(\bar z\) as \(\bar z = a - bi\). Show that the product of a complex number with its conjugate is a real number.
   3. Show that every non-zero complex number has a multiplicative inverse.

2. For this question assume the following for integers, which is the analog of "every integer is odd or even" but using 3 instead of 2 as a factor. We have not proven this assumption, but you may, and will need to, use it. The assumption is: Every integer \(n\) can be written in exactly one of the following 3 ways:

   Type A

   In the form \(3k\) where \(k\) is some integer.

   Type B

   In the form \(3k+1\) where \(k\) is some integer.

   Type C

   In the form \(3k+2\) where \(k\) is some integer.

   Answer the following questions:
   1. Show that if \(n\) is of type C, then \(2n\) is of type B.
   2. Show that if \(m\) is of type A and \(n\) is any integer, then \(mn\) is also of type A.
   3. Show that if \(n\) is an integer, then \(n^2\) cannot be of type C.
   4. Show that for any integer \(n\), the product \((2n+1)(n+1)n\) is of type A.