Week 2: September 18-22, 2016.
Tutorials:
The tutorials start this week. Although the rooms are booked for two hours,
the actual tutorial go much over one hour. Following the `formal' part of the
tutorial, the TA will be available for additional questions.
Required reading: Re-read Appendices A-D, and read Chapters 1.1 and 1.2.
In class, we covered: Meaning of the notation $P \Rightarrow Q$ and $P \Leftrightarrow Q$. More properties of fields, especially
$ab=0 \Rightarrow\ a=0 \mbox{ or } b=0$. Introduction to
complex numbers: Real and imaginary part, complex conjugate, absolute values, and their geometric interpretation in the complex plane. Basic properties; especially, how to take the inverse of a complex number. Geometry of multiplication of
complex numbers, using polar coordinates. The fact that $\mathbb{C}$ is a field. Triangle inequality $|z+w|\le |z|+|w|$. Fundamental theorem of algebra (without proof). First examples of vector spaces (without formal definition).
Homework: Assignment #2 will probably be posted on Thursday.
Additional homework (not to be handed in):
- Do examples with complex numbers! Problems with solutions can be found
on the web, for example
here
and many other places.
- Section 1.1: 3; Section 1.2: 1, 12, 18, 21; Section 1.3: 1, 19, 20, 22, 31
-
Click
here for another problem regarding the field with four elements. You don't have to do it, but you may find it interesting.
So, what's wrong with
$ 1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1\ \ \ \ \ ??\ \ \ $
Well, it assumes that $\sqrt{z w}=\sqrt{z}\sqrt{w}$ for complex numbers $z,w$, which we never proved. It's true for positive real numbers, using the positive square root. The square root of a complex number is only given up to sign, and there's no consistent convention `fixing the sign' for which this formula would always be true.