Upon my reading of the preface, I am apprehensive to Penrose's claims. I do not have faith that this book is suitable to a lay audience, even one with prior experience to calculus. I do not believe that his explanation of the rational number system sheds much light to readers without logical backgrounds, and I can only extrapolate that his prose will be similar to this. Maybe this book isn't for the lay reader, but that's a plus to me. It's better suited to me!<br><br>I did not enjoy the first three chapters. Here the book mainly focuses on philosophy. I hold my own opinions on the subject, but I am glad see it included for those readers who haven't thought about these topics.<br><br>Chapter 3 and 4 were interesting and even enlightening at some points. I now see how a slide rule works is actually very simple! Though the most important thing I learned (which I'm a little embarrassed that I didn't learn this earlier) is that the exponential operation, say z' is multivalued for arbitrary complex entries. I spent quite some time playing around with this and seeing where all the solutions lived in the complex plane (these spirals are quite pretty).<br><br>Here is something that I have been thinking about as well: Let clog denote what I will call the complex logarithm, and log the standard logarithm. The difference is how I interpret the input, as a real number or a complex number without an imaginary part. Recall then that<br>clog(z)=log(|z|)+i'+i2'k<br><br>and thus we immediately see that  clog(1)=i2'k. This isn't difficult, but it is unintuitive*. A second example of interest is<br>ii=eiclog(i)=ei'(i'/2+i2'k)=e''/2e'2'k<br><br>If I ignore how I constructed these solutions (i.e. forget which k corresponds to the principal logarithm), I do not see which solution should be the "intuitive" solution, as they are all real! Penrose promises that the multivaluedness of these complex functions will become apparent later. Apart from winding, I'm not sure how.<br><br>* Apparently unintuitive is not a word. Even Merriam-Webster was not able to help me find the correct wording!