Why does Rudin say "the rational number system is inadequate as a field"? In the INTRODUCTION of chapter 1 of Baby Rudin, he says


  The rational number system is inadequate for many purposes, both as a field and as an ordered set.  


Addition and multiplication of rational numbers are commutative and associative, and multiplication is distributive over addition. Both 'zero' and 'one' exist.

Plus, as I recall, rational numbers are the smallest subfield of $\mathbb{C}$. So exactly what does Rudin mean by &ldquo;inadequate as a field&rdquo;?
Rudin isn't questioning $\mathbb{Q}$'s status as a field &mdash; he's questioning it's suitability for doing algebra and analysis.

For the purposes of algebra, $\mathbb{Q}$ is not a very nice field to work with: for example, many (most?) polynomials don't even have a root, let alone factor completely, and dealing properly with this deficiency is the subject of the frontiers of mathematical research!

(for comparison, while not every polynomial over $\mathbb{R}$ has a root, it has a very simple relationship with the complex numbers $\mathbb{C}$, a field over which every nonconstant polynomial has a root)

Similarly, for the purposes of analysis, $\mathbb{Q}$ is not a very nice ordered set; for example, they form a totally disconnected space, making them nearly useless for capturing even basic familiar geometric notions such as continuity.
The real numbers are a subfield of the complex numbers. Not every sequence of rational numbers converges to a rational number: rational numbers aren't a complete set.