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 “inadequate as a field”?