Complex numbers

This chapter builds the complex numbers $\mathbb C$, the number system in which every polynomial has a root and multiplication acts as rotation and scaling. Those roots later describe the growth, decay and oscillation of recurrences and differential equations. The chapter assumes only real arithmetic and basic trigonometry.

Lessons

1. Complex arithmetic: Cartesian form $a+bi$, conjugation, modulus, the Argand plane and the field axioms.
2. Polar form: modulus and argument, products as rotations, De Moivre's formula and $n$th roots.
3. Trigonometric and geometric applications: trigonometric identities from exponential forms and loci in the Argand plane.
4. Complex polynomials: roots, factors and multiplicity, the fundamental theorem of algebra and real factorisation through conjugate pairs.
5. Stability of systems: characteristic roots as modes of recurrences and differential equations, and the root tests for stability.
6. Seen questions: past exam questions on complex roots, factorisation and root counts.

Where it leads

The next chapter, vector spaces, uses $\mathbb R$ and $\mathbb C$ as scalar fields and the choice of field changes independence, dimension and the eigenvalues available later.

Vocabulary checkpoints

• Number and point: the Argand plane pictures $a+bi$ as the point $(a,b)$; the picture represents the number but does not replace its algebra (complex arithmetic).
• Modulus and argument: the modulus is a size; an argument is a direction, determined only up to multiples of $2\pi$ (polar form).
• Roots, factors and multiplicity: a root $\alpha$ gives a factor $z-\alpha$ and multiplicity counts how often that factor repeats (complex polynomials).
• Discrete and continuous tests: a mode $r^n$ decays when $|r|<1$, while a mode $e^{rt}$ decays when $\operatorname{Re}(r)<0$ (stability of systems).
• Stable and bounded: in these notes stable means every solution decays to zero, so bounded solutions that do not decay are a separate boundary case (stability of systems).