Complex numbers

The real numbers, although they form an immense set, are not enough to solve every polynomial equation: already $x^2+1=0$ has no real solutions, because $x^2\ge 0$ for every $x\in\mathbb{R}$. In the sixteenth century the Italian mathematicians — Cardano and Bombelli — began to “pretend” to have a number $i$ with $i^2=-1$ in order to solve cubic equations even in the “irreducible” cases. What seemed an algebraic trick, at first judged imaginary or impossible, proved over the following centuries to be one of the most fruitful ideas in mathematics. The complex numbers are the natural closure of the reals with respect to polynomial equations and open the door to the analysis of the fifth year and to the physics of electromagnetism and quantum mechanics.

The chapter introduces the Cartesian form $z=a+ib$ with its operations, the modulus and the conjugate, the geometric representation in the Argand-Gauss plane, the trigonometric and exponential forms with De Moivre’s formula, the $n$-th roots and the fundamental theorem of algebra, up to the generalised Vieta’s relations.

Sections

• Definition and Cartesian form
• The Argand-Gauss plane
• Trigonometric and polar exponential form
• nth roots
• Fundamental theorem of algebra

Exercises

• All the worked exercises of the chapter