Complex number and Cartesian form

To solve equations such as $x^2+1=0$ one introduces a “new” unit $i$, called the imaginary unit, which satisfies $i^2=-1$. Starting from it, all complex numbers are built.

Definition — Complex number

A complex number is an expression of the form $z=a+ib$ with $a,b\in\mathbb{R}$ and $i$ a “new” unit satisfying $i^2=-1$. The number $a$ is called the real part of $z$ ($\Re z$), the number $b$ the imaginary part ($\Im z$). The set of all complex numbers is denoted by $\mathbb{C}$.

The real numbers are embedded in the complex ones as those $z$ with zero imaginary part: $a+i\cdot 0 = a$. In this sense $\mathbb{R}\subset\mathbb{C}$, and the notation $z=a+ib$ is called the Cartesian form (or algebraic form) of the complex number.

Links

Topics: Complex numbers
Concepts: Cartesian form · Complex number · Imaginary part · Real part · Cartesian plane · Imaginary unit