Complex number

The equation \(x^2+1=0\) has no real solution, and the idea is simple: add a symbol \(i\) with \(i^2=-1\) and compute with it like with any letter, using that fact to simplify. Numbers \(a+bi\) then add and multiply by the usual rules: \((a+bi)+(c+di)=(a+c)+(b+d)i\) and \((a+bi)(c+di)=(ac-bd)+(ad+bc)i\). Two complex numbers are equal exactly when their real and imaginary parts agree.

The reals are the complex numbers with \(b=0\), so \(\mathbb{R}\subset\mathbb{C}\) and the operations extend the familiar ones. Numbers of the form \(bi\) are called pure imaginary. Since \(\mathbb{C}\) is a field, every \(z\neq0\) has an inverse: multiply numerator and denominator by the conjugate, the next topic.

Geometrically, \(a+bi\) corresponds to the point \((a,b)\) of the plane, with the horizontal axis for the real part and the vertical for the imaginary part. Adding complex numbers is adding vectors. Multiplication will get its own geometric reading (rotation and scaling) once we have modulus and polar form.

Two common mistakes: calling \(bi\) the \u201cimaginary part\u201d instead of \(b\), and writing \(i^2=1\) or leaving \(i^2\) unsimplified. Also, unlike \(\mathbb{R}\), there is no order on \(\mathbb{C}\) compatible with sum and product: expressions like \(z<w\) make no sense; only moduli are compared.