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Definitions and main properties

The basic complex number is $$i$$. It is the imaginary unit and is defined as the square root of $$-1$$, i.e. $$$ i^2 = -1$$$ There are two solutions to $$x^2=-1$$, namely $$i$$ and $$-i$$, so a choice is made and $$i$$ is used.

A complex number is of the form $$$ z = x + i y,$$$ where $$x$$ and $$y$$ are real numbers. $$x$$ is the real part of $$z$$ and is denoted $$\Real(z)$$ while $$y$$ is the imaginary part of $$z$$ and is denoted $$\Imag(z)$$.

The set of all complex numbers is denoted $$\mathbb{C}$$. This includes the real numbers.

A complex number is a real number if its imaginary part is 0; it is a (pure) imaginary number if its real part is 0. Zero has an ambiguous position.

A complex number $$$z=x+iy$$$ can be represented geometrically in the plane as the point $$(x,y)$$.

The real part is its $$x$$-value. The imaginary part is the $$y$$-value.

The $$x$$-axis represents the real numbers. The $$y$$-axis represents the pure imaginary numbers.

The plane is called the complex plane, and the diagram is called the Argand diagram .

Representation of a complex number in the plane
Representation of a complex number in the plane

The operations for complex numbers are the same as for the real numbers, with $$i^2=-1$$. Addition and multiplication are both commutative and associative (order and position of brackets do not matter). Multiplication is also distributive (we multiply out brackets in the usual way).

  • Addition: $$$(x_1+iy_1) + (x_2+iy_2) = (x_1+x_2) + i (y_1+y_2)$$$
  • Multiplication: $$$(x_1+iy_1) (x_2+iy_2) = (x_1x_2-y_1y_2) + i (x_1y_2+x_2y_1)$$$
  • Inversion: $$$\displaystyle \frac{1}{x+iy} = \frac{x-iy}{(x+iy)(x-iy)}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2} $$$
  • Division (combining inversion and multiplication) $$$\displaystyle \frac{x_1+iy_1}{x_2+iy_2} =\frac{ x_1x_2+y_1y_2}{x_2^2+y_2^2}+i\frac{ -x_1y_2 + x_2y_1}{x_2^2+y_2^2} $$$
Complex addition in the Argand diagram
Complex addition in the Argand diagram

The conjugate of the complex number $$z=x+iy$$ is defined by $$$ \zo= x-iy.$$$ It can also be denoted $$z^*$$.

We have some simple formulae for conjugation: $$$ \overline{z_1\pm z_2} = \overline{z_1} \pm \overline{z_2},\quad \overline{z_1z_2} = \overline{z_1}\; \overline{z_2},\quad \overline{z^{-1}} = (\zo)^{-1}, \quad\overline{\zo}=z$$$

In the complex plane, $$z$$ and $$\zo$$ are symmetric across the real axis.

Modulus of a complex number
Modulus of a complex number

The modulus of a complex number $$z = x + iy$$ is $$$\displaystyle\vert z\vert = \sqrt{x^2+y^2}$$$ It is the length of the vector $$(x,y)$$.

Since complex numbers add like vectors in the complex plane, and vectors satisfy the triangle inequality, the triangle inequality holds for complex numbers: $$$\vert z_1+z_2\vert \le \vert z_1\vert+\vert z_2\vert .$$$

We also have the identities $$$ \vert \zo \vert = \vert z\vert,\quad \vert z\vert ^2 = z \zo, \quad \vert z_1z_2\vert = \vert z_1\vert \vert z_2\vert. $$$

The triangle inequality
The triangle inequality