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# Definitions and main properties

## Definition of a complex number

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.

## Cartesian representation of complex numbers

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

## Arithmetic operations for complex numbers

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 conjugates

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