AHL 1.12—Complex numbers – Cartesian form and Argand diag

Consider the equation $x^2+4=0$. Rearranging gives $x^2=-4$, and since no real number squared gives a negative result, we say there are no real solutions. Mathematicians in the 16th century pushed past this barrier by introducing a new mathematical object: the imaginary unit.

Imaginary Unit: The imaginary unit $i$ is defined by the property:
$i^2=-1$
Equivalently, $i=\sqrt{-1}$.

This single definition unlocks an entirely new number system. Despite the name "imaginary", these are rigorous mathematical objects , just as valid as the real numbers you've always worked with.

Using $i$, we can now write solutions to equations like $x^2=-4$:
$x=\pm \sqrt{-4}=\pm 2i$

Complex Number: A complex number is any expression of the form
$z=a+bi$
where $a,b\in \mathbb{R}$ and $i^2=-1$. The value $a$ is called the real part, written $\text{Re}(z)$, and $b$ is called the imaginary part, written $\text{Im}(z)$.

This is called the Cartesian form (or rectangular form) of a complex number. The set of all complex numbers is denoted $\mathbb{C}$.

Special cases:

• When $b=0$: $z=a$ is a purely real number. Every real number is a complex number.
• When $a=0$: $z=bi$ is a purely imaginary number.
• When $a=0$ and $b=0$: $z=0$, the only number that is both real and purely imaginary.