2. Imaginary Numbers

L3CALC

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Complex Numbers

Upto now, only ‘real’ numbers were dealt with in maths. Real numbers include:

• Rational numbers,
• Irrational numbers,
• Natural numbers,
• etc..

However, in L3 Complex Numbers, we deal with ‘complex’ numbers — also known as imaginary numbers. Imaginary numbers cannot be plotted on a standard catesian plane with planes $x$ and $y$.

Imaginary numbers are expressed as $i$.
$i$ is equal to $\sqrt{-1}$.

This means that $i^2=-1$, and so on.

So, to ‘plot’ complex numbers such as $i$, we use the Argand plane (or Argand diagram).
An Argand plane consists of 2 axes, similar to the cartesian plane.
Intead of $x$-axis, it is the REAL axis, and instead of the $y$-axis, it is the IMAGINARY axis.

In complex numbers, instead of writing the points in coordinate form like:

$(2,4)$

we plot points in ‘real’, and ‘imaginary’ parts like this:

$2+4i$

No brackets, or commas are needed. And that point, we refer to as $z$. Every complex number $z$ can be written in the form of:

$z=x+iy$

.. and can be plotted on the Argand plane.

These four complex numbers are plotted on the argand plane:

$\left[2+4i,2-4i,-2+4i,-2-4i\right]$

On quadrant 1, there is $2+4i$,
on quadrant 2, there is $-2+4i$,
quadrant 3, $-2-4i$,
and quadrant 4, $2-4i$.

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