# Complex Numbers and their Properties

Imaginary numbers as we have discussed in Tuesday’s post are numbers of the form bi where $b$ is a real number and $i = \sqrt{-1}$. The term imaginary as (opposed to real) was first used by Rene Descartes, the mathematician who invented Coordinate Geometry — the Cartesian plane in particular.  Leonhard Euler was the one who introduced the symbol $i$ for $\sqrt{-1}$.

Leonhard Euler

Gerolamo Cardano, a pioneer in probability, was the one who suggested the   use of numbers of the form $a + bi$ where $a$ and $b$ are real numbers and $i = \sqrt{-1}$.  Numbers of this form were named complex numbers by Carl Frederich Gauss.The real part of $a + bi$ is $a$ and the imaginary part is bi.

Note that if we treat these numbers as binomials, then, to add $2 - 3i$ and $4 + 5i$ will result to $(2+4) + (-3i+5i) = 6 + 2i$ .  Similarly, multiplying $(6 - 5i)(3 + i)$ gives us

$18- 9i - 5i^2 = 15 - 9i - 5(-1) = 23 - 9i$.

The following follow from the operations that we have done above: (1) $(a + bi) + (c+di) = (a +c) + (b+d)i$ and (2) $(a + bi)(c+di) = (ac - bd)(ad + bc)i$ Prove this!

We can also observe that $(a+bi)=(c+di)$ if and only if $a = c$ and $b = d$.