Let’s start with the basics. At a young age, we were taught how to count with positive numbers, such as one, two or three. Later in primary school, we were also introduced to negative numbers: for example, -15 is a negative number. I’m also going to assume that you are familiar with square roots (if not, you should revise). It is commonly taught to students that one cannot take the square root of negative numbers. But what if we could?

You may be wondering, “How is it possible to take the square root of a negative number?” In fact, mathematicians before the 16th century would’ve thought so as well. This was until Italian mathematician Gerolamo Cardano broke the convention by inventing imaginary numbers, in a desperate attempt to solve cubic equations. Throughout history, mathematicians have always loved to break their own rules: apart from taking the square root of a negative number, Ramanujan once proved that 1 + 2 + 3 + 4… all the way up to infinity is equal to -1/12. Another mathematician, Georg Cantor, proved that there are as many even numbers as positive integers. Therefore, what Cardano did was not uncommon (at least in historical records).

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and -5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits. Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, vibration analysis, and many others.

Imaginary numbers, also called complex numbers, are used in real-life applications, such as electricity, as well as quadratic equations. In quadratic planes, imaginary numbers show up in equations that don’t touch the x axis. Imaginary numbers become particularly useful in advanced calculus.

So, this course is based on an important topic in mathematics named as 'Complex Numbers'.

You will Learn this Course from Scratch.

Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers.

In real numbers, we can represent this number as a straight line.

A real number can store the information about the value of the number and if this number is positive or negative. But in complex number, we can represent this number (z = a + ib) as a plane.

More precisely, the fundamental theorem of algebra asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number.