**Hi, I am Emon and yes, I am gonna write something today, after a long period of time, maybe on some imaginary numbers and in turn, complex numbers?**

Okay, so let's look back into the history of complex numbers which first evolved mainly in the country of Italy...

Back in the $16$th century, a famous Italian mathematician, Niccolo Fontana Tartaglia posed the following problem in a journal :

Can you find a number $x$ such that $x^3+px=q$, where $p, q$ are given numbers?In fact, he had a secret formula to this question : $$\boxed{x=\sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \sqrt[3]{\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}}$$

Later, Italian Mathematician, Gerolamo Cardano proposed the same problem, with the only modification $p\mapsto -p$ :

Can you find a number $x$ such that $x^3=px+q$, where $p, q$ are given numbers?

He went on to give a second problem by asking :

"Can you divide $10$ into two parts such that their product is $40$?"

The people of those days, having no idea of imaginary numbers whatsoever said, "No, it's not possible!".

But, Cardano said, "Why? Just consider two numbers $5+\sqrt{-15}$ and $5-\sqrt{-15}$."

A few more problems involving the same idea were proposed by some other mathematicians as well. Mathematician Rafael Bombelli, another Italian, proposed a problem: Find $x$ such that $x^3=15x+4$.

Now, everyone, at very first sight answered, "$4!$". Bombelli announced that the answer was correct, but he, at the same time, gave a really intriguing solution, which involved the secret formula of Tartaglia...

**Solution of Bombelli.**By the secret formula of Tartaglia, we get, $$\begin{aligned} x &=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}\\ &= 2+\sqrt{-1}+2-\sqrt{-1}\\ &= 4.\end{aligned}$$

So, basically, the solutions are exactly the same...

Later on, $\sqrt{-1}$ was defined as $i$, and was originally coined by Rene Descartes in the $17$th century and in the future, got wide acceptance from Leonhard Euler in the $18$th century and Augustin-Louis Cauchy and Carl Friedrich Gauss in the $19$th century. The imaginary number was combined with a real number to form the complex number $z=a+ib$, where $a, b\in \mathbb{R}$ and $i=\sqrt{-1}$. Hence, the largest set of numbers $\mathbb{C}$, overtaking the reals $\mathbb R$, came into being and the era of complex numbers started.

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