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Numbers, and that too IMAGINARY!!

 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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