Mathematical Duels

Mathematics has always been a special subject for every single of us, some of us are passionate about it and the others passionately hate it.

Mathematics is often called the most pure and natural of the sciences and we often perceive that rest of the science we know emerges from mathematics but is it all pure and clear, probably not.

Mathematics is a field where we evolve stuff where we confirm things, for example if we think about a phenomenon in our everyday nature, it is also possible to visualize it mathematically along with all laws of nature and how it behaves. If a phenomenon is mathematically possible, then it may be practically possible in real life.

On the other hand, what are the things that are only possible mathematically?

Infinity is a very good example of such idea, we often here that the universe is infinite but is it true? Partially, universe is constantly expanding from everywhere and that’s how we consider it infinite.

But true infinity Is just a huge number, a gigantic number that we cannot imagine it, thus we name it infinity and when we encounter such things in mathematics, we play with them.

An incredibly famous example is the ‘The Infinite Hotel Paradox’, it is difficult to understand this paradox in layman’s terms but, think of a hotel which has infinite number of rooms and a manager to manage the hotel. Easy, huh? As the number of rooms is infinite, we can just keep on accommodating customers. Now let’s say a customer comes for a night stay in the hotel. The manager now plans to shift the people in hotel to the adjacent rooms. So, the customer in room 1 will shift to room 2 and so on. But what if there is a bus containing infinite number of passengers come for a night stay in the hotel. The manager would have to think smartly and make room for those infinite passengers in the infinite hotel containing infinite customers. The manager can ask the customer in a room to shift to a room of room no. twice of that of the original room they were living in. So, all the odd no. infinite rooms would be empty and all the even no. infinite rooms to be occupied. The manager can now escort the passengers in the infinite bus to move to all odd no. infinite rooms. Hence the problem is now solved. Easy?

Courtesy of TED talk

Mathematical Duels

Today, Mathematicians publish their theories and proofs in order to establish their efficiency in their field. About a few hundred years ago, mathematicians used to participate in a deadly duel in order to fight against each other’s brilliance, how?

They used to publically challenge each other and present a set of problems for the other to solve which decided the fate of many awards and positions of many prominent universities. Truly harmless, weren’t they?

As you might have guessed when stakes were so high, there were all kind of dirty tricks played. One of the most prominent stories is of the Origin of the Cubic Equation.

The story follows two mathematicians, one Nicole Fontana who actually discovered the cubic formula (Along with one another mathematician who actually died with the formula. So I don’t really think he matters much in this story) and the other is Gerolamo Cardano who actually took the formula out from Fontana with a pledge to never spill it out. Result? Of course, he broke his pledge, blood was spilled and disputes took place.

Moreover, this is but one story of how weird mathematical history has been and telling you every weird aspect of mathematics might not be possible in one article.

The Unreal Real Number

A Quadratic Equation, ax2+ bx + c, said to have at most two roots and in some cases no roots! How is it possible for an equation to be never zero and have such versatility at the same time?

Well, the answer is of course it is possible cause its math and it is sometimes weird but it’s not the case here.

We get almost get 2 roots for this equation every time, how?

For this we will have to think how we constructed our number line in the first place, we started with naturals, added fractions, zero and in the end the negatives.

Now, let’s go back to Fontana and Cardano’s story. So, Cardano ended up the victor of the dispute amoung the two and was happily leaving until one day he was playing with his equation in its slightly different form which was x3= cx + d .

He ended up with a negative root soon into his equation. Now, if I were doing a quadratic equation and ended up with a negative root in 1500s, I would simply say it has no roots but the case was different here. Cardano was dealing with a Cubic Equation here and as per the behavior of a cubic, it must have at least one root. Cardano was completely astonished, so we know we have a real zero as an answer and we have an equation which should work fine with any and everything. What was wrong then?

Cardano knew there has to be an answer but sadly it took one more generation to find that out, Cardano passed his quirky problem to his student Bombelli one day thought about the history of our dear mathematics and its number line and simply said, No existing number can solve this problem and thus, root of negative has to be a new number of itself.

So basically, Bomballi just took it as a new number itself but if that’s the case, this new number isn’t in our real root of the cubic equation and thus Bomballi decided that this new number must cancel out. So Bomballi tried trial and Hit method to first find out the root of the cubic equation and then simple went reverse back and plug in the root of negative one along with a and b in it so that once it solves completely they cancel each other out, now I can’t explain you the mathematics efficiently here but I am sure you will get a better version of it on internet anytime for those who are super curious about this intuitively.

Soon after, He did get his root as the Answer! He got a Real number from a number that never before existed!

What did he do with his discovery?

He basically regarded it as a new toy and not an extraordinary wonder of mathematics.

Little did he knew that almost 400 years later, his discovery will fit into the very fundamental unit of this universe.

Mathematics has always been a very absolute subject and probably will remain this way forever, but this subject hides many wonders that are both mind blowing and just completely not explainable.

So next time when you pick up the Math book of yours, remember, it has the most unique secrets hidden in it trying to squeeze out into your mind to amaze you for the rest of the eternity. Other than that, Math is probably going to be the biggest pillar of Logic and Science anyway