Transcription
A middle-class guy decided to save money. On the first day, he saved one rupee. On the second day, two rupees. On the third day, three rupees. He continued saving like this. Now, after infinite days, when he checked his bank account, there was a negative balance. It feels a bit weird, right? Exactly like this, in 1913, a Cambridge mathematician felt the same weirdness. He saw a letter. Who wrote this letter? Where did it come from? No formal education, no degrees. Schools rejected him. But still, the whole world accepted him. In 1913, at Cambridge University, a famous mathematician, G.H. Hardy, a very prominent mathematician, was sitting at his desk after completing his classes for the day. Every day, he receives a lot of letters from all over the world. Many people send him letters. But the letter he received that day was special. He was going through all his letters. In one letter, the symbols and everything were a bit different. The handwriting was also not very good. There were direct theorems without any proofs. So he thought it was just some random letter and, looking at it superficially, he just put it aside and went to sleep. But that changed his life. How? Though he neglected it that day, for one or two days, some of the theorems he had seen inside kept haunting him the whole day. So, once again, the next day, he went there and looked at the letter again, in detail. In that letter, there was an interesting equation: 1 + 2 + 3 + 4 + ... so on up to infinity = -1/12. He felt that this equation was something special. So, he called Littlewood, a famous mathematician he worked with at that time. He called him and showed him this equation. The proof for that equation was also in the letter. So, seeing that, they were very surprised. And after that, there were many different theorems along with it, without proofs, but many different theorems. Seeing these, what Hardy said to Littlewood was, "Either this man is a genius or he is a big fraud." Where did that letter come from? It came from India. That letter came from India. And the person who wrote that letter was a person without any formal education, without any degree, a simple clerk. That letter was written by our Srinivasa Ramanujan. And this story is about that equation. Now, let's try to prove that equation. It's very interesting, anyone can understand it. So, to prove 1 + 2 + 3 + 4 + ... up to infinity, first, let's take this as S. Okay. But before solving this, let's take a few more equations. What is the first equation we should take? 1 - 1 + 1 - 1 + 1... so on up to infinity. Okay. Let's call this equation S1. Let's prove S1 first. See, it's very simple. S1 = 1. Now, if we take out the minus sign from these, if I take minus common, what will we get? - (1 - 1 + 1 - 1 + ... so on up to infinity). If you look carefully, what is this? This is S1 itself. So, I can directly write S1 = 1 - S1. Now, if this S1 comes here, then 2S1 = 1. So, from this, I will get S1 = 1/2. This is one of the proofs which Ramanujan has given. This is also one of the equations for an infinite series. Ramanujan has given many theorems on such infinite series. In fact, I'll tell you an interesting story. Ramanujan, in the evening, after all his work, was solving different different books. One day, Hardy entered the room and saw a lot of papers solved and folded like this, just thrown into the bin. He looked at it, wondering what he was solving. When he looked, he saw that these were equations, very prominent unsolved equations. That is, equations that no one had solved. He just solved them roughly, folded them, thinking "it's done," and put them aside. He had that level of knowledge. Then Hardy asked Ramanujan, "Actually, how are you getting these ideas to solve this?" You won't believe what Ramanujan said. He said, "When I go to sleep at night, the goddess Namagiri comes in my dreams and tells me the integrals of this, the integral of that, these series, those series. I just write them down as they are. Those are my theorems." Surprising, isn't it? In fact, if you ask someone, if I ask, if this happens, any psychologist will say that it is hallucination. But what Hardy said is, "This is intuition beyond training." Some minds work with logic, some minds work with patterns. But somehow, Ramanujan's mind works with reality. He used to feel the numbers. He used to feel the numbers. Understood. Coming back to our series. Here, we need to prove one more interesting series before proving that. Okay. 1 - 2 + 3 - 4 + 5... so on up to infinity. Let's call this S2. I will take this as S2. I will write it below the same S2. And what I will do is, I will write the same equation, leaving one step. So, 1 - 2 + 3 - 4... so on up to infinity. Now, let's add these two. If we add these two, S2 + S2 = 2S2 = See here, look carefully. 1 - 2 + 1 - 1 + 3 - 2 + 1 - 4 + 3 - 1 + 1... so on up to infinity. This looks familiar, right? This is the proof for S1, which is 1/2. So, 2S2 = This we have already proved. This is 1/2. So, S2 came out to be 1/4. So, using this S1 and S2, we will prove S. See, we will take one more thing now. We know S. We know 1 + 2 + 3 + ... so on up to infinity. Now we know S2, right? I will write S2 here. S2 is 1 - 2 + 3 - 4 + ... so on up to infinity. Okay. Now, what I will do is, I will just subtract. I will subtract these two. If we subtract, what will we get? S - S2. I will get. See, 1 cancels, 3 cancels, 5 cancels. So, 2 will become plus. This will become plus. So, 2 + 2 = 4. Here, next is 8. Next, here is 12. It will come like that. Now, you see, S - S2 = 4 common. If we take 4 common, what will come? 1 + 2 + 3 + 4. What is this equation? This equation is S. So, S - S2 = 4 * This is S. So, if I bring this side, I will get 3S = If it comes here, 3S = - S2. And we already have S2. We have proved S2. We know S2. We have already done S2. 2 is 1/4. If we put S2 here as 1/4, -1/4. What are we getting? We are getting S = -1/12. These will come down to 3. So, what it means is, it means 1 + 2 + 3 + ... so on up to infinity = -1/12. So, at the beginning, an example was given: 1 + 2 + 3 + 4 + ... so on up to infinity is negative. So, this is that proof. It doesn't sound intuitive to hear. Because this is infinity. Okay. Ramanujan is called. I always call Ramanujan the "Bap of Infinity." Okay. I consider him a man who knew infinity. There is even a movie about it, "The Man Who Knew Infinity." He died at a very young age, like 30-32 years, because that climate, Cambridge University, the food, the different climate, his body couldn't withstand it. Imagine if he had lived another 30-40 years, he could have been compared to great mathematicians like Euler, Renard Descartes, or even Newton. Even Hardy, when asked by the press to rate mathematicians, was asked, "How do you rate yourself?" He gave himself 25 out of 100. Littlewood? He gave 30 out of 100. At that time, one of the most famous mathematicians, Hilbert, "How do you rate him?" He said 80. But when they asked about Ramanujan, he said, "I would rate 100." See? Without any formal education. Okay. And from a very rural background, if he can think this much, if he can feel the numbers, he can feel the patterns. Don't you think the people of this generation also have to take some lessons from him? It's not always about learning the subject, how many marks you got. It's not about writing answers. It's about asking questions. Think about being a pathbreaker. Marks guarantee a job. But the curiosity inside your brain will guarantee discovery. So, run behind this curiosity and this discovery.