What was the very first mathematical fact you learned that blew your mind?

 1+2+3+4+5…..= -1/12

Yeah! The fact that the sum of natural numbers till infinity is equal to -1/12 completely blew me.

It's weird and nonsensical to even think that the sum of positive integers till infinity can give a definite value, that too a negative fraction.

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Well, this equation was given by Ramanujan and is known as The Ramanujan Summation. Let's look over the proof now.

Let, A = 1-1+1-1+1-1….

The summation of the above series can be 0 or 1, depending upon the number of terms it has, even or odd respectively.

Now subtracting A from 1 we get,

1-A = 1-(1-1+1-1+1….)

1-A = 1-1+1-1….

Eventually, the Right Hand Side equals A.

So, 1-A = A

Or A =1/2

Now let's take another series—

B= 1-2+3-4+5-6……

Therefore,

A-B = (1-1)+(2-1)+(-3+1)+(4-1)….

A-B = 1-2+3–4+5–6…. = B

A=2B

B= 1/4

Now coming to our main interest, let—

C = 1+2+3+4+5….

C-B = (1-1) + {2-(-2)} + (3-3) + {4-(-4)}….

C-B = 4+8+12….

C-B = 4(1+2+3+4…..)

C-B = 4(C)

3C = -B

C = -1/12

Amazing! Isn't it?

But we know this isn't true. The sum of infinite positive integral series must be an infinity. Hence, the result makes no sense.

Then where are we wrong?

We were wrong at assuming the value of 1+2+3….=C. By doing this, we had already assumed it's value to be a well defined where all the algebraic calculations could be done. But this is untrue. How can we assume it's value to be a real quantity when it is infinite?

But this equation has it's use in String Theory and is still a debatable equation of all time!

SS_

Sources:-

The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12?

Is sum of Natural Numbers upto Infinity -1/12??

Ramanujan summation - Wikipedia

Why is the first AC coach train fare so high?

 Want to take a bath? Yes in the train? You have a shower option in the first Ac coach. That's different will it be clean or not?

Are you a couple? Need privacy? Indian Railway trains give you the option for a coupe and an exclusive private room on the train.

You don’t have to say excuse me sir during the lobby in the First AC train? None legs and hands are spread in the lobby.

You don’t have to beg for lower birth because you are aged or scared to climb on difficult steps of trains because First AC will give you a comfortable ladder.

Wanna taste good foods? Or fresh food direct from the Kitchen? The first AC provide the option of fresh food cooked at the train's pantry and delivered to you before it gets cold.

So tell this much facility that too on train will not be costly? Yea same train where in 3rd AC or sleeper people are fighting that in morning their is no reservations and non reservation people can sit on your seat.

What is Kajol’s superhit first film in the Bollywood industry?

 Kajol made her debut at the age of seventeen. Her first film was Bekhudi which released in the year 1992. The film bombed at the box office but Kajol was noticed.

People term Dilwale Dulhaniya Le Jayenge as her first hit. Dilwale Dulhaniya Le Jayenge is an all time blockbuster. Basically Kajol’s first super hit film is Baazigar which released in 1993.

The film was a commercial hit along with chartbuster music. This was the first time that Kajol and Shahrukh Khan were seen together in a film. Later the pair delivered several hit films for the next two decades.

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What was Amrish Puri's first and last movie?

 Amrish Puri’s has acted totally in 450 films, his first appearance happened in 1955 in an unrecognisable or in an unimportant charachter in the film Bhai-Bhai in which he did the role of a goon.

It surprises that such an iconic figure of Bollywood did very small unimportant roles in several films in the early 1970. Amrish Puri didn't even have proper dialogues in many films. In the 1970’s film Prem Pujari he had a small role of Jerry, a hench man. In this film his brother Madan Puri also acted.

In 1971 he did a supporting role of Rehmat Khan in Sunil Dutt's film Reshma Aur Shera.

Amrish Puri had struggled very much in his early career before he established himself as an iconic villain of Bollywood. In his 50s he got the opportunity to act as a main villain in the films. Throughout 1970s he appeared as an associate to main villain.

For the first time he received wider recognition in 1980s film Hum Paanch in which he performed as a main villain for the first time. He did the role of Thakur Veer Pratap Singh and for the first time his strong performance, impressive baritone voice got noticed..

After this film he got several opportunities to exhibit his dominating screen presence as a main villain in several films like Vidhaata, Shakti, Hero, Meri Jung, Mr. India, Ghayal and Damini throughout 1980s and 90s.

Amrish Puri's last acted film acted is Kachichi Sadak which was released in 2006 in which he did the role of a jailor Hasin Kairanvi.

But his another film Aahat: Ek Ajeeb Kahani was released posthumously in 2010 as he had passed away on 12th January 2005. But this film was shot in 1971.

Amrish Puri for the last time appeared in 2023s film Gadar 2 in a computer generated imagery. His character Ashraf Ali, father of Sakina (Ameesha Patel) was briefly featured in a scene as he was the main villain in Gadar first part which was released in 2001.

So, one can consider that Amrish Puri's last acted film is Kachichi Sadak which was released in 2006.

Amrish Puri had given numerous memorable performances throughout his career as a dominating villain or as a supporting actor. He also acted in Telugu, Kannada, Tamil, Malayalam, Marathi and Bengali films.. His character Mogambo in the film Mr. India and Mola Ram in the Hollywood film Indiana Jones and The Temple of Doom are very popular. Amrish Puri is one of the greatest villains of the Indian cinema.

In which film did Anushka Sharma make her first appearance in Bollywood?

 Anushka Sharma made her first appea rance in the movie,”Rab Ne Bana Di Jo di"(2008) opposite Shahrukh Khan.She played the role of Taani Sharma.The st ory of Rab Ne Bana Di Jodi goes like th is,Surinder Sahni (Shahrukh Khan),see s Taani,who is very naughty and active in a wedding and falls in love with her. After this,her fiance dies in a car accid ent.Taani's father arranges marriage to her rid of grieve shock.Taani agrees an d nods but she tells Surinder that she cannot love him.Now Surinder wants t o win her heart so he transforms hims elf to a Machoman,Raj.Taani immediat ely gets attracted to him while Surinder was simple and sober type.The movie revolves among both characters and S uinder tries to win the love of Taani.Raj is full of Ego but Surinder is Down to Ea rth.Surinder now falls in a difficult situa tion as he wants to reveal his real ident it.Taani slowly start realising that she l oves Surinder not the Egoistic Raj.She joins Surinder in a Dance Competition and the duo wins.The couple jets off to Japan as Mr and Mrs Surinder Sahni.T he film was superhit and directed by A ditya Chopra.She had no formal experie nce in acting but her performance in th e movie was applauded.Anushkas nam e was nominated for Filmfare Award fo r Best Female Debut and Best Female Actress.She started her career with a modelling assignment for the Fashion Designer,Wendel Rodricks (2007).She shifted from Bangalore to Bombay to pursue her career in modelling.Anushk a was spotted by Aditya Chopra in a tal ent hunt programme and selected her. She gave her best in Rab Ne Bana Di Jo di although she was not very beautiful or attractive as other Bollywood heroin es.There was unusual attraction in Tan ni.I saw this movie and became Anush ka fan.

Who is the first Indian to act in a Hollywood movie?

 Sabu Dastagir (Selar Sabu) was an Indian born American actor who was discovered by documentary filmmaker Robert Flaherty at the age of 13.

Sabu was casted in the British film Elephant Boy (1937). He played the role of an elephant driver in the film.

Sabu is best known for the role of Abu in the film The Thief of Bagdad. Some of his other films are The Drum, Jungle Book, Arabian Nights, White Savage, Black Narcissus and Sabu and the Magic Ring.

After becoming an American citizen Sabu also joined the U.S. Army Air Forces. His career declined after World War 2.

Sabu was also considered for the role of Birju (played by Sunil Dutt) in Mother India but was denied a work permit. He never got a chance to appear in a film made in his native country.

He died of a heart attack at the age of 39, three months before the release of his film, The Tiger Walks.

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Who was the first person to receive a Nobel Prize? And in which category?

 When Swedish chemist Alfred Nobel died on 10 December 1896, he left his fortune to individuals or organisations that provide the "greatest benefit on mankind". The very first awarding of five Nobel Prizes took place five years after his death in 1901 – and they remain some of the most prestigious awards in the world. On the anniversary of the first prize-giving ceremony, we look at who won the first Nobel Prizes.

Physics

German physicist Wilhelm Conrad Rontgen circa

In November 1895, German physicist Wilhelm Conrad Rontgen produced and detected electromagnetic radiation in a wavelength range known as X-rays, or Rontgen rays, an achievement that earned him the very first Nobel prize in physics. In 2004, the International Union of Pure and Applied Chemistry named a radioactive element roentgenium after the scientist.

Chemistry

Chemist Jacobus Henricus van ‘t HoffNICOLA PERSCHEID

Dutch chemist Jacobus Henricus van't Hoff was awarded the first Nobel Prize in chemistry for his work on rates of chemical reaction, chemical equilibrium and osmotic pressure – the measure of the tendency of a solution to take in water by osmosis. His work defined the field of chemistry as we know it today.

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Physiology/Medicine

Professor Emil von Behring (1854 - 1917), a German Bacteriologist and pioneer in immunologyGETTY

German physiologist Emil von Behring received the first Nobel Prize in physiology or medicine for his discovery of diphtheria antitoxin, used to prevent or treat against the potentially fatal bacterial infection. Von Behring was known as the "saviour of children" as diphtheria was a major cause of child death.

Literature

French poet Sully PrudhommeBIBLIOTHÈQUE NATIONALE DE FRANCE/WIKI COMMONS

French poet Sully Prudhomme was awarded the first Nobel Prize in literature "in special recognition of his poetic composition, which gives evidence of lofty idealism, artistic perfection and a rare combination of the qualities of both heart and intellect".

Peace

Swiss businessman Henry DunantWIKI COMMONS

Henry Dunant and Frederic Passy were the first recipients of the Nobel Peace Prize. Swiss businessman Dunant was a witness to the battle at the town of Solferino in Italy in 1859. After organising humanitarian aid work, he wrote the book A Memory of Solferino, which stated all countries should form associations to help the sick and wounded on the battlefield – regardless of what side they belong to. The result was the foundation of the International Committee of the Red Cross in 1863 and the adoption of the Geneva Convention the following year.

As an economist and politician, Passy maintained that free trade between independent nations promoted peace. He founded the first French Peace Society and an independent leftist republican in the French Chamber of Deputies, he opposed France's colonial policy as it went against the ideals of free trade. Passy was also one of the founders of the Inter-Parliamentary Union, an organisation for co-operation between the elected representatives of different countries.

Which is the first CinemaScope movie of Hindi and also Indian cinema?

 It’s none other than Guru Dutt’s melancolic masterpiece Kaagaz Ke Phool which released in 1959.

The film which is considered a cult classic marked a technical revolution in Indian cinematography. Along with cameraman V.K. Moorthy, Guru Dutt was able to showcase stunning visuals. through the cinemascope format.

The artistic use of light and shadow generated a great film through the cinemascope format which produces a widescreen picture using an anamorphic lens.

Guru Dutt had promised full creative liberty to VK Murthy regardless of the time it consumed. Anamorphic lenses were brought from abroad for this ambitious project.

The film was a commercial failure but gained popularity with time. As Raj Kapoor had rightly commented to Guru Dutt, “You have produced the picture 15 years too early.”

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What was the first movie of Rekha?

 Being a starkid, the daughter of Tamil actors Gemini Ganesan and Pushpavalli, Rekha faced the shooting camera for the first time as an infant when she was just one year old child. It was a small role in a Telugu film Inti Guttu which was then released in 1958, so this is her first movie.

After that she did a brief role in an another Telugu film Rangula Ratnam in 1966, for this film she got introduced as Baby Bhanurekha, her mother Pushpavalli also did a supporting role.

As a leading actress Rekha‘s first movie is Kannada film Operation Jackpotnalli CID 999 which was released in 1969. Dr. Rajkumar was the leading hero in this film. At the age of around 15 years she became a leading actress.

Her first Hindi movie is Sawan Bhadon in 1970. She did the role of Chanda, a village girl, Navin Nishchol was the leading hero who appeared as a CID officer. This film established Rekha as a rising star.

Though she gave some successful films in her initial stage of her career, she got criticized widely for her outlooks and weight but she took it as a challenge and she changed her outlooks significantly, she improved her acting abilities and also got the command over the Hindi language.

With her amazing transformation she became a popular actress of Bollywood after acting in the films like Ghar, Muqaddar Ka Sikandar in 1978. She is considered as one of the classic and finest actresses of Bollywood, some of her big performances are in the films like Khoobsurat in 1980, Umrao Jaan in 1981, which brought her National Film Award, Utsav in 1984 and Khoon Bhari Maang in 1988.

She acted in over 200 films and won 3 Film Awards and one National Film Award for Best Actress. She is a realistic actress well suited for both commercial and art films. She is known for portraying strong female characters in several films.

Rekha was born in Madras (Chennai) as Bhanurekha in 1954. Being a daughter of a Tamil father and Telugu mother she has fluency in Tamil and Telugu languages. She mentioned that at home they used to talk in English, so she is fluent in English and ofcourse in Hindi also. Currently she is not working in the films, in 2014 in the film Super Nani she did the role of a grandmother who transformers herself into a glamorous model. In 2015 she made her last appearance in the film Shamitabh, playing herself.