Skip to main content

A never Ending Contradiction

 Things can get weird when we deal with infinity. Consider the following sum:

Grandi

S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...


It’s called Grandi’s series, after Italian mathematician, philosopher, and 

priest Guido Grandi (1671-1742).


It is weird but grouping the series in different ways gives different results. 


Let’s look at the first way,


  1. S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...

S = (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + (1 ...

S = 0


Here, it is easy to say that the answer is 0 as each individual bracket is equal to 0.


There’s another way of solving this,


  1. S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...

S = 1 + (- 1) + 1 + (-1) + 1 + (-1) + 1 + (-1) + 1 …

S = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + …

S = 1


Now, we have an answer which is equal to 1.


Again,

What if we subtract 1 from both sides of the equation? it is obvious that it will remain the same. Or will it?

Let’s find out,


  1. S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...

1 - S = 1 - ( 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 …)

1 - S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 … 

1 - S = S

2S = 1

S = ½


We’ve got S = ½ 


So, what is the actual answer to this series? Actually, no one knows and it is quite controversial because one can reach towards 2 conclusions,

  • The series 1 − 1 + 1 − 1 + ... has no sum.

  • The sum is 1/2 .

 

Infact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts. 

After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.

 

-Niraaj Ojha (niraaj.ojha@gmail.com)

Comments

Post a Comment

Popular posts from this blog

Mathematics in Criminal Justice

            “There’s a dead body inside the hotel room!!! I’m really scared… Please arrive on time.”   The police did arrive, but not quite on time. By the time they arrived, people had already gathered outside Hotel Chandani with a stressed look on their faces. The police entered the room and saw a man stained with blood on his head, and on his back — they immediately declared him dead seeing no signs of life whatsoever. “Definitely not a gunshot” - a police officer insisted as if he was a hundred percent sure. Immediately after, another police officer reached the dead body and measured his temperature while the other officers looked around if the murderer left some clue. The police stayed in the hotel for another hour and a half: some closely inspecting the crime scene, some talking to the owner (who found it difficult to even utter a word), and some asking the gathered mass to return to their places. Before leaving the hotel, the same officer who to...
6 comments
Read more

Remembering Katherine G. Johnson : Mother, Mathematician, and a Pioneer for Space Exploration

Many people spend decades writing about an important figure, who has impacted not the life of a single person but an entire race. Writing about this important of a person is really tiresome and it takes sometime to process the fact that one has been endowed with this amazing opportunity.      For me, It all began with a simple google search of Katherine Johnson's name.   Portrait of Katherine Johnson via. NASA   I like to view this as a simple google search that changed the way I perceived the world of Mathematics. An African-American lady who was rejected the first time she applied to NACA ( National Advisory Committee for Aeronautics, which was later changed to NASA ) changed the way I view Mathematics.      Katherine Johnson was simply put an extraordinaire. She was an amazing Mathematician who helped NASA lay it's foundations for Space Travel and also is an inspiration to many and one of the very few awardee of Presidential Medal of Freedom ( ...
1 comment
Read more

Why is ‘6174’ so magical?

  Why is ‘6174’ so magical? Let's do some magic today. The special number for this is ‘6174’ . Yes, you heard it correctly. You may be asking what makes this number so significant when it appears to be just another number. Let me tell you that this is a magical number since it connects virtually all four-digit numbers. You can easily understand this if you follow the simple steps of subtraction. Let's unravel the mystery: Step 1:   Choose any four-digit numbers, using at least two different digits (1111,7777 are not allowed). Numbers having leading digits (like 0321,0045) are also allowed.         I take the current year: 2021 Step 2:   Arrange the digits in ascending and descending order, which can have leading zeros if necessary.            Ascending order: 0122            Descending order: 2210 Step 3:  Subtract th...
3 comments
Read more