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

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

When I was first told by my friend about this, I was just blown away. I thought he is joking around, but then I got to know that this is real. Then I thought that most of the scientists must have rejected this, but no. I was wrong, it is even used in higher physics.

This was and is one of the most mind blowing mathematical facts that I came to know.

It is amazing how each positive numbers are added and the result you get is negative.

Here's the prove which is even more interesting.

The prove given below is just for fun and it is not valid. So if you want to enjoy then only read and if you are a mathematician then please don't read.

Consider

S1= 1-1+1-1+1-1+1-1…..

Now, this sum should be 0 or 1 based on number of natural numbers taken. If infinite numbers are even, S1=0, if odd then S1=1. But, Riemann zeta function gives it a value of ½. Mathematical community too agrees that the sum is ½.

Let

S2=1-2+3-4+5-6+7…..

So, S2=1-2+3-4+5-6+7-8+9…..

S2= 1-2+3-4+5-6+7-8……. I have shifted RHS by a unit position

+2S2=1-1+1-1+1-1+1…..

Hence, 2S2=S1

Therefore, S2=1/4

Let’s come back to our sum of infinite numbers.

S=1+2+3+4+5+6+7+8+9…..

S2=1-2+3-4+5-6+7-8+9….

S-S2=4+8+12+16+20…..

Hence,

S-S2=4(1+2+3+4+5+6+7+8….)

S-S2=4S

So, -S2=3S

And, S = -S2/3 = -1/12

This shocking result is not known to many non-mathematicians. Number-theorists call it “One of the most remarkable formulae in science”. This summation is a secret of mathematics kept away from layman. Further, it is interesting to know ‘S’=-1/12 has been used to derive the equations in “string theory”, quantum field theory and in some complex analytics.

EDIT: here is a new proof for all mathematicians:

I have given the following propositions. If you want a prove of that the please comment.

Proposition 1.

(2.1)

Proposition 2.

(2.2)

We define the function Sn and Hn (x) as follows.

(2.3)

(2.4)

Then we define the following symbols.

(2.5)

(2.6)

(2.7)

(2.8)

Then we have the following propositions.

(2.9)

(2.10)

(2.11)

The double quotes mean the analytic continuation of the sum of natural numbers.

The traditional sum diverges for the infinite terms. On the other hand, the new “sum” is equal to the traditional sum for the finite term. In addition, the “sum” converges on -1/12 for the infinite term.

What mathematical problem was solved in an improbable way?

 In 1939, George Dantzig was a student at the University of California, Berkeley...

… And like many students, he was sometimes late.

One day, when his math class had already begun, Danzig entered class and saw two statistical problems written on the blackboard. Thinking that it was homework for the next lesson, he hurried to write them down without question. And a few days later, he gave the solution to each of these two problems.

Six weeks passed when his statistics teacher (Jerzy Neyman) visited him with unexpected news. Danzig learned that he had solved two of the most famous statistical problems that have not yet been solved.

He later said that the problems "seemed a little more difficult than usual."

---

Even though the name Danzig remains relatively unknown, this anecdote has become popular over time. It may remind you of that famous scene with Matt Damon in the movie Good Will Hunting.

What are some surprising mathematical facts?

 Clean bottle

Many of us do not know what they tell or how to describe them mathematically.

A clean bottle is an object that has no inside and no outside. It is a fixed shape where if you walk through the starting point of a surface, you will never cross one end, and will return to the original place. It is an object of 1 dimension with no edges.

This photo shows how to make a clean bottle

Clean bottle is a surface shape that cannot be made in 3D. It may just look like a closed bottle but you cannot drink water from it. All the water will fall from it.

This is a beautiful picture of a clean bottle. Makes a surprise.

Thank you for reading.

Images: Google image

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.

Google Image

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

What are some of the most interesting mathematical coincidences?

 THE MOST INTERESTING AND MYSTERIOUS MATHEMATICAL COINCIDENCE

The circle have 360 degrees

Well, this is not at all RANDOM

If we go on bisecting the circles into sectors the sum of digits of an angle of every sector always reduces to 9.

360 degree = 3+6+0 = 9

Bisect the circle, now you have

180 degrees = 1+8+0 = 9

Bisecting further

90 degrees = 9+0 = 9

Again bisecting

45 degrees = 4+5 = 9

22.5 degrees= 2+2+5 = 9

11.25 degrees = 1+1+2+5 = 9

5.625 degrees= 5+6+2+5 = 9

This finally ends to a SINGULATITY

Now the another aspect

The sum of digits of sum all angles of a regular polygon inscribed in a circle always reduces to 9.

Triangle - 60+60+60 = 180 = 1+8+0 = 9

Rectangle- 90×4 = 360 = sum of digits = 9

Pentagon- 108×5 = 540 = sum of digits = 9

Hexagon- 120×6 = 720 = sum of digits = 9

And it goes on and on…

CONCLUSION

When we bisected a circle IT ENDED INTO A SINGULARTIY.

Whereas inscribed polygons revealed the EXACT OPPOSITE i.e. “THE VACCUM”

Nikola Tesla said

If you only knew the magificience of numbers 3, 6 and 9 then you would have the key to the universe

This is the beauty of number 9. It simultaneously represents “EVERYTHING” and “NOTHING” . The list is endless, call it a coincidence or mystery but we have not been able to crack the DIVINITY that is embedded in our Mathematics system.