Number Devil

This blog post is inspired by another blog post – Number Devil.

I really enjoyed reading the book “The Number Devil”. A lot of interesting mathematical concepts are explained in very friendly way – a boy named Robert who always dreams of fishes and ants and falling into a hole suddenly starts dreaming of numbers and there he meets the number devil. But he is not a devil at all, he shows him some of the beautiful things that numbers can do. And here I am sharing some interesting stuffs I found while reading the book.

1. In the very first dream of numbers that Robert gets, the number devil shows him the following mathematical trick.

$1 \times 1 = 1$

$11 \times 11 = 121$

$111 \times 111 = 12321$

$1111 \times 1111 = 1234321$

The pattern seems to go on forever, but it doesn’t.

$1111111111 \times 1111111111 = 1234567900987654321$

2. Take any number larger than $2$ and multiply it by $2$. Let’s take $9$. $9 \times 2 = 18$. Now between $9$ and $18$ there are three prime numbers $11$, $13$ and $17$. Similarly there will always be at least one prime number between a number ‘$n$’ which is larger than $2$ and another number ‘$m$’ that you get by multiplying ‘$n$’ by $2$.

3. Take any even number greater than $2$, you can always find two prime numbers which add up to that number (Goldbach Conjecture). For example take $96$. The two prime numbers that add up to $96$ are $43$ and $53$. $43 + 53 = 96$

4. In one of his dreams the number devil talks about his friend Bonacci and the Bonacci numbers. A sequence of numbers that starts from 1. The sequence appears when you count the growth of rabbits and also when you count the number of branches in a tree!

Fibonacci devised a thought experiment to figure out the growth of rabbits in a year. So he assumed that no rabbit dies in one year span and a rabbit takes a month to grow up from its soft white fur to brown. These assumptions are not realistic but it made the problem easier.

So according to his assumption next month two rabbits are born. Then in a month they also grow up and the first mother rabbit gives birth to another pair of rabbits which add up to 3 pairs and so on.

So the sequence that we get is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… The numbers follow a simple pattern. Every number can be obtained by adding two previous numbers. So the next Fibonacci number will be $55 + 89 = 144$.

5. In another of Robert’s dream he and the number devil build a large triangle by piling small cubes. The cubes are arranged as the bricks are arranged in a wall. On the very first cube number devil writes 1. And then in another row two 1’s. The number in each cube is the sum of the numbers written in the cubes above it.

When they completely fill the triangle with numbers, the number devil lights up all the even numbers in the triangle and there emerges a beautiful pattern that Robert watches in amazement.

He then shows some other patterns made by numbers when he lights up all odd numbers and then all the numbers divisible by 5. He tells Robert that you can also find natural numbers, triangle numbers, Fibonacci numbers etc. in the triangle. Another interesting things that you can do is if you have to add all the numbers, say from 1 to 9 then you just have to check what’s the 9th number in the row of triangle numbers which is the third row in Pascal’s triangle. And the number is 45. So all the numbers from 1 to 9 add up to 45 and you get the answer without actually adding every single number!

2018

To celebrate the new year, many posts have been written to share the properties of number $2018$ . And its been pointed out that since its a semi-prime number (a natural number that is product of two prime numbers) its somewhat less interesting.

But still there are lots of beautiful properties and in this blog post I am compiling some properties related to the number 2018 which I found interesting.

-2) In the blog Math with Bad Drawings, Ben Orlin have written about the special days in this year, like the golden ratio day 1-6-18. And we can celebrate it on 6th January or 1st June.

Golden ratio Φ can be approximately written as $1.618...$

And what I found most interesting was a comment on this blog post by michael tamblyn. He points towards the date 2-7-18 which is actually the exponential number ‘e’.

e can approximately be written as $2.718...$

So we can celebrate e-day on 2nd July or on 7th February.

[Both of these dates for phi day and e day are not official but e day is celebrated every year on 2nd July or 7th February]

-1) In the blog Life Through A Mathematician’s Eyes authored by Ioana, she has shared another interesting property. If you write the number 2018 in reverse i.e. 8102 it can be written as the multiplication of 2 prime numbers just like 2018.

$2018 = 2\times1009$

$8102 = 2\times4051$

0) Allex Bellos in the gaurdian posted an awesome puzzle which is to turn the numbers written below into an equation using any one of the operations ÷,+,-,×. And the numbers must be written in the countdown sequence.

${1 0 9 8 7 6 5 4 3 2 1 = 2018}$

Same puzzle can be solved with some different rules e.g. forget about the sequence and use any number you want.

1) In the blog Learn Fun Facts Edmark M. Law has shared this really amazing equation.

2) I tried to find prime numbers using the digits 2, 0, 1, 8

$2+0+1+8 = 11$ (prime number)

$2{^2} + 2{^0} + 2{^1} + 2{^8} = 263$ (prime number)

Constellations

There are 88 official constellations according to International Astronomical Union.

Recently 8 unofficial constellations were invented by the University of Birmingham UK in partnership with The Big Bang Fair.

All eight constellations can be seen here.

I am only including pictures of some constellations from the Twitter profile of University of Birmingham Observatory.

Five days ago when I went to see the night sky, it was very clear. I watched the Orion constellation and Pleiades star cluster after a long time.

But due to bad weather we were not able to watch the Geminid meteor shower yesterday.

Even though these pictures are giving a sense of how Harry Potter’s glasses or Paddington’s boots can be seen, I have to wait to see them in a real sky. Because even today the sky is not clear so I can only imagine how these constellations will look like!

Particles

How many fundamental particles are there in the universe? Answer to this question was different in past and can also be different in future. Because particles were discovered gradually. Many-a-times the theory predicted a new particle and other times the results from particle detectors required a new particle.

What is interesting is that each particle has its own story. And I want to share some of them here including a brief about the heroes of those stories – the physicists and experimentalists.

1.Murray Gell-Mann.

Born -15 September 1929 (age 88)

Gell-Mann arranged hadrons in beautifully simple geometric patterns. Hadrons include mesons and baryons. He arranged these particles in octets and decuplets. And he called this arrangement “Eightfold way” in a joking manner. It was independently discovered by Yuval Ne’eman (14 May1925 -26 April 2006) also.

Here Gell-Mann is holding in his hands the bubble chamber photograph which proved the existence of the Omega minus baryon. It was predicted by his “Eightfold way” of arranging hadrons. And with him is the physicist Yuval Ne’eman.

In this picture Ω¯ (omega minus) baryon is produced when K⁻ meson interacts with proton. And it also shows the decay of particles. Ω¯ decays into ≡° (Xi not) and a π¯ (pi minus) meson. The dashed lines are the path of neutral particles.

But why Hadrons can be arranged in such a way? And the answer came again from Gell-Mann and independently from Geroge Zweig.

2.George Zweig

Born: 20 May 1937 (age 80)

He discovered that the reason that hadrons can be arranged in such a way is because of a deeper symmetry. That is the hadrons are made of fundamental particles and he named them “aces”. Gell-Mann called them “quarks”. Even though Gell-Mann didn’t believe that they were real particles but George did.

At that time only three quarks were predicted: up, down, strange.

But the real problem was the quarks themselves could not be detected. A hypothesis called “quark confinement” was proposed which simply says that quarks are confined inside hadrons thus cannot be detected separately. But it was still a hypothesis and why it should be true was not known.

Later a new heavy meson was discovered in the year 1974 by two different detectors and it was found to be bound state of fourth quark named “charm”.

3.Samuel Chao Chung Ting

Born: 27 January 1936 (age 81)

At Brookhaven laboratory with his team he discovered the new heavy meson particle and named it (psi) ψ.

4.Burton Richter

Born: 22 March 1931 (age 86)

At SLAC with his team he discovered the new heavy meson and named it “J”.

We now call the particle J/Ψ (because the two experimentalists discovered it independently and gave it different names). It’s bound state of charm and anti-charm quarks.

5.Sheldon Lee Glashow

Born: 5 December 1932 (age 84)

He already predicted the existence of the new quark and named it “charm” quark using the symmetry of leptons and quarks. Only 4 leptons were known at that time so he thought there should be 4 quarks also, not 3.

According to their spin, particles are described as Fermions (spin 1/2) and Bosons (integer spin). So leptons and quarks are Fermions.

So the question that I asked in the beginning of this post can now be answered.

There are 61 fundamental particles. Leptons, quarks, force carriers and Higgs Boson. The total number comes as follows,

6 leptons + 6 anti-leptons = 12

6 quarks (each with 3 different colours)+6 anti-quarks (with 3 different anti-colours) = 36

8 gluons + W(±) + Z° + Υ = 12

Graviton, the hypothetical particle for gravitational interaction is not included in the standard model. A theory that unifies all 4 interactions is yet to be discovered.

Reference

Griffiths, David J. (1987). Historical Introduction to the Elementary Particles. Introduction to elementary particles (pp. 11-48). Retrieved from Internet Archive website: Introduction to the Elementary Particles

Rainy Day

It was a holiday. So I got the advantage of taking an afternoon nap. But after half an hour, I woke up to hear the voice of heavy rain from my window. I suddenly remembered my clothes must have been wet till now. But after coming out of the room, I saw them hanging in the lobby. So I thanked mommy for it.

After a long time it was raining so heavily. I started thinking how would it look like from space. There will be a lot of clouds over Haldwani right now. And I thought how much area these clouds must be covering? It’s for sure that not all of Uttarakhand will be experiencing rain. Are there other places on earth where it would be raining right now? Then out of the blue many questions flooded in my mind. How high these clouds will be? I remembered reading in a science activity book as a child that we can calculate the amount of rain in our region.

So the first thing I wanted to do was to measure the amount of rain. So I started reading about how it can be done at home. And while I was doing this the rain started slowing down. And when I finished reading some articles and went upstairs with the bottle, it was only drizzling. The time was 3:30 PM.

So I kept it there. And it again started raining at nearly 5:25 PM. Now I only have to check the water level after 24 hours. So I will update the post tomorrow.

On this website I saw how much clouds were over my city. It also showed at the same time the data for all other countries on this planet.

Update: 22nd September

I kept the bottle for 24 hrs. And using the formula from this website:

Volume of rainfall/ Area of the bottle I got 50 mm which indicates the depth of rainfall. In today’s newspaper I got the data to be 55mm. So I am almost right. Another thing that caught my attention was, the rainfall is believed to increase 70 to 75mm for today.

It’s not very comforting to write about everything you do. But now I think it’s still somewhat convincing to write because I will not be able to remember everything.

Also, for me telling someone about these things is very different from writing them in words. So these blog-posts are like time capsules in which I keep some of my favourite memories and works.

Partition function 2

2. In additive number theory the basic problem is how an integer can be written as the sum of different integers.

And as an example we can take one of the most famous unsolved problems in mathematics,

Goldbach conjecture: Every even integer $n>2$ is the sum of two odd primes.

Other terms related to the conjecture are,

Goldbach number: All those positive integers that can be expressed as the sum of two odd primes. And the smallest number that can be written in such a way is $4$. So, every even $n>4$ are Goldbach numbers.

Goldbach Partitions: The expression is called the Goldbach partition of that number.

1. The partition function I wrote about last month is just one of such problems in additive number theory, which tells us the number of ways an integer can be written as the sum of different positive integers.

$P(1) = 1$

$P(2) = 2 = 1+1$

$P(3) = 3 = 1+2 = 1+1+1$

This particular problem has a specific name, it’s called unrestricted partitions.

Because the number of integers that can be used to write the sum is unrestricted. Repetition is allowed and the order of numbers does not matter.

So, $P(n)$ is called unrestricted partition function.

Partition Function

An integer $n$ can be written as a sum of different positive integers. Its also called partitions of an integer or $P(n)$.

For $4$, there are $5$ partitions.

$4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1$

So $P(4) = 5$.

Partitions were studied by many mathematicians like Euler, Ramanujam, Hardy and also by Freeman Dyson. But the finite formula for partitions was discovered by Dr. Ken Ono and Jane Bruinier at Emory University. You can plug in any number in that formula and get the partitions for it.

I first came to know about partitions when I watched this beautiful video by Dr. Ken Ono.

2 months ago we were assigned to give a presentation on some specific topics in physics.

Even though the topics I got were from statistical mechanics, I included the mathematical partition function also.

The slides can be seen here – Partition Functions by Shivani Mishra

I have not shared the remaining 20 slides which contain derivations of statistical partition functions and other statistical mechanics definitions.