Amazing discoveries in Science!

1.  Electricity and Magnetism are one and the same thing.

 

This was clear from Faraday and Oersted's experiments but was theoretically unified completely by Maxwell's great equations. Now it underlies all of modern electricity generation and hence, modern economy. Still thinking about it, it is amazing. What has Magnetism of Compass, magnets have to do with electric current, light bulbs and so on. This is one of the results which separates a learned man from an illiterate to me.

2. Light is an electromagnetic wave.

Nowadays this looks like a normal thing everyone knows. Still think about it. What has light from sun, moon and candle have to do with electricity or magnetism. 

This was the result which made me realize the beauty of science!

This is science at its most beautiful and pure, elevates one spiritually on to a different plane and no religion or other superstitious nonsense can event match remotely. 

This is again from Great Maxwell who purely by theoretical work and completing his equations purely in terms of mathematical elegance came up with this discovery.

3. E=Mc^2

When I first saw this equation [in high school] I thought it must be a printing mistake as what does light has to do with mass or energy?

When I understood special relativity and implications behind this it was truly amazing. To truly appreciate the beauty of this one should read Einstein's original paper which is very readable even for a non physicist. To start from 2 simple assumptions and to derive something this powerful, well I guess that is genius!

This also has the non-intuitive implication that there is no absolute notion of time or simultaneity which is amazing in its own right!

4. Evolution and natural selection

This shook me to the core when I first learned Darwin's theory in my higher secondary school. This meant we are not "special" and just one among so many animals. When I saw the proof it was hard to deny it with so many things pointing in its way - DNA, common proteins, blood, bones and embryology among so many others. You really have to be an idiot to reject it after being shown the evidence.

But it is one of the foundational results in all of science.

5. Gravity

It is unfair not to include Newton in this list as he pretty much started all of modern physics and quite a bit of mathematics including calculus. His many discoveries are important in so many ways. 

That said the discovery that moon is bound by the same force which breaks one's bones when falling from tree is truly amazing and enlightening!

Proof - PI is irrational

                                                    [image source: wikipedia]
 Lets assume PI is rational.
Then PI = a/b, a and b are integers.

We also know pi is not an integer as its value lies between 3 and 4.

We know the equation:

e^(i PI) = -1

substituting its rational form:
 e^(i * (a/b)) = -1

raising both sides to bth power.
e^(a i) = (-1) ^b

now there are two cases:

1. b is odd.
in this case e^(a i) = -1
this means cos a = -1
this implies a is an odd multiple of PI. That is PI, 3PI, 5PI etc. As cosx  takes -1 only in these points.
but we know PI is not an integer. so a is not an integer either.

2. b is even
in this case e^(a i) = 1
this means cos a =1
this means a is an even multiple of PI. That is 0, 2 PI, 4 PI etc.
but again 2 PI etc. are not integers. So only option left is a = 0 which would mean PI = 0!

This proves by contradiction that PI is irrational.

Godel's theorem for layman

In mathematics we use:

1. a set of assumptions (axioms) and 
2. use a set of rules for deriving conclusions (inference)
3. Proofs are statements which are derivable from axioms using these rules.

A system is consistent if we can only prove true statements.  That is we can't prove a statement and its opposite.

For example, we can't prove both 2+2 = 4 and 2+2 != 4 in the system. Only one can be proven.

Now we know we can't prove both. Do we know we can prove the true one in all cases?

Godel theorem says no. In a consistent system, there will always be statements which are true but not provable! Sounds odd, but that's exactly the reason the theorem is puzzling and popular!

              THIS STATEMENT IS NOT PROVABLE


The implication is although mathematicians may be working on things like Fermat's last theorem they may not provable even if they are true. So they may be just wasting time [in some sense] trying to prove it. Of course Fermat's last theorem itself was provable and proved by Andrew Wiles.

Parking in a narrow parking space in parking lot

                                   courtesy: http://www.aviewfromthecyclepath.com

I had the problem of parking in a spot which has poles for roof support on one side [left] and another car at the other side [right] and space itself was narrow. The opposite side had the cars as well [in the parking lot] and the passage was not very wide to go to the other end and make a 90 degree turn into the space.

My main problem was avoiding scratching or hitting the other car to my side. So I tended to go close to the pole and turn sharply in. I kept scratching my car against the pole.

Suddenly the idea struck me.

Go past the spot and in front of the car to your side. turn in front of the car [of course, it is not possible to do 90 degree turn. If it were possible you may as well do it in front of your own spot] as much as you can and back up in front of your own slot making you turn 90 degrees in front of your own slot. Now you can go straight in.

So the key idea: Turn not in front of your space but past the space in front of the next car and back up till you are perfectly aligned 90 degrees!

Hope this helps someone in same situation!

Continuum hypothesis for layman

We have at least 2 different kinds of infinities:

1. Integers and all the other countable numbers - That is things you can pair with integers.

For instance pairs: {1, 1}, {1,2} ...

These can be easily paired integers along diagonals:


                            {1   1}     {1 2}    {1 3}

                            {2   1}     {2  2}
                            
                             {3   1}

We can easily pair like so:
1 -> {1 1}
2 -> {1  2}
3 -> {2  1}
4 -> {1  3}
5 -> {2  2}
...

So pairs  [fractions being one of them with equivalence class notion on top] are no 'bigger' than integers.

2. On the other hand real numbers are really bigger than integers!

Courtesy:Wikipedia

There is a nice way to prove this due to George Cantor.

In fact we need to only consider real numbers in {0, 1}

if it is countable, We can write this as:

1-> 0.0000...0
2-> 0.0000...1
   0.0000..2
   ..
   0.9999..9
Now if we go along the diagonal and change 0 to 1 and everything else [1, 2,...9] to 0 we get a number which differs from all these numbers in atleast one position and it is not mappable to an integer!

So real numbers are indeed bigger, in fact way way bigger! In fact they are not even real despite their name!

Now the continuum hypothesis asks whether there is an infinity strictly between integers and reals or reals are the next big infinity. Of course hypothesis claims it is true.

No one has been able to prove or disprove this in more than a century!

How do we know neutrinos exist

I was wondering today why should we postulate neutrinos and make life more complicated than it already is.

This article explains why neutrinos are required and in addition as a bonus explains how they are detected. Apparently it is must for accounting for conservation of energy in beta decay.

Story of the Neutrino

Income inequality of US or hail capitalism US style!

This is from wikipedia and made me fall out of my chair!

1. Top 1% owns 35% of wealth.
2. Top 5% owns 60% of wealth or 3 out of every 5 dollars.
3. Bottom 40% owns  just 0.2% of wealth.
4. Bottom 60% owns just 4% of wealth or 96% is owned by top 40%

Really hail capitalism US style - 60% is poor because they don't try and Big banks/wall street firms will get saved by government if they fail [of course with fat bonuses for their executives].