Monthly Archives: July 2002

WHY COMPLEX NUMBERS?

 

 Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was younger, I always did it for half an hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

      Today’s VERITAS is inspired by a brilliant essay by the eminent Physicist Eugene Wigner : “The unreasonable effectiveness of mathematics in the natural sciences”. In this essay Wigner expresses surprise at how successful a tool created by the human brain(Mathematics) can be at explaining nature(Physics).

        How come our brains are able to understand so much. How come the mathematics invented in a year x becomes useful in explaining physical phenomenon in the year x+50? Isnt it a wonder.

      In his essay Wigner tells the story of two friends who land into different fields: one becomes a statician and the other a shopkeeper.

After some years they meet again and the statistician shows the shopkeeper his latest research paper on population statistics. The shopkeeper does not understand much but finds some funny looking formulae in the paper. He turns to the statician and asks him ” what is this?”.

The statistician says ” This is pi”. The shopkeeper says “what is pi?”.

The statistician says ” it is the ratio of the circumference of a circle to its diameter”. The shopkeeper is shocked ” SURELY THE CIRCUMFERENCE OF A CIRCLE HAS NOTHING TO DO WITH POPULATION!!”.

    Is it now strange how our small set of mathematical contants find themselves everywhere.

    In VERITAS today we ask ” If imaginary numbers are not “real” then why are they used everywhere(electrical engineering, Physics etc)?”

    A complex number(or an imaginary number) is the square root of -1. They were invented by Mathematicians for their own enjoyment(maths). The Maths guys had invented them because they wanted to solve all algebric equations including the one : x^2 + 1 = 0

   Okay so these are not real numbers. So why do we use them?

   They are useful because of a most beautiful formula(Feynman calls it a jewel):

   exp(ix) = cos(x) + i sin(x)

  where i is the square root of -1.

  What this means is that whereever you see a sin or a cos you can replace it with exp(ix) and do all your calculations and in the end take the real part of the result.

  But why would anyone want to do that?

  We do this because calculations with exp(x) etc are very easy compared to the calculations with sin , cos and such. for example the differential of exp(ax) is a*exp(ax): so differentiation is as simple as multiplication. Similarly exp(x) * exp(y) = exp(x+y) : multiplication is as easy as addition.

  So in any equation we can change the sin and the cos to the exp(ix) and do all calculations and in the end take the real part as the answer.

  So what I am saying here is that complex numbers make Physics simple but they are not necessary. Even if complex numbers were not invented things would still have been okay(**well almost! see later :-)**)

 

 Now when I first read about negative numbers I felt uneasy. How can there be less numbers than 0? Later we understood that negative numbers mean that you have made loss  or that you have less than an arbitrary assigned 0.

 So in case of negative numbers we assigned a number not to a quantity but to a concept( the concept of loss etc). You cannot see a negative number in terms of set of pencils etc.

 Similarly imaginary number is a number assigned to a concept. The concept that there is a “something” whose square is -1. By itself it may have no meaning. But the concept is useful when it is a part of a larger concept( a larger equation perhaps).

 The human brain has gone beyond the number of fingers in our hands. Now we can think in terms of more abstract things and that has helped us understand nature.

 Now I will tell you something that makes me uneasy. I said earlier that complex numbers are used only to simplify the equations of Physics. And that they have no “real” meaning. But the equations of Quantum Mechanics(the study of subatomic particles) NEEDS the concept of imaginary numbers! Without complex numbers Quantum mechanics cannot be formulated! I find this very very crazy! Is God making a fool of us by making us play with the toys that we invent.

 And I ask myself and the rest of the readers of VERITAS the question: “WHY IS MATHEMATICS SO SUCCESSFUL??”

Alice: If I had a world of my own, everything would be nonsense. Nothing would be what it is, because everything would be what it isn’t. And contrariwise, what it is, it wouldn’t be, and what it wouldn’t be, it would

-kanwar

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  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                          

  Correct old time and regulate the Sun;                             

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Nicola Tesla

 

      Today’s VERITAS is about a genius who changed the way we live. And yet very few people know about him.

      What inspires a genius? What does he work for? Why is he happy without sleep and without money? Why does he seemingly torment himself and live so close to madness? All these questions baffle us. The answer to these questions seems to me this: genius is the fire that needs no other fuel. Genius(the faculty) needs no outside reason to work. It is a self driven force that works without an end in mind. This theory of mine is supported by the life of Nicola Tesla.

        Tesla was born in 1856 in Yoguslavia. His early education was in a village school. When he was seven the family moved to the city of Gospic. It is at this age that Tesla showed his first sign of genius.

A new fire brigade decided to have a public demonstration of its new pump that would replace the old buckets of water. The whole town came to see. Telsa and his parents were also there. Tesla became instantly attracted to the big red water pump. The pump was huge. The aim of the demonstration was to connect one side of the pump to the river and from the other end a stream of water would be shot skywards by 16 firemen.

But when the demonstration began there was not a drop of water coming out of the nozzle. The firemen starting pumping faster but nothing happened. It was becoming a great embarrasment. Tesla was inspired by the problem. He took off his clothes and jumped into the water. At the end of the pipe he found that the pipe had twisted and kinked. He straightened it and the water shot skyward from the other end!

Tesla became a hero. He had solved a problem at the age of 7. The joy of problem solving would guide the rest of his life. 

      Tesla studied math, physics, and mechanics at the Polytechnic Institute at Graz. He displayed an extraordinary memory and was able to solve most math problems without writing them down. He also mastered 6 languages.

      During college Tesla first saw a DC machine: a machine that could be used as a dynamo or as a motor. Tesla was impressed but something told him that the DC machine was not the best way to produce electricity. After several years of research he invented the A.C induction motor. He however did not reveal his results to anyone.

       Tesla Joined the French branch of the Edison Electric company. He was given the duty of repairing an electric plant at Strassburg. AFter the completion of the work Edison’s company refused to pay him the money. Tesla left and migrated to America. He started working for Edison. Edison had several problems with his DC motors and told Tesla to improve them. He also offerred a prize of 50,000$ if he could. Tesla improved Edison’s motors but in the end Edison refused to pay and said that the promised prize was just “American Humour”. Tesla quit immediately.

      However Tesla could not find a job and he had to dig ditches for two years to support himself. However he knew that he was worth more. He would dig ditches but his brain was full of dreams of experiments he would do if he got the money.

       Tesla found work in the Westinghouse’s Pittsburgh labs. He told Westinghouse about his idea of AC electricity. He demonstrated that AC current can be used to transmit power to large distances which DC failed at. Westinghouse immediately liked the idea and paid a good sum to Tesla for the idea.

      Edison had invested a lot of money on DC power and he tried his best to show that AC power is not a good idea. He even burnt animals using AC power to show that AC is too dangerous to be used in homes. The press believed Edison …. he was really famous! Tesla did not give up. At a fair in Chicago he even passed AC current through his own body to light electric bulbs to show the world that it is not so dangerous.

     Tesla also invented transformers to step up and step down Electric power. Westinghouse with Tesla’s help started generating electric power from Niagra falls. Soon America’s homes had electricity!

    With the money Tesla set up his own electric company and started inventing a lot of exotic electric machines. But he was careless with money. Westinghouse one day decided to pay Tesla no more( The initial contract was to pay $2.50 for each KW of energy sold). Tesla did not bother. He only wanted enough money for his experiments. 

    In 1900 he got finance from JP Morgan to setup the world’s first wireless broadcasting system tower. The dream was to be able to transmit information to the whole world. JP Morgan soon realized that this system would mean free information to the whole world and they would not be able to get any money out of it. So they stopped the finance. And the project stopped. Tesla had created wireless about 10 years before Marconi!

   In his lab Tesla was using fluorescent bulbs in his lab some forty years before industry “invented” them.

    He even created an artificial earthquake in New York. He did this by vibrating a stream driven oscillator at the same frequency as the natural frequency of the earth!

     In his Colorado Springs lab in 1899, he sent waves of energy all the way through the Earth, causing them to bounce back to the source (providing the theory for today’s accurate earthquake seismic stations). When the waves came back, he added more electricity to it.The result? The largest man-made lightning bolt ever recorded – 130 feet! -a world’s record still unbroken! The accompanying thunder was heard 22 miles away.

   However all his life Edison kept says things against Tesla. He made sure that the press and the people think of Tesla as an eccentric crank and not as a great inventor.

   In the year 1912 Edison and Tesla were to share the Nobel Prize in Physics. But they did not want to share the prize with each other. So none of them got the prize.

   In his life Tesla recorded over 800 patents. He was only stopped from Edison’s record number by a constant lack of money.

  His notes even contain the principles of superconductivity. And this was about 30 years before it was discovered.

 Tesla however was also a bit eccentric. He never married and had only a few friends. A lot of his proposed inventions were more like flights of fancy.

 At this time when power distribution seems to be the talk to the town(New Delhi 🙂 ) lets remember the great Genius who made this all possible.

-kanwar

 

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  Go, wondrous creature! mount where Science guides:                 

  Go, measure earth, weigh air, and state the tides:                 

  Instruct the planets in what orbs to run,                          

  Correct old time and regulate the Sun;                             

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Mysteries of the Brain Part : 2

Brain

This article was written on 2 July, 2002 
        I would like to thank Sakharwade for making me read some very interesting articles on the Brain published in a special edition of the Scientific American. Today’s VERITAS is inspired by one of these articles.

Have you ever wondered if your emotions are permanent or temporary?
If you are afraid of something would you fear that thing for a long time or would you be able to control it? Why do emotions always hit you before reason does? Where are the emotions stored in the brain?

The emotional aspects of the brain have been studied for about a decade now and Scientists have some up with some pretty interesting facts.
The brain pathway of one emotion has been studied in detail: fear.

 Joseph LeDoux and his team did some very interesting experiments on rats called fear conditioning. A set of rats were made to hear a sound and
the sound was followed by a mild electric shock on the base of the cage. 
After 2-3 sound-shock pairs the rats began to associate the particular sound with shock. And later even if the shock was not given the mice
would still show fear(high blood pressure, freezing) on hearing the sound. The Le Doux and his team made lesions on various parts of the rat’s brain to study where emotions are stored and how they can be removed.

There are four parts that interests us : thalamus, Amygdala , Cortex and the Hippocampus. The purpose of these various parts is :

1)Thalamus: The stimulus(from the outside brain enters from here). It sends these signals to various other parts for processing.

2) Amygdala: This is an almond shaped part deep in the brain. It stores emotional memories. In our rats it stores “sound means shock so RUN!!”
kind of simple associations. It activates emotional response(high BP,fight , run , heartbeat etc)

3) Cortex:  The part involved in planning and thinking.

4) Hippocampus: This stores declarative memories(facts). The emotional part of the memories is not here.

The basic pathways of emotions are these:

outside stimulus  -> thalamus -> amygdala     ..1

outside stimulus  -> thalamus -> cortex -> amygdala     ..2

outside stimulus  -> thalamus -> cortex -> Hippocampus -> amygdala  ..3

  So this is what happens when you see something: Lets say you see a wooden stick that resembles a snake. The message from the eyes
reaches the amygdala from 2 different paths: from the thalamus directly(pathway 1) and from the thalymus->cortex(pathway 2).
But pathway 1 is faster because it has one link less. Pathway 2 involves the cortex and involves thought. So when you see
the stick the first thing that happens is that the amygdala forms the association “Snake so RUN!”. But after a short time
the cortex (having examined the object and having thought) sends a message to the amygdala : “Dont bother… it is a stick”
So Emotions always hit you before thought does. Our evolution has made us this way. You would be better off confusing a
stick for a snake than examining everything before deciding to act. SUrvival is the primary need.

   AFter this the scientists started giving the rats the sound but not following the sound with a shock. Slowly the response of the rats started diminishing. They realized that sound is not followed by shock. So their emotions to the sound decreased. Now the team removed the cortex of these mice. Then they delivered the sound to the mice again and the mice showed the old HIGH level of fear! The scientists realized something:
the emotional associations that form in the amygdala are PERMANENT! You cannot change your emotions! The cortex merely changes your
response to your emotions. So if you like or dislike something you will do that all your life. However your response to it may change because of conscious thinking.

     Now something interesting: the Hippocamus stores factual memory. For every emotion stored in the amygdala there is a set of facts stored in the Hippocamus. When you see something both these memories are retreived together. For example if you had an accident a long time back you would remember how you felt and also remember the facts about how it happened and what happened after that. Each of these memories can trigger off the other.

   When we are children our Hippocamus is not developed. But our amygdala is. So if you go through a disturbing experience during
childhood you may develop a lifelong emotion(fear maybe) towards it without remembering the actual incident!

  
    The scientists found that people with irrational fears can never be completely cured. A person who has been treated for a phobia can go back to his initial state if he goes through an emotionally disturbing experience. The fear association in the amygdala is permament. The cortex may behave differently but you cannot change the amygdala.

    The brain is a complex and wonderful thing. The more we study it the more we are surprized at ourselves. When I got to understand the permanent nature of emotions I, in a way understood myself. I feel less responsible now for being me…

Kanwar

|======================================================|
  Go, wondrous creature! mount where Science guides:                 
  Go, measure earth, weigh air, and state the tides:                 
  Instruct the planets in what orbs to run,                          
  Correct old time and regulate the Sun;                             
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