Complex numbers (imaginary numbers) are numbers of the form:

                       a + ib, where a and b are real numbers and i is

                       the square root of minus one.

Not only are complex numbers extremely useful in the subject of mathematics, they play an essential role in many areas of Physics - Electromagnetic Theory, Electronics, AC Circuit Theory, Atomic & Nuclear Physics, Quantum Physics, Wave Theory and Signal Analysis just to name a few.  In many situations in Physics, the real solutions to a problem would be inaccessible without the use of complex numbers.  For instance, Schrodinger's Equation, the basic equation of wave mechanics expressing the behaviour of a particle moving in a field of force, contains the imaginary number i (see below).

The following is taken from Michio Kaku's excellent book titled "Parallel Worlds".

There are several different types of paradox that can occur when one considers the possibility of time travel.  Most people are familiar with the "Grandfather Paradox" in which you alter the past in a way that makes the present impossible - eg you shoot your grandfather when he was just a boy, thus ensuring that your father was never born and therefore that you were never born.

Then there is the "Information Paradox" where information comes from the future, which means that it may have no origin.  The "Terminator" movies hinge on an information paradox - a microchip recovered from a robot from the future is studied by scientists, who then create a race of robots that become conscious and take over the world.  The design for these robots was never created by an inventor; it simply came from a piece of debris left over from one of the robots from the future.

"Bilker's Paradox" is where a person knows what the future will be and does something that makes the future impossible - eg you know you are going to marry a particular person, so you deliberately decide to marry a different person instead.

The "Sexual Paradox" is perhaps the most interesting of all the different types of time paradox.  In this paradox, you go back into the past and father yourself, which of course is biologically impossible.  In a science fiction story written by Jonathan Harrison in 1979, a young woman by the name of Jocasta Jones finds a freezer.  Inside the freezer she discovers a handsome young man frozen alive.  She thaws him out and learns that his name is Dum.  Dum tells Jocasta that he has a book that tells how to build a freezer that can preserve humans and how to build a time machine.  Jocasta and Dum fall in love, marry and have a son, whom they call Dee.

Years later, when Dee has grown to be a young man, he follows in his father's footsteps and builds a time machine.  This time both Dee and Dum take a trip into the past, taking the book with them.  Sadly, they find themselves stranded in the distant past and running out of food.  Realizing that the end is near, Dee does the only thing possible to stay alive.  He kills his father and eats him.  Dee then follows the instructions in the book and builds a freezer.  To save himself, he enters the freezer and is frozen in a state of suspended animation.

Many years later Jocasta Jones finds the freezer and thaws its occupant.  To disguise himself, Dee claims to be Dum.  They fall in love, marry and have a baby, whom they call Dee.  And so the cycle continues.

Clearly, there is a serious biological paradox here.  Since half the DNA of an individual comes from the mother and half from the father, this means that Dee should have half of his DNA from Jocasta Jones and half of his DNA from his father, Dum.  However, Dee is Dum.  Therefore, Dee and Dum must have the same DNA because they are the same person.  But this is impossible, since by the laws of genetics, half their genes come from Jocasta Jones.

Weird eh?

Top of Page

CHOCOLATE MATHEMATICS 2008

An interesting piece of mathematical fun that has appeared on the Internet in recent years is something called "Chocolate Maths".  While it is fun just to follow the instructions below and marvel at the fact that it works, it is even more fun to explain mathematically how it works.  It is a good exercise in thinking!

On the Internet this set of instructions often ends with words to the effect of:

TASK:

Explain mathematically how this mathematical novelty works and comment on whether it will only work this year.

Go To Answer

Top of Page

			Answers To Brainteasers & Puzzles

Answer to Hollywood Physics question:

Clearly, the momentum of the bullet is given by

               p = m.v

           p = 0.025 x 500

           p = 12.5 Ns  (or kgm/s)

If all of this momentum is transferred to the victim, we have

               12.5 = 100 x v, where v = speed of victim after impact of bullet

            v = 0.125 m/s

So, we have that the speed of the victim after the impact of the bullet is 0.125 m/s or about 0.45 km/h.

Clearly, 0.45 km/h is not a very high speed.  The speed of a person walking for exercise can be as much as 10 km/h.  It is therefore very unlikely that a person shot as described would be lifted off his feet and thrown backwards.  Thus, the Physics of such scenes in movies is often inaccurate.

Thinking about things from another perspective - when a gun fires a bullet, the gunpowder in the cartridge explodes and vaporizes.  The gas produced expands rapidly and applies a force to the bullet, pushing it in the direction in which the gun is pointing.  By Newton's Third Law, the bullet applies an equal but oppositely directed force on the gun and therefore on the gunman.  This is called the recoil of the gun.  It stands to reason that the force the bullet applies to the target can at the very most be equal to the force applied backwards on the gunman.  Otherwise, the bullet has gained extra momentum from some unknown source.  So, if the gunman is not lifted off his feet and thrown backwards, neither is the victim.

Careful analysis reveals that in reality, the force that the bullet applies to the target is always less than the force it applies backwards on the person firing the gun.  You might like to think about why this is so.

Note that the stimulus for this Brainteaser came from Dr Karl Kruszelnicki's book "Dis Information".  Also have a look at the Physics Web article "Hollywood Physics".

Top of Page

Discussion related to "Beam Me Up Scotty":

In one form, Heisenberg's Uncertainty Principle states that it is impossible to know both the position and the momentum of a quantum particle (eg an atom, electron etc) simultaneously.  In attempting to measure say an electron's position, the technique we use must interact with the electron to obtain the measurement.  In so doing, we alter the velocity of the electron and therefore its momentum.  So, after the measurement we may know more about the electron's position but less about its momentum.

Clearly, the Uncertainty Principle seems to forbid the possibility of a "transporter".  Whenever you make a measurement on an atom, you disturb the quantum state of the atom, so how can you possibly make an exact copy of that atom?

Quantum entanglement is a phenomenon in which the quantum states of two or more objects have to be described with reference to each other, regardless of how far apart the objects may be spatially.  This means that there are correlations between the observable physical properties of the objects.

Let's consider an example.  Experimentally, it is possible to prepare two particles, say two electrons, in a single quantum state such that their spins are correlated.  In other words, when one electron has its spin axis pointing up, the other electron has its spin axis pointing down, so that the total spin is zero.  Until we make a measurement however, we do not know which electron is spin up and which is spin down.

Now imagine that the two electrons move away from each other at close to the speed of light for several years.  By then the two electrons are many light years apart.  Now, if at this time we make a measurement of the spin of one of the electrons, we see the startling consequence of entanglement.  If we find that the measured electron has spin up, then instantly we know that the other electron has spin down.  Think about this for a moment.  By measuring the spin of one electron, we instantly determine the spin of the other electron, even though it is many light years away.  Information seems to have travelled faster than the speed of light, in apparent contradiction of Special Relativity Theory.  Albert Einstein, Boris Podolski and Nathan Rosen proposed this very scenario back in 1935 as a demonstration that quantum mechanics must be wrong, since it clearly led to absurd results.  This scenario became known as the EPR Experiment and was the focus of much philosophic and scientific investigation over many years.

The EPR Experiment was actually performed for the first time in the 1980's by Alan Aspect et al and has since been performed several times over distances as great as 11 kilometres.  The results show that the predictions of quantum mechanics are accurate.  Yes, if you do measure the spin of one entangled particle, you instantly know the spin of its entangled twin regardless of how far away that twin is.  Entangled particles act as if they are one single object.  What happens to one particle automatically affects the other, regardless of the distance between them.

Please note, although Einstein was not correct about the implications of the EPR Experiment for quantum mechanics, he was still correct in saying that communication cannot occur faster than the speed of light.  The EPR Experiment does allow you to know something instantly about the other side of the galaxy but it does not allow you to send a stream of useful information (that is, to communicate) in this way.  You cannot, for instance, send a short-wave radio message in this manner.  There is a distinct difference between knowing a piece of information and communicating it to someone else.  Given the right piece of evidence you can know a particular fact instantly but you cannot communicate that information from one point to another faster than the speed of light.

Now, getting back to the "transporter" idea - In 1993, Physicists suggested that it might be possible to perform some sort of Star-Trek-type transportation using quantum entanglement.  The technique is called quantum teleportation and aims to send a quantum state from one place to another.  This was first experimentally demonstrated in 1997 when the quantum state of a single photon was teleported across a table top.  In 2004 the quantum characteristics of an entire atom were successfully teleported from one place to another.

How does this work?  Let us consider two eminent Physicists, Angela and Rebecca.  Angela & Rebecca produce a pair of entangled electrons - electrons 1 & 2.  Rebecca takes electron 2 to the Moon, while keeping it completely isolated from any other influences.  Angela remains on Earth with electron 1, which she also keeps isolated from all other influences.  One day, Angela wishes to teleport the unknown quantum state of a third electron - electron 3 - to Rebecca on the moon.  To do this, Angela firstly allows electron 3 to interact with her entangled electron 1 and measures the quantum state of the combined system of two particles.  This measurement causes Rebecca's twin entangled electron 2 on the Moon to adopt a quantum state resembling electron 3.  Now in order for Rebecca to "know" the quantum state of electron 3, she needs to be informed of the result of the measurement of the quantum state of the combined electron 1 & 3 system.  It is this result that tells Rebecca how to adjust the state of her electron 2 to obtain an exact copy of the state of electron 3.  Once this is done, the quantum state of electron 3 has been teleported from Earth to the Moon.

If you would like a more detailed explanation of the actual process of quantum teleportation, Chapter 23 of Roger Penrose's "The Road To Reality", specifically section 23.9 on Quantum Teleportation, is an excellent reference.  There are also some good references on the Internet.  Search for quantum entanglement, quantum teleportation or Bell state measurements for a start.

A few points of clarification are needed!  Firstly, note that quantum teleportation is not a means of transferring physical systems from one point to another, but rather the information encoded by those systems.  Secondly, note that the original quantum state of electron 3 in our example, is destroyed in the process.  It is combined with the quantum state of electron 1 during the interaction of those particles.  What you end up with at the other end is an exact copy of the quantum state of electron 3.  These two points taken together imply that if we ever get to the stage of teleporting a whole person, we will actually be teleporting the entire sum of quantum states that make up that person NOT the physical person themselves.  In other words, each time Captain Kirk is to be teleported, he will be scanned to obtain his complete quantum state information, destroyed in this process and then reassembled from appropriate raw materials at the other end.  Still want to try this?

Thirdly, the teleportation process must occur at less than the speed of light.  Even though quantum entanglement is involved here and in our example, the measurement of the quantum state of the combined electron 1 & 3 system instantly affected the quantum state of electron 2 on the Moon, Rebecca could not produce the final copy of electron 3's quantum state until Angela informed her of the result of her measurement.  This would take place for instance via computer, say by email, or by some other classical (ie non-quantum) communication channel.  It is this fact that ensures that the whole teleportation process must occur slower than light speed and is therefore not in contravention of Special Relativity.  In terms of the teleportation of human beings, this means that as quantum state information was determined for the human being teleported, it would have to be communicated via a classical communication channel to the device at the other end, to enable the human to be re-assembled.  So, if we wished to teleport someone to the mars, the quickest it could be accomplished is about five minutes - not instantaneously!

Now, to what is probably the biggest barrier to the teleportation of human beings.  Humans are incredibly complex systems.  One of the Physicists who actually worked on the first successful teleportation of the quantum state of a photon, Samuel Braunstein of the University of Wales, has calculated that even if all the technical barriers to the teleportation of humans could be overcome tomorrow, it would take at least the age of the universe just to teleport all of the quantum information that describes just one human being.

So, I suppose the "Beam-me-up-Scotty Transporter" will remain in the realm of science fiction for some time to come.

The above discussion was prepared using information contained in Roger Penrose's "The Road To Reality" chapter 23 and Michio Kaku's "Parallel Worlds" chapter 6.

Top of Page

Discussion of Schrodinger's Cat:

Be aware that Schrodinger's Cat is one of the most famous & thought about paradoxes in Physics and Philosophy.  It raises many deep questions about the nature of the real world and what we can know about it.  Indeed whole PhD Theses have been written in an attempt to answer the questions raised by the paradox.

The traditional question to ask at the end of the Schrodinger cat story is: “Is the cat dead OR alive or dead AND alive before we open the box to check?”.  At the simplest level Schrodinger’s cat is meant to point out to us that the quantum mechanical answer to this question is that the cat is BOTH dead AND alive at the same time until we perform a measurement on the system by opening the box and observing the cat. 

Strange, eh?  Quantum mechanical systems (eg atoms, nuclei, quarks etc) actually exist in all possible states at the same time until we make a measurement on the system and in so doing “collapse” its wave function, that is force it to adopt a particular state at that particular time.  The wave function of a system is the mathematical equation that contains all the information about the system that is knowable. 

When we speak of a system being in all possible states simultaneously we say that the system exists in a “superposition” of states.  For example, let’s consider the spin property of electrons.  An electron can spin up or spin down.  Since electrons are quantum mechanical systems, this means that all the electrons in each atom of your body, for instance, exist in both the spin up and spin down states simultaneously.  The only way we can say that a particular electron is in a particular state is if we measure it at a particular time and force it to adopt a particular state of spin.  Until then the electron exists in BOTH states at the same time. 

On a deeper level, Schrodinger’s cat asks the question “At what stage does a system become too big or too complex to exist as a whole in a superposition of states?”.  For example, our experience of the macroscopic world tells us that even before we look in the box, the cat is EITHER dead OR alive.  Opening the box (measuring the system) cannot force the cat into one state or the other.  Our experience tells us that the cat is already in one state (dead) or the other (alive).  We know even before we open the box, the cat can never have been BOTH dead AND alive in our macroscopic world. 

The famous physicist Niels Bohr (1885-1962) did a lot of work trying to establish where along the chain of increasing size from atoms to macroscopic objects did it become unnecessary to use quantum mechanics to give an accurate description of the state of a system.  To cut a long story short, Bohr came to the conclusion that for systems of the order of the size of the atom (10^-10m) and smaller, quantum mechanics is essential to accurately describe the system.  For systems bigger than this, ordinary “classical” physics is adequate to calculate the state of the system.  This idea is called the Correspondence Principle. 

So, for anything as big as a cat, although the atoms of which it is composed will be in all manner of superposition of states, the actual macroscopic entity that we call the cat always and only exists in one definite state (dead) or in the other (alive) – never BOTH at the same time.  For our cat, or any other macroscopic entity, ordinary classical physics is capable of accurately describing any action that it may (or may not) perform. 

Top of Page

The Domain of Quantum Mechanics:

The following is a summary of pages 8-11 of Basic Quantum Mechanics by R. L. White (McGraw-Hill, 1966) – an excellent text commonly used in third year university Physics courses on quantum mechanics.

How do we know precisely when the use quantum mechanics is essential?  An accurate criterion for establishing the appropriate domain for quantum mechanics is based directly on Planck’s constant, h = 6.626 x 10^-34 joule-second.  Planck’s constant has the dimensions of energy x time, a quantity known in classical physics as action.  The dimensions of action can be shown to be the same as momentum x distance, which is another physical quantity called angular momentum. 

In terms of Planck’s constant, our criterion for the use of quantum mechanics can be stated as follows: 

If the action or angular momentum involved in a given physical event is of the order of Planck’s constant, then we must use quantum mechanics to describe the event accurately.  If the action or angular momentum involved in a given physical event is orders of magnitude larger than Planck’s constant, classical mechanics will describe the event with satisfactory accuracy. 

Quantum mechanics can be used to describe systems where classical mechanics is valid, but the effort involved would usually be substantially greater than that required to describe the system using classical physics.  The reverse, however, is not true.  In general, classical mechanics cannot be extended to provide accurate descriptions of atomic or smaller systems. 

The difference between quantum and classical mechanics is much more profound, however, than just that of scale.  Classical mechanics is deterministic.  Given a completely specified initial state, a set of forces and constraints, the values of the various system parameters can be determined at any subsequent time with arbitrary precision. 

Quantum mechanics, on the other hand, is fundamentally a probabilistic theory.  Where classical mechanics will predict a specific result for the measurement of some dynamic quantity (momentum, velocity, etc), quantum mechanics will predict that any of several values might be the result of an accurate measurement of the quantity involved and will assign a relative probability to the various results.  In some instances the “allowed” results may be grouped closely together, and the coarseness of the measuring apparatus or the large size of the dynamic system may mean that it is impossible to tell the difference between the prediction of the classical theory and the quantum theory.  In this limit, called the correspondence limit, the two theories merge, as they must. 

The basic philosophic difference between classical mechanics and quantum mechanics can be understood as follows: All physical theory is directed at explaining the results of experiments and at correlating these results in the simplest and most aesthetically pleasing manner possible.  Fundamentally the theory must explain the results of measurements.  In the classical limit, the measurement on a system can in principle be made so as to produce a vanishingly small reaction upon the system.  The interaction of a physical system with the measuring apparatus is a physical event, which must involve action in the amount of at least Planck’s constant, h, if any change of state of the measuring apparatus is to be effected – that is, if any reading is to be obtained.  For large systems the resultant alteration of the total state of the system (itself involving for example 10³⁰ h) is relatively so small as to be entirely negligible.  So, the measurement is presumed to have no effect whatsoever upon the state of the system being measured. 

For atomic systems, however, the action involved in the measurement is of the same size as the action involved in the system’s own evolution.  By making a measurement on such a system, we inevitably interact with the system in such a way as to significantly alter the state of the system.  The alteration of the system by the measurement can be treated only on a statistical or probabilistic basis.  The theory that deals with the results of such a statistical measurement must also be statistical in character. 

It is also worth mentioning here that a statistical theory is not an intrinsically inaccurate one.  While quantum mechanics cannot generally predict unambiguously the result of a single measurement on a single atom, it can generally predict the distribution of results of a large number of measurements.

Top of Page

Answer to Complex Number Magic:

To derive the first two trigonometric identities start with:

               e^{i(A + B)} = e^iA . e^iB

and proceed from there.

To derive the three theta identities start with:

               e^i3q = (e^iq)³

and proceed from there.

If you have difficulty with this exercise, I suggest you ask your Maths Teacher for assistance.

Top of Page

Answer to Diophantus' Age Problem

There are at least two quick ways to solve this problem.  The first uses algebra (in honour of Diophantus).  The second uses common factors and in this case is probably even easier and quicker than the algebraic way.

Have you actually done this problem yet?  If not, go back & do it before you read the answer.  It's more fun that way.

Method 1: Let x be Diophantus' age at death.  Then we have

              

 

Solving this equation gives: 

               

Method 2: The only number divisible by 6, 12, and 7, and also a number in the human lifespan is 84.  So, he spent 14 years in childhood, seven in youth, and 12 as a bachelor.  He married at 33, and five years afterward, at age 38, a son was born who later died at age 42, when Diophantus was 80.  Four years after that, Diophantus himself died at 84, twice the age to which his son lived.

Top of Page

 

 

 

Answer to Sharing Birthdays question: To solve the shared birthday problem, we need to make use of one of the basic rules of probability: the sum of the probability that an event will happen and the probability that the event won't happen is always 1.  If we can work out the probability that no two people will have the same birthday, we can use this rule to find the probability that two people will share a birthday.

P(two people share birthday) = 1 - P(no two people share birthday).

So, what is the probability that no two people will share a birthday?

The first person can have any of the 365 days in the year as his birthday and the probability that he has a birthday is 365/365 or 1. The second person's birthday has to be different. There are 364 different days to choose from, so the chance that two people have different birthdays is 364/365. That leaves 363 birthdays out of 365 open for the third person, and so on.

To find the probability that both the second person and the third person will have different birthdays, we have to multiply:

(365/365) * (364/365) * (363/365) = 0.9918.

If we want to know the probability that four people will all have different birthdays, we multiply again:

(364/365) * (363/365) * (362/365) = 0.9836.  I have left out the (365/365) since it is just 1.

Following the pattern we can develop a formula for the probability that n people have different birthdays:

((365-1)/365) * ((365-2)/365) * ((365-3)/365) * . . . * ((365-n+1)/365).

If you know permutation notation, you can write this formula as

               ³⁶⁵P_n / 365ⁿ

which can be re-written as:

 

                

which when calculated for n = 23 equals 0.493.

 

So, the probability that no two people share a birthday at a party of 23 people is 0.493.  Therefore the probability that two people at a party of 23 people do share a birthday is (1 - 0.493) = 0.507 or 50.7%. 

So, next time you are at a party with more than 23 people, it is more likely than not that two people at the party share a birthday.  Obviously, the more people present, the higher the probability that two people share a birthday.  Once you get to 32 people present, there is a 75% chance that two people at the party will share a birthday.  With 41 people present there is a 90% chance that two of them will share a birthday.  What good odds.

Top of Page

 

 

Answer to Chocolate Maths Task for 2008:

call the number you thought of x

multiply by 2 gives 2x

add 5 gives 2x +5

multiply by 50 gives 100x + 250

then add 1758 gives 100x + 250 + 1758

NOTE that 250 + 1758 = 2008 which is the current year.

Subtracting the year of your birth from the current year just leaves you with your age.  That's why you add 1757 if you have not had a birthday yet this year.  Otherwise you would work out to be a year older than you actually are.

The 100x (that is 100 times the number you first thought of) just allows the number you first thought of to be in the hundreds column.  It becomes the first digit of the 3 digit number you have finished up with.

The process will work each year.  Just keep adding 1 to the 1758 (and 1757) for each new year.

Cute eh?  Mathematics is truly very beautiful.

Top of Page