Q: What the heck are imaginary numbers, how are they useful, and do they really exist?

Posted on December 23, 2009 by The Mathematician

Mathematician: Imaginary numbers arise quite naturally when you start asking certain basic mathematical questions. Probably the best example is the following: Once we know how to multiply and add, we might ask ourselves “are there any numbers x that satisfy the equation x^{2} = 1 ?” The answer is yes, x=1 and x= -1 both satisfy it. If we try some more equations of this form, we might at some point ask the similar question, are there any numbers x that satisfy the equation x^{2}= -1 ? None of the ordinary (real) numbers satisfy this property. If we are feeling lazy we can stop there and say that this equation can’t be satisfied. But, if we are feeling creative, we might assume for a moment that there is a (special) number that satisfies this equation, (we’ll call it the number i ), and then see if we can derive its properties. By definition, of course, our number i satisfies i^{2}=-1 , but we can ask other questions about this bizarre number, such as “what is i^{n} for any positive integer n?” Some simple calculations show us that i^{n} is either equal to i , -i , 1 , or -1 depending on the value of n. We will now say that i is an “imaginary number”, as is any multiple of i  , such as 3 i and 14.2 i . We can now also introduce what we are going to call “complex numbers”, which are formed by the sum of a real and an imaginary number, such as 3+2 i  , 6.2-4.1 i  , 0+2 i and   9 + 0 i . It is not too hard to show that complex numbers yield a consistent theory, in the sense that when you add them, subtract them, multiply them, or divide them, you get a complex number back as the result of the operation. Furthermore, it turns out that all polynomial equations (which includes the equations x^{2} = 1  and   x^{2} = -1 that we considered before, as well as others like x^{3} - 2 x^{2} = 6 ) have at least one complex number solution (notice that real numbers are also complex numbers because, for example, we can write 3 as 3 + 0 i ). That means that no other weird types of numbers need to be introduced in order to find solutions to these equations, so complex numbers are “enough” to find every polynomial equation solution.

Okay, so maybe imaginary and complex numbers make sense to introduce and lead to a reasonable theory, but how could they possibly be useful? After all, in the real world we have real numbers of things (e.g. 3 frogs) and real amounts (16.2 dollars), not imaginary numbers of things. As it turns out, complex numbers are fantastically, staggeringly useful. This is especially true once we allow ourselves to start plugging complex numbers into functions (like e^{x} , sin(x) , x^{2} , etc.) , and see what output they produce. For example, imaginary numbers give us a useful and surprising link between the exponential function e^{x} and the sine and cosine functions, in Euler’s beautiful formula:

e^{i x} = cos(x) + i sin(x)

We see this funny object, e^{i x} surface again in an essential way in the subject of Fourier Analysis, which finds applications in the study of heat flow, signal processing, music filtering, data compression, image processing, and many other areas (in fact, one of the most referenced applied math papers of all time relates to how to quickly approximate a sum involving e^{i x} on a computer in order to do a discrete version of Fourier Analysis). Complex numbers pop up again in the theory of Taylor Series, one of the most used mathematical tools ever invented. One way to think about this is that Taylor Series provide a method for locally approximating a function using polynomials, and the region in which such approximation succeeds depends in a crucial way on the function’s behavior when it’s thought of a function of complex rather than real numbers. Complex numbers make an important appearance yet again in the theory of matrices. When we have a matrix (which is like a grid of numbers) that has only real numbers, and we want to split it apart into multiple matrices (via an eigen vector decomposition, let’s say) we discover that in general the matrices we divide it into may have complex (non-real) entries! What’s more, questions about prime numbers, solutions to problems in electrical engineering and the differential equations in quantum mechanics are deeply connected to complex numbers. The list of ways in which complex numbers appear (in a fundamental way) throughout mathematics is truly enormous.

Fine, so imaginary and complex numbers arise naturally and are extremely useful, but do they really exist? Well, this raises deeper (or should I say, different) questions, like whether numbers in general exist. Sure, numbers (complex or otherwise) exist as concepts in our brains, but no numbers ever appear in the physical world. We can see 3 cats, or the symbol “3” scrawled on a sign, but we will never see the actual number 3. Imaginary numbers, like real numbers, are simply ideas without any physical existence. They are both very useful (though with real numbers, it is much more obvious why that is so). But it is hard to see how one would (convincingly) argue that real numbers actually exist while imaginary numbers do not.

Some commentators have taken my argument above to imply that I think numbers (including imaginary numbers) actually exist (perhaps in some platonic sense). However, as was pointed out, this hinges very much on your definition of “exist”. Although the idea of numbers is very natural (you might almost say “obvious”), and probably would have been invented by almost any highly intelligent beings that may have happened to inhabit our planet, they are indeed still simply just an idea that we created. They lack physical existence, but “exist” in our minds as much as any ideas do.

Posted in -- By the Mathematician, Math, Philosophical | 19 Comments

Q: What’s that third hole in electrical outlets for?

Posted on December 16, 2009 by The Physicist

Physicist: Ground.

You can put a paperclip practically anywhere.

You can put a paperclip practically anywhere. All the same, don't try it.

The zero volt, large-slit wire is called the “return” or “neutral” line.  If it seems strange that the power company would supply you with a wire that has no voltage, keep in mind that what you really need is a voltage difference.  That’s why everything that uses electricity has at least two wires connecting it to its power supply.

Green=ground, Black=power, White=neutral

Green=ground, Black=120v, White=neutral. Notice that you can easily remove the green lines on the right without actually affecting anything.

The difference between the neutral line and the ground line is that the neutral line carries current (all the current that enters through the 120 volt line has to go back out), while the ground line is literally connected to the ground (usually through your plumbing) and carries no current.  The ground and neutral lines are spliced together at the fuse box (and generally again out by the local transformer).  You’d expect that since the lines are connected there’s no difference between them, and that’s almost entirely true.  The ground line, having no current and a shorter path to the ground, is always at exactly 0 volts, which is important information for many electronic devices to have access to.

In your house it’s important for most electronics to have a wire to dump energy out of in case of a short.  This can be the ground or neutral lines, since they’re practically identical.  If a device needs to know the difference between the 120v and neutral lines, then it will have one big tine and one small one at the end of its cord.  So when you’re wiring up a house it’s important to keep track of which line is the 120v line and which is the neutral line.  A good way to keep track is to wire up outlets with a temporary connection between the ground and neutral lines during installation, so that  if the electrician accidentally switches the 120v and neutral lines there will be a flash, a pop, a puff of smoke, and no lights.  So that’s one use for the nearly useless ground line.

It’s a bad idea in general to connect the neutral and ground lines.  Nothing bad will happen (if the wiring is up to code), but you will be creating a loop (check the figure above and connect the white and green lines at an outlet).  Loops are bad because they turn changing magnetic fields into changing electric current (and vice versa).  There are plenty of random magnetic fields out there, so you’d be introducing an unpredictable source of current to your electrical system.  Still, you’d probably never notice if the loop is relatively small (say, smaller than an entire house).

So the reason that ground lines run to outlets is that every now and then it’s nice to have access to a 0 volt, 0 current wire.  But it’s not really that important, which is why so many outlets don’t have a third hole.

Posted in -- By the Physicist, Engineering | 9 Comments

Q: Do physicists really believe in true randomness?

Posted on December 15, 2009 by The Physicist

Physicist: With very few exceptions, yes.  What we normally call “random” is not truly random, but only appears so.  The randomness is a reflection of our ignorance about the thing being observed, rather than something inherent to it.

For example: If you know everything about a craps table, and everything about the dice being thrown, and everything about the air around the table, then you will be able to predict the outcome.

Not actually random.

Not actually random.

If, on the other hand, you try to predict something like the moment that a radioactive atom will radioact, then you’ll find yourself at the corner of Poo Creek and No.  Einstein and many others believed that the randomness of things like radioactive decay, photons going through polarizers, and other bizarre quantum effects could be explained and predicted if only we knew the “hidden variables” involved.  Not surprisingly, this became known as “hidden variable theory”, and it turns out to be wrong.

If outcomes can be determined (by hidden variables or whatever), then any experiment will have a result.  More importantly, any experiment will have a result whether or not you choose to do that experiment, because the result is written into the hidden variables before the experiment is even done.  Like the dice, if you know all the variables in advance, then you don’t need to do the experiment (roll the dice, turn on the accelerator, etc.).  The idea that every experiment has an outcome, regardless of whether or not you choose to do that experiment is called “the reality assumption”, and it should make a lot of sense.  If you flip a coin, but don’t look at it, then it’ll land either heads or tails (this is an unobserved result) and it doesn’t make any difference if you look at it or not.  In this case the hidden variable is “heads” or “tails”, and it’s only hidden because you haven’t looked at it.

It took a while, but hidden variable theory was eventually disproved by John Bell, who showed that there are lots of experiments that cannot have unmeasured results.  Thus the results cannot be determined ahead of time, so there are no hidden variables, and the results are truly random.  That is, if it is physically and mathematically impossible to predict the results, then the results are truly, fundamentally random.

What follows is answer gravy: a description of one of the experiments that demonstrates Bell’s inequality and shows that the reality assumption is false.  If you’re already satisfied that true randomness exists, then there’s no reason to read on.  Here’s the experiment:

The set up: A photon is fired at a down-converter, which converts it into two entangled photons. These photons then go through polarizers that are set at two different angles. Finally, photo-detectors measure whether a photon passes through their polarizer or not.

The set up: A photon is fired at a down-converter, which converts it into two entangled photons. These photons then go through polarizers that are set at two different angles. Finally, photo-detectors measure whether a photon passes through their polarizer or not.

1) Generate a pair of entangled photons (you can do this with a down converter, which splits one photon into an entangled pair of photons).

2) Fire them at two polarizers.

3) Randomly change the angle of the polarizers after the photons are emitted.  This prevents information about one measurement to affect the other, since that would require that the information travels faster than light.

4) Measure both photons (do they go through the polarizers (1) or not (0)?) and record the results.

The amazing thing about entangled photons is that they always give the same result when you measure them at the same angle.  Entangled particles are in fact in a single state shared between the two particles.  So by making a measurement with the polarizers at different angles we can measure what one photon would do at two different angles.

It has been experimentally verified that if the polarizers are set at angles \theta and \phi, then the chance that the measurements are the same is: C(\theta, \phi) = \cos^2{(\theta-\phi)}.  This is only true for entangled photons.  If they are not entangled, then C = .5 = 50\%, since the results are random.  Now, notice that if C(a,b) = x and C(b,c) = y, then C(a,c) \ge x+y-1.  This is because:

\begin{array}{l}P(a=c)\\= P(a=b \cap b=c) + P(a \ne b \cap b \ne c)\\\ge P(a=b \cap b=c)\\= P(a=b) + P(b=c) - P(a=b \cup b=c)\\\ge P(a=b) + P(b=c) - 1\end{array}

We can do two experiments at 0°, 22.5°, 45°, 67.5°, and 90°.  The reality assumption says that the results of all of these experiments exist, but unfortunately we can only do two at a time.  So C(0°, 22.5°) = C(22.5°, 45°) = C(45°, 67.5°) = C(67.5°, 90°) = cos2(22.5°) = 0.85.  Now based only on this, and the reality assumption, we know that if we were to do all of these experiments (instead of only two) then:

C(0°, 22.5°) = 0.85

C(0°, 45°) ≥ C(0°, 22.5°) + C(22.5°, 45°) -1 = 0.70

C(0°, 67.5°) ≥ C(0°, 45°) + C(45°, 67.5°) -1 = 0.55

C(0°, 90°) ≥ C(0°, 67.5°) + C(67.5°, 90°) – 1 = 0.40

That is, if we could hypothetically do all of the experiments at the same time we would find that the measurement at 0° and the measurement at 90° are the same at least 40% of the time.  However, we find that C(0°, 90°) = cos2(90°) = 0 (they never give the same result).

Therefore, the result of an experiment only exists if the experiment is actually done.

Therefore, you can’t predict the result of the experiment before it’s done.

Therefore, true randomness exists.

As an aside, it turns out that the absolute randomness comes from the fact that every result of every interaction is expressed in parallel universes (you can’t predict two or more mutually exclusive, yet simultaneous results).  “Parallel universes” are not nearly as exciting as they sound.  Things are defined to be in different universes if they can’t coexist or interact.  For example: in the double slit experiment a single photon goes through two slits.  These two versions of the same photon exist in different universes from their own points of view (since they are mutually exclusive), but they are in the same universe from our perspective (since we can’t tell which slit they went through, and probably don’t care).  Don’t worry about it too much all at once.  You gotta pace your swearing.

As another aside, Bell’s Inequality only proves that the reality assumption and locality (nothing can travel faster then light) can’t both be true.  However, locality (and relativity) work perfectly, and there are almost no physicists who are willing to give it up.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 177 Comments

Q: Could a simple cup of coffee be heated by a hand held device designed to not only mix but heat the water through friction, and is that more efficient than heating on a stove and then mixing?

Posted on December 13, 2009 by The Physicist

Physicist: You could definitely make a device that heats water through mixing.  In fact, that’s exactly how scientists (Joule) figured out how to equate heat energy and kinetic energy in the first place.

Joule's device, which turned to energy of a dropping weight into heat.

Joule's device, which turns the energy of a dropping weight into heat.

When you introduce turbulence to a system the energy flows from large scale eddies into smaller and smaller scale eddies.  At some point the eddies are about the size of molecules (this takes about one minute).  At this point you’re no longer talking about the flow of a fluid, and are instead talking about the random motion of molecules (heat).

Fun fact!: In two dimensions turbulence actually starts at small scales and moves up into larger scales!  You can see this exhibited in weather systems larger than about 15 km across in the atmosphere (at these scales the atmosphere is effectively flat).

A good way to induce large-scale eddy currents.

A good way to induce large-scale eddy currents.

From this point of view the difference between a mixer and a heater is that a mixer induces large eddy currents, and a heater induces the smallest possible eddies.  Ultrasonic heaters fall neatly in between.

Efficiency is defined as \eta = 1 - \frac{E_{heat}}{E_{in}}, where \eta is efficiency, E_{in} is the energy put in, and E_{heat} is the energy lost to heat.  You’ll notice that no lost heat means 100% (\eta = 1) efficiency, and if all the energy in is lost to heat then the efficiency is 0%.  So the nice thing about trying to create heat intentionally, is that you’ll always be 100% efficient (if you lose some heat to heat, would you notice?).  Or close enough at least.  What you have to worry about is accidentally heating up the wrong thing.  Mixers generate large eddies which can move the cup, makes noise, and what-have-you.  In other words, some of the energy is wasted heating up stuff near the cup (if you can hear it, then some of the energy is being wasted in your ear).

An electric stove top pushes energy into water at a rate of about 1kW.  A normal blender (mixer) draws power at about 400W, and loses almost all of it to noise and vibration.

So, to actually answer the question, you can heat coffee through mixing, but you’ll get plenty of splashing, it’d be slow, and it’d lose a fair amount of energy through noise and vibration.  You’d be better off with a normal mixing device that has a heating element built in, or heating on a stove first.

Posted in -- By the Physicist, Engineering, Physics | 2 Comments

Q: What did Einstein mean by: “Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

Posted on December 9, 2009 by The Physicist

Albert Einstein:
“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

Mathematician: It is impossible to know precisely what Einstein meant by this quote without being able to ask him. It is often taken to mean that he did not have a strong understanding of math or that he was bad at it when he was young. It is simply wrong, however, to say that Einstein was bad at math. Some of his papers were quite mathematically sophisticated, involving advanced subjects such as stochastic differential equations and tensor calculus. What’s more, he excelled in math as a youth.

The following Einstein quote may help us gain a bit more perspective:

“Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.”
(source: In A. Sommerfelt “To Albert Einstein’s Seventieth Birthday” in Paul A. Schilpp (ed.) Albert Einstein, Philosopher-Scientist, Evanston, 1949.)

Perhaps what Einstein meant when he claimed to have difficulties in math is that he felt as though it was a struggle to learn some of the very advanced math necessary for formulating his theories, or that (compared to mathematicians or mathematical physicists) his math skills were not exemplary. However, he certainly was far, far more gifted at math than your average person. All of this being said, the original quote was actually directed at young students, so it may have reflected little more than an attempt to encourage them to persevere despite their perceived difficulties.

Physicist: Here’s my guess. Einstein witnessed the end of what might be considered the “intuitive physics” of the 19th century and before.  Most of that was his own damn fault.  In 1905 (Einstein’s “miracle year“) he introduced the world to both quantum mechanics (QM) and special relativity.  Up until that time (most of) modern physics was something that could be easily pictured and intuited.  You can draw the trajectories of moving objects, electric and magnetic fields can be modeled using field lines, you can picture heat flowing like a fluid, etc.

Electric field lines

Electric field lines (in blue). Easy to picture.

Moreover, the math involved is fairly simple.  As in High School simple.  With the advent of relativity and QM came an age of science where the conclusions being reached fell entirely out of the math.  Nowadays, not only is intuition pretty useless, but it will actively lead you wrong more often than not.
Some of the first things we learned in relativity and QM are: there’s no such thing as “now” for two different points, particles are actually waves, those waves are actually particles, going fast makes time slow down, going fast makes lengths shorter, energy and matter are the same stuff, space and time are nearly the same, sometimes things will suddenly appear on the other side of barriers they can’t get through, and on and on…
None of these things could be predicted intuitively, and many of them were the result of some pretty tricky math.  The brand of physics that’s practiced today (the Standard Model, String Theory, and whatnot) involve math that’s so crazy hard that it takes many years of training before you can even start to grasp what’s going on.  Einstein had many years of training, but could not have imagined the tremendous variety of different maths that would come into play because of his theories.
It was this last leap in complexity that threw Einstein.  Homeboy was really smart, really really good at math, and a snappy dresser.  However, even the people you would consider to be unimaginably intelligent (I’m talking about the ‘Stein here) are almost always over shadowed by someone way smarter, and with better, more specific, preparation.  Although he had a solid physics background, it was up to the mathematicians (with their math and their glasses) to move the science forward.

Another Physicist: I am not familiar with the quote. I only have a wild guess. In the period between the publication of special relativity and general relativity he took some time to learn enough differential geometry to develop his ideas. This apparently did not come easily to him, and involved a lot of consultation with other people in Europe.  He and Levi-Civita were in very frequent communication for example. So he may have been responding to someones comment of the sort we all hear that they we’re having a hard time with math.

Posted in -- By the Mathematician, -- By the Physicist, -- Guest Author, Philosophical, Physics | 61 Comments

Q: Why does saliva boil in the vacuum of space?

Posted on December 6, 2009 by The Physicist

Physicist: This is my new favorite question.  Blood, saliva, lymph, and whatever’s in your eyeballs (eyeball fluid?) all boil in space because they’re all basically water.  Both liquids and gases are made up of molecules that fly about at random, the difference is in the balance of kinetic energy and binding forces.  In a liquid the binding forces win so it stays together, and in a gas the kinetic energy wins so it flies apart.  This is why you can boil water.  You add heat energy (heat is really just random kinetic energy), the balance is tipped toward kinetic energy, and the water flies apart.

In the case of water the binding forces include: dipole-dipole attraction, surface tension, and atmospheric pressure.  Atmospheric pressure manifests as air molecules pounding on water molecules that make it to the surface.  That is to say, air pressure is a force that acts only on the water’s surface.  It turns out that water can’t hold itself together without external pressure.

The "Phase Diagram" for water. For a given external pressure and temperature this tells you what phase water will be in.

The "Phase Diagram" for water. For a given external pressure and temperature this tells you what phase water will be in.

In fact, you would have to cool water down to nearly zero before you can expose it to space without vaporizing it.  Even then it would be ice, not liquid water.  There are plenty of youtube videos of water boiling in vacuum chambers.

So, the moral of the story is: don’t leave the atmosphere.  You’ll boil too much.

Posted in -- By the Physicist, Physics | 3 Comments