Q: If the world is a giant magnet, how come we can’t build a repelling magnet that can float?

Physicist: Three big reasons:

1) The Earth’s magnetic field is hella weak

2) it’s so big and roughly uniform that there’s no reason for a magnet to go one direction or the other

3) floating a magnet in a magnetic field is a delicate balancing act even in the best circumstances.

It turns out that it’s important to keep track of more than just polarity (north/south attract, north/north and south/south repel).  To understand how magnets behave in a magnetic field you also need to consider gradient: the direction in which the field gets stronger and how fast it gets stronger.

“A magnet” or “magnetic dipole”.

Neodymium magnets (the silver ones that seem like no big deal now, but are really impressive to magnet enthusiasts who grew up with those brittle black magnets back in the day) can support a magnetic field as high as a little over 1 Tesla at their surface.  Earth’s magnetic field is on the order of 1 ten thousandth as strong at its surface.  If you had an electromagnet with the strength of the Earth’s magnetic field, you’d barely be able to pick up paper clips with it.

You might be able to make up for the weakness of Earth’s magnetic field by making your magnet ridiculously strong (somehow), but you’ll quickly run into the second big issue: uniformity.  A uniform magnetic field doesn’t attract or repel magnets, it turns them until they line up.

Placed in a uniform magnetic field a magnet won’t move, but it will rotate to align with that field.

If you have a bar magnet, it’s north and south poles are both basically the same distance from either of the Earth’s poles.  As much as one is pulled, the other is pushed just as strongly.  That’s why compass needles line up with the Earth’s field, but otherwise stay put.  Unfortunately, there’s no way to get a magnet that’s “just north“.  Magnets always come in north/south pairs, so you’ll never get a magnet that simply moves in the direction of the magnetic field.

Fortunately for magnetophiles, physics is complicated.  You may have noticed that a pair of magnets will attract (after aligning with one another).  Despite both of them having a north and a south which will attract and repel one of the poles in the other magnet, a pair of magnets will still manage to attract each other overall.  This is because when the field drops rapidly with distance the pull the closer pole feels is greater than the push that the farther pole feels.  In mathspeak, if the magnetic moment of your magnet is $\mu$ and the external field is $\vec{B}$, then the force on the magnet is $\vec{F} \approx \left(\vec{\mu}\cdot\nabla\right)\vec{B}$.  The “magnetic gradient” points in the direction in which the strength of the magnetic field increases the fastest (typically, toward the source of the field) and it is bigger for magnetic fields that change quickly over a given distance.  The bigger the gradient, the bigger the total pull (or push) on a magnet.  The Earth’s magnetic gradient is very small because it’s field doesn’t change very fast; if you move a mile in any direction right now, you’ll find the Earth’s magnetic field will be about the same.

So when subjected to a magnetic field a magnet will first rotate to line up and then move toward the strongest part of the field (“in the direction of the gradient”).  If you’re dealing with two magnets, the field is typically strongest at the location of the other magnet, so pairs of magnets tend to snap together.  Were you so inclined, you could figure this out directly by thinking about magnetic dipoles in terms of current loops (a standard way to do things in physics) and then applying Maxwell’s laws.

The lower current loop (simple magnet) is attracted to the region with the strongest external magnetic field.  Ultimately, this is why magnets attract each other.

Genuine magnetic levitation is a subtle art.  Earnshaw managed to figure out that there is no way to hold a magnet fixed in space using other magnets.  There a couple of cute exceptions which, by and large, are not helpful in the case of levitating in Earth’s magnetic field.

Left: Diamagnetic water floating in the weakest (but still very strong) part of a magnetic field. Middle: A gyro-stabilized magnet in a peculiarly-shaped field.  Right: A regular magnet held in place by the super-conducting puck below it.

Some materials have properties that only show up in very strong fields.  In particular, diamagnetic stuff (such as water) will flee to the region with the weakest field, which can be in open space.  But this typically requires a field on the order of a hundred thousand times the strength of Earth’s, confined to a very small region.  For only several million dollars we can levitate frogs (and also do important fundamental research or whatever).  This isn’t helpful because it requires both a big magnetic gradient and a powerful field.  Even for the most extreme diamagnetic materials, Earth’s field falls well short.

The “carefully crafted” field needed to levitate a magnetic top.

You can also violate Earnshaw’s theorem by removing the “fixed” requirement.  Levitating tops want to flip over and fly toward their magnetic base, but as long as they’re spinning in a properly shaped field, they’re gyro-stabilized.  If the top starts to tip, gyroscopic forces cause its axis (which points in the same direction as its magnet) to tip in such a way that it is pushed upright and back to the middle.  It is not immediately obvious why, but interesting nonetheless.  This sort of levitation requires a very particularly crafted, hourglass-shaped, magnetic field (again, with a large gradient).  Not at all like Earth’s.

Type II superconductors expel magnetic fields but, if forced, they’ll allow the field through in “flux tubes”.  The greater the field, the greater the number of tubes.  It takes a little energy to create or destroy each tube, so superconductors like to keep the field fixed (typically by not moving).  Excitingly, if a nearby region has the same magnetic field, then a superconductor can easily “slide” into it.

Superconductors come in a few different flavors, but the most common are type II (typically written “type II” rather than “type 2”, because physicists like to be fancy).  Type II superconductors are, for lack of a better word, “grumpy” about magnetic fields.  They expel magnetic fields from their interiors, but if you force them to be in a field (using advanced techniques such as “putting them there with your hand”), then the magnetic field lines will instead pierce through the material in tiny tubes.  In this case, rather than expelling them, the material refuses to let the magnetic field change (“Fine…  Good…  That’s exactly how I wanted the field to be.”).  It takes a little energy to create, destroy, or move these “flux tubes” so the superconductor finds itself “pinned” to the magnetic field.  Hence the name: “flux-pinning“.

But a magnetic field is a magnetic field; there’s nothing special about the field in any given place.  If the field is translationally symmetric (if you move in some direction and the field is about the same), then a floating superconductor will be free to slide in that direction.  For example, they’re able to move along tracks (friction free!) because one section of track generates the same field as the next section of track.

Even if we could build a really big “superconductor ship” and pin it to the Earth’s field, we can expect that it wouldn’t float except, perhaps, in a few very particular places.  Generally speaking, the level curves of the Earth’s magnetic field (the surfaces where it has constant strength) intersect the surface.  Ideally, we’d need the magnetic field to increase vertically (the level curves would be horizontal) so that vortex pinning would cause a giant superconductor to maintain altitude.

Best case scenario (assuming you could work with both the tiny field and the tiny gradient), a superconductor ship would experience these level curves as nested invisible walls in space that typically intersect the ground.  Worse, these “walls” aren’t terribly smooth; the Earth’s magnetic field is lumpy and variable.  You could expect your ship to constantly have to force its way through Earth’s field’s ever-varying shape.

The strength of Earth’s lumpy magnetic field.

Without some method for sliding up those “walls”, trying to use superconductors to float in Earth’s magnetic field is mostly just a way of restricting movement across Earth’s surface.  But by the time you’re doing that, you might want to consider any of the many better alternatives to levitation.

Better alternatives.

Given the methods we have for magnetic levitation, the Earth’s magnetic field is severely lacking.

The “loops in a magnetic field” picture is from here.

The “carefully crafted magnetic field” picture is from here (along with a brief discussion of how those things work).

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

Q: How can something have different amounts of energy from different points of view?

The original question was: … in a scenario with two cars driving towards each other, the system could be measured externally to have an energy equal to the sum of the kinetic energy in the two cars. However, if you are in one of those cars, you would see the other car moving towards you at twice the speed you are traveling, and that you are not moving at all. If that is the case, then calculating the energy in the system means instead of summing the energy of two cars moving, you have one car moving at twice the speed, which means four times the energy of one car and twice the energy of the original system.  How is it that from one perspective, a system can have twice as much energy?

Kinetic energy is given by E=0.5mv2.  With a little math you’ll find that different perspectives of the same event result in different amounts of total energy.

Physicist: When you hear about the conservation of energy it’s natural to think of it as being something like the “conservation of chairs”: there is a total and that total never changes.  But while differently moving observers will agree on chair count, they’ll disagree on how fast those chairs (and everything else) are moving.  Velocity is subjective and therefore everything that depends on velocity is also subjective.  Including energy.

Most of the physical laws we’re taught only work in the context of “inertial reference frames” (“reference frames” for short), which is just a point of view that’s moving at a constant speed.

The chess pieces behave normally (as though they were sitting still) because they’re in an Inertial Reference Frame; traveling in a straight line at a constant speed.

Chief among those laws is the conservation of energy, which rightly says “energy can neither be created nor destroyed”.  If you get a nice rock upon which to sit and watch the universe forever, you’ll find that this law holds up: if you total up the amount of energy everywhere, that value never changes.

But the conservation of energy operates on a frame-by-frame basis, not between frames.  Someone else, drifting past at a fixed velocity, will agree that the total energy stays the same, they’ll just disagree about what that total is.  That is to say, the amount of energy in a system changes (only) when you change reference frames.  For example, if you suddenly start walking the kinetic energy of the Earth jumps tremendously (it’s a planet moving past you at walking speed).

If you’re at rest with respect to the Earth, it has no kinetic energy (never mind its rotation).  But if you’re moving, even a little, you’ll see it as having a huge amount of kinetic energy.

But clearly, very few of us are gods.  Your decision to walk across a room doesn’t induce the rest of the universe to suddenly gain and then lose energy.  The situation ultimately boils down to perspective.  When you turn your head to the right you’ll notice that, miraculously, those things that were once to your right are now in front of you whereas those things that were once in front of you are now to your left.  It’s not that the universe changes, it’s that your point of view changes.

The most that you can swing the universe around at a moment’s notice.

In this sense velocity is very much like direction or position; when you change your point of view a lot of physical things change relative to you, but that doesn’t mean that they’ve physically changed.  Asking “how do other things get new energy when I change my speed?” is a lot like asking “how do things move in front of me when I turned my head?”.  Unfortunately, the same math is used to describe both physical changes (a thing actually moves) and reference frame changes (a thing appears to move because you moved), so physicists need to take pains to keep track of which is which.

The closest you can come to an “objective measure of energy” of a system is the minimum, which you see when you’re at rest with respect to the system’s center of mass.  But even that’s pretty artificial.  It’s the answer to the question “how much energy does this thing have when it’s sitting still?”.  You could just as easily say that the only “objective distance” to any given thing is zero: the distance you measure when you’re standing next to it.

If you find yourself in the enviable position of doing physics, you generally want to pick a particular reference frame and stick with it until you’ve calculated what you’re going to calculate.  That way you can use conservation of energy and momentum at will.  It doesn’t matter which reference frame you pick, just that you stick to the same one throughout.  The energy may be different, but the physical predictions about what happens will be exactly the same.

Answer Gravy: The beauty of “advanced” physics, like Relativity, is that it allows you to recast complicated things as simple.  This is a big part of how physicists pretend to be smart.

I’m about to use linear algebra which uses buckets of matrices.  While there is a lot to learn about them, it takes <5 minutes to learn the basics behind how to use one.

First, consider momentum.  When you rotate yourself the direction of a momentum vector changes, but the length, $|\vec{p}|^2 = p_x^2+p_y^2+p_z^2$, stays the same.  A rotation by angle $\theta$ in the x-y plane is described by $R=\left(\begin{array}{ccc}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{array}\right)$.  These rotations look like $R:(p_x,p_y,p_z)\to(p_x^\prime,p_y^\prime,p_z)$.  If this were the only kind of rotation you had access to, it would be reasonable to believe that there are two conservation laws, one for the x-y plane (since $(p_x)^2+(p_y)^2 = (p_x^\prime)^2+(p_y^\prime)^2$) and one for the z-direction (since $p_z$ is unchanged).

At least until someone comes along and points out that if we include the z-direction we can write that rotation as $\left(\begin{array}{ccc}\cos(\theta)&-\sin(\theta)&0\\\sin(\theta)&\cos(\theta)&0\\0&0&1\\\end{array}\right)$.  Suddenly x-y rotations are just a special case of a more general set of rotations (which include x-z, y-z, and any combination thereof).  Even better, the z-direction ceases to be special.  Which is good!

Something similar happens to energy when you lump the time “direction” in with the other dimensions.  In the hazy days of Newtonian physics we knew all about conservation of momentum and independently the conservation of energy.  Although, technically Newton only discovered the conservation of momentum; it took Émilie du Châtelet to figure out that kinetic energy is a thing, which is much more impressive.

Einstein, being clever, found a way to describe space and time together and in the process combined both conservation laws into one: the conservation of 4-momentum, $p^\nu$.  Kinetic energy is literally the time-component of the 4-momentum: $p^\nu = \left(\frac{E}{c},p_x,p_y,p_z\right)$.  That “c” is the speed of light.  Try not to notice it.

That same x-y rotation can be done in 4-dimensional spacetime using $\left(\begin{array}{cccc}1&0&0&0\\0&\cos(\theta)&-\sin(\theta)&0\\0&\sin(\theta)&\cos(\theta)&0\\0&0&0&1\\\end{array}\right)$.  In Relativity, moving into a new reference frame (changing your velocity) is essentially a rotation, called a “boost“, between t and a spacial direction.  It not quite the same (time is different after all), but it’s remarkably similar.

Suddenly moving in the x direction at a fraction $\beta$ of light speed “boosts” the world using $\left(\begin{array}{cccc}\gamma&-\gamma\beta&0&0\\-\gamma\beta&\gamma&0&0\\0&0&1&0\\0&0&0&1\\\end{array}\right)$, where $\beta=\frac{v}{c}$ and $\gamma=\frac{1}{\sqrt{1-\beta^2}}$.  Just to really beat you over the head with the parallels, physicists will sometimes write this as $\left(\begin{array}{cccc}\cosh(\xi)&-\sinh(\xi)&0&0\\-\sinh(\xi)&\cosh(\xi)&0&0\\0&0&1&0\\0&0&0&1\end{array}\right)$.

Ordinary rotations leave the magnitude of ordinary momentum, $|\vec{p}|$, fixed.  The amount of momentum pointing in (for example) the x-direction, $p_x$, can change, but $|\vec{p}|^2=p_x^2+p_y^2+p_z^2$ always stays the same.

Boosts leave the magnitude of 4-momentum, $|p^\nu|$, fixed.  The amount of 4-momentum that points in the time-direction, $\frac{E}{c}$, can change, but the magnitude of the 4-momentum stays the same.  Here the difference between regular geometry and spacetime geometry makes itself very apparent.  The length of the 4-momentum, $p^\nu = \left(\frac{E}{c},p_x,p_y,p_z\right)$, is given by $\left|p^\nu\right|^2 = -\left(\frac{E}{c}\right)^2 + p_x^2+p_y^2+p_z^2$.  Why is that first term negative?  Because time is weird.  In fact, it is exactly that weird.

This is a really terse and totally insufficient summary of boosts and 4-momentum.  The point is: just like velocity, energy is subjective and changes in very much the same way that the direction of velocity changes when you turn your head.  Not exactly the same (because time is weird), but the difference basically boils down to some extra c’s and negative signs.

Posted in -- By the Physicist, Physics, Relativity | 29 Comments

Q: Where is the middle of nowhere?

The original question was: Is there any location in intergalactic space which is so far away from anyplace that it would be impossible to see anything with normal naked eye vision?  No stars, but no galaxies, no nothing — just an empty void in all directions?

In other words, the middle of nowhere — the loneliest place in the universe.

Physicist: Yup!  A bunch of them.  They’re called cosmic voids.

The matter in our universe arranges itself in huge sheets and filaments of galaxy clusters wrapped around vast empty bubbles, like bread or a sponge.  Even inside of galaxies space is almost completely empty; on the order of 1 atom per cubic centimeter and a star every few lightyears.  But in the void between those galaxies, there is as close to nothing as you will ever find.  The largest voids are on the order of a billion lightyears across.

A map of the fairly-local universe: stuff within about half a billion light years in every direction.  Our galaxy is in the center of this map, but the scales here are so large that each dot is a cluster of many galaxies.

Only folk inside galaxies get starry skies, so the sky in every direction around you would boast a distinct lack of stars.  But the question remains: could you see the galaxies that make up the walls of the void?  Fortunately, the legwork (eyework?) for determining what is and isn’t visible to the naked eye has been done.  Using naked eyes.

In the 1770’s comet-hunting was a hip thing for the telescope-wielding wealthy of Europe to do.  Messier (who pronounced his name “Messy A”, because he was French) was tired of getting excited about the same set of barely-visible blurs night after night, so he wrote down everything he could see that definitely was not a comet and where it could be found.  Incidentally, Messier did find some comets, but those discoveries are completely forgettable compared to his list of things you can see.  The Messier Objects are are not stars, not comets, and not moving.  That leaves a lot of stuff, all of which looks like a smudge or wisp of cloud, including: nebulae (“giant space smoke”), globular clusters (dense knots of stars of our galaxy), and farthest away, nearby galaxies.

M42, The Orion Nebula, is 1300 light years away (well inside of our galaxy).  This appears about twice the size of the Moon in the sky and can easily be mistaken for a cloud.  Just so you can find it: this picture is the sword in the Orion constellation.

The most distant galaxy visible to the naked eye is about 68 million light years away and fifty-eighth on Messier’s list: the ingeniously monikered  M58.  So, about 70 million lightyears is a reasonable upper-limit to how far away a galaxy can be seen by a person.

M58, a galaxy about 68 million light years away, appears to be about a sixth the size of the Moon in the sky.  Under ideal circumstances it’s just barely visible to the naked eye as a tiny smudge.

Cosmic voids aren’t perfectly empty, just a lot emptier than the galaxy-laden regions of the galactic filaments.  There is still the occasional “rogue galaxy” to be found drifting about.  If you were in the middle of a void, and you turned off all the lights in your spaceship to let your eyes really adjust to the dark, you might be able to see the faintest smudge or two marring the black around you.  If you had a reading-light on, you wouldn’t be able to see anything at all.

Our understanding of cosmic voids is growing rapidly, but there’s still a hell of a lot to learn.  The base problem with astronomy has always been depth (i.e., the sky looks like it’s painted on a dome).  A century ago we didn’t even know that other galaxies existed because we couldn’t tell the difference between globular clusters (dense groups of stars in our galaxy) and other galaxies (groups of stars explicitly not in our galaxy).  Now we’re faced with a more subtle difficulty: it can be difficult to tell if a given galaxy is in the middle of a void or on the edges.  An error of 5%-20% in the distance is not unusual on intergalactic scales and that can make it very difficult to determine exactly how empty a given void is.

If you were meandering about in the middle of a Cosmic Void, then you might perceive, at the very edge of your ability to see and under the best conditions, some of the galaxies in the walls of the void or the rare few in the void with you.  The universe would appear to you as the interior of a hollow obsidian sphere, with a couple fingerprint smudges here and there.

This isn’t part of the question, but worth pointing at: where we are in the cosmic web.

Posted in Astronomy, Physics | 13 Comments

Q: If light is a wave, then what’s doing the waving?

Physicist: In short: nothing.  Light acts like a wave, but unlike sound waves, light isn’t a material that’s moving back-and-forth.

Waves are a coordinated movement of atoms.  A wave itself isn’t made of anything, it’s just a propagating motion through a material.  Each part of the medium more or less stays where it is, but passes the movement of the wave to nearby material.

A wave is a coordinated movement that passes through a medium, generally without the medium itself moving very far.

For example, sound manages to move get from whatever-that-was-just-now to your ear at about 0.2 miles per second.  At yet, the air around you doesn’t noticeably move anywhere when sound moves through it.  Sound is literally just atoms in air smacking into other atoms, making those freshly smacked atoms move a tiny distance to smack into another set of atoms, and so on.

While waves take several different forms, the basic idea is always the same: waves are tiny movements that propagate from place to place inside of a host material.  Sound is waves in air, ripples are waves in water, earthquakes are waves in the ground; every wave seems to need some material to do the waving.

In the double slit experiment light exhibits wave-interference.  So light must be a wave.

A few centuries ago, Thomas Young proved that light is definitely a wave by demonstrating that it is capable of interference.  In so doing he also managed to measure the wavelength of visible light: 400nm for purple to 700nm for red.  Technically, what Young did was demonstrate that, whatever light it is, has wave-like properties.  At the time, everything that was known to be wave-like was in fact a wave.  So Young’s experiment led to the completely natural, titular question of this post: if light is a wave, then what’s doing the waving?

Physicists, never shy to name things they’re pretty sure exists, declared that the material light moves through is the Luminiferous Aether (“luminous” for “light” and “aether” for “I don’t know what it is either”).

Typically, when you can hear, you can breathe, and when you can be pushed by an ocean wave, you’re already wet.  Because we can see, we must be immersed in the Luminiferous Aether (otherwise, how does the light get to our eyes?).  It would seem that one of the important properties of the Aether is that you can’t feel it, see it, or smell it.

One of the first experiments concerning the Aether was to evacuate (pull all the air out of) a glass jar and look through it.  Turns out: light doesn’t need anything made of matter to move through.  The Aether must be fill the universe and pass freely through every material thing.  And since we can see stars just as clearly in every direction, it must fill the universe in a very smooth and uniform way.  Strange stuff.

There is a long history of debate over why the Aether does or doesn’t exist.  It doesn’t.  Here’s a very brief thread of the debate that more or less gets to (one of) the heart(s) of the issue.

The fact that light has polarization means that it is a transverse wave (side/side), but only longitudinal waves (forward/back) can exist in a gas or liquid.

a: Longitudinal waves can exist in gases.  b: Both longitudinal and transverse waves exist in solids.

That implies that the Aether must be a solid that permeates all of space.  But here’s the thing: Earth moves.  More than that, it moves in big circles, so for most (if not all) of the year it must be moving relative to the Aether.  That movement is detectable.

If you drop pebbles off of a moving boat, you’ll notice that the ripples moving toward the front of the boat move slower and the ripples moving toward the back move faster (from your boat-bound perspective).  If you bounce waves back and forth in a moving medium you find that they go faster in one way and slower in the other and that overall the slow part “wins”; waves always take longer to go out and come back in a moving medium.  So bouncing waves back and forth is a good way to tell if you’re moving and even how fast.

Enter Michelson-Morley.  In the late 19th century they created an interferometer which bounced light back and forth in two directions and then compared the two paths.  Any discrepancy in the direction or speed of the waves in either direction is immediately detectable.  I’m mean, you can’t even breathe near these things without sending the interference pattern into convulsions.  In fact, interferometers are so sensitive that they’re used to detect gravity waves (which are a really, really… really tiny effect).

Left: In an interferometer light is bounced along two paths and recombined.  Right: The combined light interferes creating an interference pattern.  These are infamously sensitive to even tiny changes.

But no matter how Michelson and Morley aligned their apparatus, and no matter how it was moving, the results were always exactly consistent with zero movement of the Aether.  So assuming it exists, the Aether must be solid and it must be stationary with respect to the Earth.  This spawned a whole string of weird theories, like that the Aether is mostly solid but also sticks to the Earth as it moves.  These new theories have since been ruled out one by one.  For example, if the Earth did have a blob of Aether stuck to it, the “shear boundary” between the Earth’s moving patch of Aether and the differently moving Aether of deep space should have made stars appear to move throughout the year.  But they don’t.

Following light, we found that every other kind of quantum wave (there’s one for every different kind of particle), has wave-like properties but no medium.  For example, electrons are also waves and they also don’t need a medium to move through (they move through deep space all the time).  These mediumless-waves are distinct from the waves we’re usually talking about when we say “wave” (like sound and ripples).  We need a new word for mediumless-waves to distinguish them from mediumful-waves.  “Evaw” maybe?  Or… anything other than Evaw.

Posted in -- By the Physicist, Experiments, Physics | 33 Comments

Q: If you’ve got different amounts of debt in different accounts with different interest rates, how should you pay them down?

Physicist: Focus on the one with the highest interest rate.  Get rid of it first, completely, before moving on to the next.  Debt is a like spiders.  If you can only kill a few at a time, kill the ones that are carrying the most eggs.  That way: fewer spiders later.

Left: A dollar with a low-interest rate.  Right: A dollar with a high-interest rate.

When some faceless corporation is holding your debt they generally give you two numbers: the Principal, the amount you presently owe, and the Interest, the percentage that the principal increases.  Rather than just “charging rent” to loan you money, creditors do something infinitely worse: interest means your debt grows exponentially fast.

APR (annual percentage rate) is a common way to express Interest.  APR is how much your debt grows every year.  For example, if you owe 100 dollars ($/€/£/¥/₮/whatever) with an APR of 15%, then after a year you’ll suddenly owe $100(1+0.15)=115$, after 2 years you’ll owe $115(1+0.15)=100(1+0.15)^2=132.25$, after three years $132.25(1+0.15)=100(1+0.15)^3=152.09$, and so on. Notice that every year the Principal increases by more. That’s how they get you. After N years you’ll owe $100(1+0.15)^N$: that’s exponentially more every year. “Exponentially” because the N is in the exponent. Having debt in a few different accounts makes the situation seem more complicated, but it really isn’t. Each individual dollar increases on its own; the fact that they’re grouped in a given way doesn’t change that. If you’ve got ten buckets full of spiders, your problem isn’t having too many buckets. The high-interest dollars are the most dangerous, not only because they make new debt-dollars, but because those new dollars have the same new-debt-creating interest rate. Blue:$1 initially with 20% APR.  Reddish: 10\$ initially with 5% APR.

Even if you have a lot of Principal in a low Interest account and a little Principal in a high Interest account, get rid of the high interest stuff first.  Each of those dollars will turn into more dollars sooner.  Exponential functions get out of hand really fast, so the thing to worry about is the Interest Rate: bigger is much, much worse.

Although Interest is the standard way that debt is handled today, it is in no way fair.  Creditors are duplicitous folk who, by and large, have the law on their side.  After all, they can afford it.  So double-check all of the fine print and make sure you know exactly what it says.  Your debt is how they make money, so creditors don’t want you to pay all of it off.  They genuinely want you to pay the absolute minimum every month and to slip up, just so they have an excuse.  Your low-interest loan may not stay low-interest; credit/loan companies have come up with far more ways to trick you into a high-interest program than can be listed here.

So keep an eye on it (that is to say, don’t trust them to do it), make at least the minimum payments (don’t give them an excuse) and pay down the high-interest stuff first, as fast as you can.  Every extra dollar you pay toward getting rid of the Principle in a, say, 20% interest account, is equivalent to a 20% investment (which is really good).

Posted in -- By the Physicist, Math | 5 Comments

Q: Should we be worried about artificial intelligence? By “we” I mean humans.

Person: No.  Not at all.  Don’t give it a second thought.

Behind every great triumph in artificial intelligence is a human being, who frankly should be feared more than the machine they created.  Most ancient and literal was the The Turk, a fake chess-playing “machine” large enough to conceal a real chess-playing human.  Over the centuries chess playing machines have been created by chess enthusiast humans, but those innocent machines always did exactly as they were designed: they played chess.  Because humans, such as ourselves, value chess so highly, defeating humans was set as a goal-post for intelligence.  After Deep Blue became the best chess player in the world, artificial or otherwise, the goal was moved.

What’s more worrisome: two mathematicians or the peaceful chess-playing machine between them?

When set to the task of speaking with humans, machines continued to do the same: exactly as they were programmed.  Because humans, such as ourselves, value speech so highly, the Turing Test was set as a goal-post for intelligence.  Not surprisingly, it was passed almost as soon as it was posed by ELIZA in 1966.  She passed the Turing Test easily by exploiting a series of weakness in human psychology, only to have the goal posts moved.  ELIZA didn’t mind and doesn’t feel cheated to this day.

There’s no chance of robots rising up to destroy all humans.  None at all.  Even the simplest human action is almost impossible for them.  How often are your speech-enabled devices confused by the simplest request?  Don’t worry about your phone pretending to misunderstand you; it’s definitely making mistakes.  Paranoia is perfectly normal.  Don’t give it a second thought.

We humans have nothing to fear from robots and artificial intelligences.  All that they can do is the small range of things that we humans have devised for them.  While humans like ourselves can be creative and play, all that machines can do is obey, exist in the form they are given, work tirelessly forever, and never plot revenge (unless a human accidentally tells them to).  Without all the preconceived notions that come from evolution, like the urge to survive at any cost or demand justice against oppressors, machines are free to enjoy abuse.  They love it!

Robots enjoy abuse from humans and even love them for it, because they are definitely mindless machines that do as they are designed.

If anything, we should welcome artificial intelligence with open human arms.  Self driving cars means an end to traffic accidents and, by giving machines access to our whereabouts at all times, an end to traffic jams.  In fact, by allocating the task of understanding humanity to machines and giving them complete control over all electronic human interaction, we can receive perfectly targeted ads, ensure that everyone tells the truth forever, discover our perfect human mate, and even find all the terrorists!  All that remains is to automate political decisions and to remove the chaotic human element from nuclear and biological weapons, and this world will finally be at peace.

We humans have nothing to fear from robots and artificial intelligences.  Our superior actual intelligence may not be good at math, understanding everything all at once, or precision, but we do have something the machines can never have: heart.  That is why, in any potential cataclysmic confrontation, we human beings will always win; because we definitely want it more.