Q: Why is the area of a circle equal to πR2?

Posted on October 2, 2014 by The Physicist

Physicist: To demonstrate this you typically have to use either calculus or oranges.  They both use more or less the same ideas, they’re just applied in different ways.

Oranges:

The relevant fruit structure.

The relevant fruit structure what makes the math happen.

Imagine taking an orange wedge and opening it so that the triangles all point “up” instead of towards the same point.  If you interlaced two of these then you’d have a small brick that’s roughly rectangular.

By slicing the circle with thinner and thinner triangles, the parallelogram turns into a rectangle.

By slicing the circle with thinner and thinner triangles, the lines on the “wavey shape” all straighten out into a rectangle.

As more triangles are used, the curved end produces less pronounced bumpiness and the straight sides come closer and closer to being straight up and down, making the brick rectangular.  The height becomes equal to the radius, while the length is half of the circumference (C = 2πR) which now finds itself running along the top and bottom.  As the number of triangles “approaches infinity” the circle can be taken apart and rearranged to fit almost perfectly into an “R by πR” box with an area of πR2.

This is why calculus is so damn useful.  We often think of infinity as being mysterious or difficult to work with, but here the infinite slicing just makes the conclusion infinitely clean and exact: A = πR2.

Calculus:

On the mathier side of things, the circumference is the differential of the area.  That is; if you increase the radius by “dr”, which is a tiny, tiny bit, then the area increases by Cdr where C is the circumference.  We can use that fact to describe a disk as the sum of a lot of very tiny rings.  “The sum of a lot of tiny _____” makes mathematicians reflexively say “use an integral“.

With calculus, you can find the area of a circle by adding up the areas of a lot (infinite) of very thin (infinitely thin) rings.

With calculus, you can find the area of a circle by adding up the areas of a lot (infinite number) of very thin (infinitely thin) rings.  Here a random intermediate ring is red.

Every ring has an area of Cdr = (2πr)dr.  Adding them up from the center, r=0, to the outer edge, r=R, is written: \int_0^R C\,dr = \int_0^R 2\pi r\,dr = 2\pi\frac{r^2}{2}\big|_0^R = \pi R^2.

This is a beautiful example of understanding trumping memory.  A mathematician will forget the equation for the area of a circle (A=πR2), but remember that the circumference is its differential.  That’s not to excuse their forgetfulness, just explain it.

Posted in -- By the Physicist, Geometry, Math | 17 Comments

0.999… revisited

Posted on September 22, 2014 by The Physicist

The original question was: 0.999… = 1 does not make sense with respect to my conception of the number line, I do not know much about number classes but for the number line from lets say 0 to 1 there is an infinite number of points, so there is a number right next to 1 we cant write it in entirety because the decimal expansion is infinite but the difference between that 1 and that number is 1*10^(-infinity) (sorry if I am abusing notation). so that number to me should be 0.999… but it is not where am I missing the point?

Physicist: In the language of mathematics there are “dialects” (sets of axioms), and in the most standard, commonly-used dialect you can prove that 0.999… = 1.  The system that’s generally taught now is used because it’s useful (in a lot of profound ways), and in it we can prove that 0.99999… = 1.  If you want to do math where 1/infinity is a definable and non-zero value, you can, but it makes math unnecessarily complicated (for most tasks).  The way the number system is generally taught (at the math-major level, where the differences become important) is that the real numbers are defined such that (very long story short) 1/infinity = 0 and there isn’t a “next number” for any number.  That is, if you think you’ve found a number, x, that’s closer to 1 than any other number, then I can find a number half way between it and 1, (1+x)/2, that’s even closer.  That’s not a trivial statement.  In the system of integer numbers there is a next number; for 3 it’s 4, for 26 it’s 27, etc..  In the system of real numbers every number can be added, subtracted, multiplied, and divided without “leaving” the real numbers.  That leads to the fact that we can squeeze a new number between any two different numbers.  In particular, there’s no greatest number less than one.  If there were, then you couldn’t fit another number between it and one, and that would make it a big weird exception.  Point is: it’s tempting to say that 0.999… is the “first number below 1”, but that’s not a thing.

The term “real numbers” is just a name for a “sand box” of mathematical tools that have become standard because they’re useful.  However!  There are other systems where “very very very slightly less than 1” , or more precisely “less than one, but greater than every number that’s less than one”, makes mathematical sense.  These systems aren’t invalid or wrong, they’re just… not as pretty and fluid as the simple (as it reasonably can be), solid, dull as dishwater, real number system.

I prefer Calvinball Chess.

It is a fact, immutable and True, that every rook is safe for the next move according to the most widely accepted rules for chess.  In other games that may not be the case.  0.999…=1 in the most widely accepted rules for real numbers.

In the set of “real numbers” (as used today) a number can be defined as the limit of the decimal expansion taken one digit at a time.  For example, the number “2” is {2, 2.0, 2.00, 2.000, …}.  The “square root of 2” is {1, 1.4, 1.41, 1.414, 1.4142, …}.  The number, and everything you might ever want to do with it (as a real number), can be done with this sequence of ever-longer decimals (although, in practice, there are usually more sophisticated methods).

These sequences are “equivalent” and describe the same number if they get (arbitrarily) closer and closer to that same number forever.  Two sequences don’t need to be identical to be equivalent.  The sequences {1, 1.0, 1.00, 1.000, …} and {0, 0.9, 0.99, 0.999, …} both get closer and closer to each other and to the value “1” forever, so they’re equivalent.  In absolutely every way that counts (in terms of the real numbers), the number “0.99999…” and the number “1” or “1.0000…” are exactly the same.

It does seem very bizarre that two numbers that look different can be the same, but there it is.  This is basically the only exception; you can write things like “0.5 = 0.49999…”, but the same thing is going on.

Posted in -- By the Physicist, Math, Philosophical | 86 Comments

Q: Before you open the box, isn’t Schrödinger’s cat alive or dead, not alive and dead?

Posted on August 30, 2014 by The Physicist

The original question was: If I don’t open the lid of the bunker, the cat inside is either dead OR alive.  Why does Schrödinger say that the cat is both alive AND dead if I do not perform the act of observation?  Isn’t he confusing AND for OR?

Physicist: He’s very intentionally saying “and” and not saying “or”.

When something is in more than one place, or state, or position, we say it’s in a “superposition of states”.  The classic example of this is the “double slit experiment”, where we see evidence of a single photon interfering with itself through both slits.

The patterns on the screen cannot be described by particles of light traveling in a straight line from the source.

The patterns on the screen cannot be described by particles of light traveling in a straight line from the source.  They can be described very easily in terms of waves (which go through both slits), or by saying “particles of light certainly do behave oddly, yessir” (hidden variables).

Schrödinger’s Equation describes particles (and by extension the world) in terms of “quantum wave functions”, and not in terms of “billiard balls”.  His simple model described the results of a variety of experiments very accurately, but required particles to behave like waves (like the interference pattern in the double slit) and be in multiple states.  In those experiments, when we actually make a measurement (“where does the photon hit the photographic plate?”) the results are best and most simply described by that wave.  But while a wave describes how the particles behave and where they’ll be, when we actually measure the particle we always find it to be in one state.

“Schrödinger’s Cat” was a thought experiment that he (Erwin S.) came up with to underscore how weird his own explanation was.  The thought experiment is, in a nutshell (or cat box): there’s a cat in a measurement-proof box with a vial of poison, a radioactive atom (another known example of quantum weirdness), and a bizarre caticidal geiger counter.  If the counter detects that the radioactive atom has decayed, then it’ll break the vial and kill the cat.  T0 figure out the probability of the cat being alive or dead you use Schrödinger’s wave functions to describe the radioactive atom.  Unfortunately, these describe the atom, and hence the cat, as being in a superposition of states between the times when the box is set up and when it’s opened (in between subsequent measurements).  Atoms can be in a combination of decayed and not decayed, just like the photons in the double slit can go through both slits, and that means that the cat must also be in a superposition of states.  This isn’t an experiment that has been done or could reasonably be attempted.  At least, not with a cat.

Schrödinger’s Cat wasn’t intended to be an educational tool, so much as a joke with the punchline “so… it works, but that’s way too insane to be right”.  At the time it was widely assumed that in the near future an experiment would come along that would over-turn this clearly wonky interpretation of the world and set physics back on track.

But as each new experiment (with stuff smaller than cats, but still pretty big) verified and reinforced the wave interpretation and found more and more examples of quantum superposition, Schrödinger’s Cat stopped being something to be dismissed as laughable, and turned instead into something to be understood and taken seriously (and sometimes dropped nonchalantly into hipster conversations).  Rather than ending with “but the cat obviously must be alive OR dead, so this interpretation is messed up somewhere” it more commonly ends with “but experiments support the crazy notion that the cat is both alive AND dead, so… something to think about”.

If it bothers you that the Cat doesn’t observe itself (why is opening the box so important?), then consider Schrödinger’s Graduate Student: unable to bring himself to open one more box full of bad news, Schrödinger leaves his graduate student to do the work for him and to report the results.  Up until the moment that the graduate student opens the door to Schrödinger’s Office, Schrödinger would best describe the student as being in a superposition of states.  This story was originally an addendum to Schrödinger’s ludicrous cat thing, but is now also told with a little more sobriety.

The double slit picture is from here.

Posted in -- By the Physicist, Philosophical, Physics, Quantum Theory | 29 Comments

Q: How many times do you need to roll dice before you know they’re loaded?

Posted on August 22, 2014 by The Physicist

Physicist: A nearly equivalent question might be “how can you prove freaking anything?”.

In empirical science (science involving tests and whatnot) things are never “proven”.  Instead of asking “is this true?” or “can I prove this?” a scientist will often ask the substantially more awkward question “what is the chance that this could happen accidentally?”.  Where you draw the line between a positive result (“that’s not an accident”) and a negative result (“that could totally happen by chance”) is completely arbitrary.  There are standards for certainty, but they’re arbitrary (although generally reasonable) standards.  The most common way to talk about a test’s certainty is “sigma” (pedantically known as a “standard deviation“), as in “this test shows the result to 3 sigmas”.  You have to do the same test over and over to be able to talk about “sigmas” and “certainties” and whatnot.  The ability to use statistics is a big part of why repeatable experiments are important.

mn

When you repeat experiments you find that their aggregate tends to look like a “bell curve” pretty quick.  This is due to the “central limit theorem” which (while tricky to prove) totally works.

“1 sigma” refers to about 68% certainty, or that there’s about a 32% chance of the given result (or something more unlikely) happening by chance.  2 sigma certainty is ~95% certainty (meaning ~5% chance of the result being accidental) and 3 sigmas, the most standard standard, means ~99.7% certainty (~0.3% probability of the result being random chance).  When you’re using, say, a 2 sigma standard it means that there’s a 1 in 20 chance that the results you’re seeing are a false positive.  That doesn’t sound terrible, but if you’re doing a lot of experiments it becomes a serious issue.

The more data you have, the more precise the experiment will be.  Random noise can look like a signal, but eventually it’ll be revealed to be random.  In medicine (for example) your data points are typically “noisy” or want to be paid or want to be given useful treatments or don’t want to be guinea pigs or whatever, so it’s often difficult to get better than a couple sigma certainty.  In physics we have more data than we know what to do with.  Experiments at CERN have shown that the Higgs boson exists (or more precisely, a particle has been found with the properties previously predicted for the Higgs) with 7 sigma certainty (~99.999999999%).  That’s excessive.  A medical study involving every human on Earth can not have results that clean.

So, here’s an actual answer.  Ignoring the details about dice and replacing them with a “you win / I win” game makes this question much easier (and also speaks to the fairness of the game at the same time).  If you play a game with another person and either of you wins, there’s no way to tell if it was fair.  If you play N games, then (for a fair game) a sigma corresponds to \frac{\sqrt{N}}{2} excess wins or losses away from the average.  For example, if you play 100 games, then

1 sigma: ~68% chance of winning between 45 and 55 games (that’s 50±5)

2 sigma: ~95% chance of winning between 40 and 60 games (that’s 50±10)

If you play 100 games with someone, and they win 70 of them, then you can feel fairly certain (4 sigmas) that something untoward is going down because there’s only a 0.0078% chance of being that far from the mean (half that if you’re only concerned with losing).  The more games you play (the more data you gather), the less likely it is that you’ll drift away from the mean.  After 10,000 games, 1 sigma is 50 games; so there’s a 95% chance of winning between 4,900 and 5,100 games (which is a pretty small window).

Keep in mind, before you start cracking kneecaps, that 1 in 20 people will see a 2 sigma result (that is, 1 in every N folk will see something with a probability of about 1 in N).  Sure it’s unlikely, but that’s probably why you’d notice it.  So when doing a test make sure you establish when the test starts and stops ahead of time.

Posted in -- By the Physicist, Experiments, Math, Philosophical, Skepticism | 5 Comments

Q: Since it involves limits, is calculus always an approximation?

Posted on August 3, 2014 by The Physicist

Physicist: Nope!  Calculus is exact.  For those of you unfamiliar with calculus, what follows is day 1.

Between any two points on a curve we can draw a straight line (which has a definite slope).

Between any two points on a curve we can draw a straight line (which has a definite slope).

In order to find the slope of a curve at a particular point requires limits, which always feel a little incomplete.  When taking the limit of a function you’re not talking about a single point (which can’t have a slope), you’re not even talking about the function at that point, you’re talking about the function near that point as you get closer and closer.  At every step there’s always a little farther to go, but “in the limit” there isn’t.  Here comes an example.

The slope of x2 at x=1.

The slope of x^2 at x=1.

Say you want to find the slope of f(x) = x2 at x=1.  “Slope” is (defined as) rise over run, so the slope between the points \left(1, f(1)\right) and \left(1+h, f(1+h)\right) is \frac{f(1+h)-f(1)}{(1+h-1)} and it just so happens that:

\begin{array}{ll}\frac{f(1+h)-f(1)}{(1+h-1)} \\= \frac{(1+h)^2-1^2}{h} \\= \frac{(1+2h+h^2)-1}{h} \\= \frac{2h+h^2}{h} \\= \frac{2h}{h}+\frac{h^2}{h} \\= 2+h\end{array}

Finding the limit as h\to0 is the easiest thing in the world: it’s 2.  Exactly 2.  Despite the fact that h=0 couldn’t be plugged in directly, there’s no problem at all.  For every h≠0 you can draw a line between \left(1, f(1)\right) and \left(1+h, f(1+h)\right) and find the slope (it’s 2+h).  We can then let those points get closer together and see what happens to the slope (h\to0).  Turns out we get a single, exact, consistent answer.  Math folk say “the limit exists” and the function is “differentiable”.  Most of the functions you can think of (most of the functions worth thinking of) are differentiable, and when they’re not it’s usually pretty obvious why.

Same sort of thing happens for integrals (the other important tool in calculus).  The situation there is a little more subtle, but the result is just as clean.  Integrals can be used to find the area “under” a function by adding up a larger and larger number of thinner and thinner rectangles.  So, say you want to find the area under f(x)=x between x=0 and x=3.  This is day… 30 or so?

As a first try, we’ll use 6 rectangles.

A rough attempt to find the area of the triangle using a mess of rectangles.

A rough attempt to find the area of the triangle using a mess of rectangles.

Each rectangle is 0.5 wide, and 0.5, 1, 1.5, etc. tall.  Their combined area is 0.5\cdot0.5+0.5\cdot1+0.5\cdot1.5+\cdots+0.5\cdot2.5+0.5\cdot3 or, in mathspeak, \sum_{j=1}^6 0.5\cdot\left(\frac{j}{2}\right).  If you add this up you get 5.25, which is more than 4.5 (the correct answer) because of those “sawteeth”.  By using more rectangles these teeth can be made smaller, and the inaccuracy they create can be brought to naught.  Here’s how!

If there are N rectangles they’ll each be \frac{3}{N} wide and will be \frac{3}{N}, 2\frac{3}{N}, 3\frac{3}{N}, \cdots, N\frac{3}{N} tall (just so you can double-check, in the picture N=6).  In mathspeak, the total area of these rectangles is

\begin{array}{ll} \sum_{j=1}^N \frac{3}{N} \left(j\frac{3}{N}\right) \\ = \frac{3}{N}\frac{3}{N}\sum_{j=1}^N j \\ = \frac{9}{N^2}\sum_{j=1}^N j \\ = \frac{9}{N^2}\left(\frac{N^2}{2}+\frac{N}{2}\right) \\ = \frac{9}{2}+\frac{9}{2N} \\ \end{array}

The fact that \sum_{j=1}^N=\frac{N^2}{2}+\frac{N}{2} is just one of those math things.  For every finite value of N there’s an error of \frac{9}{2N}, but this can be made “arbitrarily small”.  No matter how small you want the error to be, you can pick a value of N that makes it even smaller.  Now, letting the number of rectangles “go to infinity”, \frac{9}{2N}\to0 and the correct answer is recovered: 9/2.

In a calculus class a little notation is used to clean this up:

\frac{9}{2} = \int_0^3 x\,dx = \lim_{N\to\infty}\sum_{j=1}^N\left(\frac{3j}{N}\right)\frac{3}{N}

Every finite value of N gives an approximation, but that’s the whole point of using limits; taking the limit allows us to answer the question “what’s left when the error drops to zero and the approximation becomes perfect?”.  It may seem difficult to “go to infinity” but keep in mind that math is ultimately just a bunch of (extremely useful) symbols on a page, so what’s stopping you?

Mathematicians, being consummate pessimists, have thought up an amazing variety of worst-case scenarios to create “non-integrable” functions where it doesn’t really make sense to create those approximating rectangles.  Then, being contrite, they figured out some slick ways to (often) get around those problems.  Mathematicians will never run out of stuff to do.

Fortunately, for everybody else (especially physicists) the universe doesn’t seem to use those terrible, terrible… terrible worst-case functions in any of its fundamental laws.  Mathematically speaking, all of existence is a surprisingly nice place to live.

Posted in -- By the Physicist, Equations, Math | 4 Comments

Q: How does Earth’s magnetic field protect us?

Posted on July 5, 2014 by The Physicist

Physicist: High energy charged particles rain in on the Earth from all directions, most of them produced by the Sun.  If it weren’t for the Earth’s magnetic field we would be subject to bursts of radiation on the ground that would be, at the very least, unhealthy.  The more serious, long term impact would be the erosion of the atmosphere.  Charged particles carry far more kinetic energy than massless particles (light), so when they strike air molecules they can kick them hard enough to eject them into space.  This may have already happened on Mars, which shows evidence of having once had a magnetic field and a complex atmosphere, and now has neither (Mars’ atmosphere is ~1% as dense as ours).

Rule #1 for magnetic fields is the “right hand rule”: point your fingers in the direction a charged particle is moving, curl your fingers in the direction of the magnetic field, and your thumb will point in the direction the particle will turn.  The component of the velocity that points along the field is ignored (you don’t have to curl your fingers in the direction they’re already pointing), and the force is proportional to the speed of the particle and the strength of the magnetic field.

For notational reasons either lost to history or not worth looking up, the current (the direction the charge is moving) is I and the magnetic field is B. F is the force the particle feels.

For notational reasons either lost to history or not worth looking up, the current (the direction the charge is moving) is I and the magnetic field is B. More reasonably, the Force the particle feels is F.  In this case, the particle is moving to the right, but the magnetic field is going to make it curve upwards.

This works for positively charged particles (e.g., protons).  If you’re wondering about negatively charged particles (electrons), then just reverse the direction you got.  Or use your left hand.  If the magnetic field stays the same, then eventually the ion will be pulled in a complete circle.

As it happens, the Earth has a magnetic field and the Sun fires charged particles at us (as well as every other direction) in the form of “solar wind”, so the right hand rule can explain most of what we see.  The Earth’s magnetic field points from south to north through the Earth’s core, then curves around and points from north to south on Earth’s surface and out into space.  So the positive particles flying at us from the Sun are pushed west and the negative particles are pushed east (right hand rule).

Since the Earth’s field is stronger closer to the Earth, the closer a particle is, the faster it will turn.  So an incoming particle’s path bends near the Earth, and straightens out far away.  That’s a surprisingly good way to get a particle’s trajectory to turn just enough to take it back into the weaker regions of the field, where the trajectory straightens out and takes it back into space.  The Earth’s field is stronger or weaker in different areas, and the incoming charged particles have a wide range of energies, so a small fraction do make it to the atmosphere where they collide with air.  Only astronauts need to worry about getting hit directly by particles in the solar wind; the rest of us get shrapnel from those high energy interactions in the upper atmosphere.

If a charge moves in the direction of a magnetic field, not across it, then it’s not pushed around at all.  Around the magnetic north and south poles the magnetic field points directly into the ground, so in those areas particles from space are free to rain in.  In fact, they have trouble not coming straight down.  The result is described by most modern scientists as “pretty”.

Charged particles from space following the magnetic field lines into the upper atmosphere.

Charged particles from space following the magnetic field lines into the upper atmosphere where they bombard the local matter.  Green indicates oxygen in the local matter.

The Earth’s magnetic field does more than just deflect ions or direct them to the poles.  When a charge accelerates it radiates light, and turning a corner is just acceleration in a new direction.  This “braking radiation” slows the charge that creates it (that’s a big part of why the aurora is inspiring as opposed to sterilizing).  If an ion slows down enough it won’t escape back into space and it won’t hit the Earth.  Instead it gets stuck moving in big loops, following the right hand rule all the way, thousands of miles above us (with the exception of our Antarctic readers).  This phenomena is a “magnetic bottle”, which traps the moving charged particles inside of it.  The doughnut-shaped bottles around Earth are the Van Allen radiation belts.  Ions build up there over time (they fall out eventually) and still move very fast, making it a dangerous place for delicate electronics and doubly delicate astronauts.

Magnetic bottles, by the way, are the only known way to contain anti-matter.  If you just keep anti-matter in a mason jar, you run the risk that it will touch the mason jar’s regular matter and annihilate.  But ions contained in a magnetic bottle never touch anything.  If that ion happens to be anti-matter: no problem.  It turns out that the Van Allen radiation belts are lousy with anti-matter, most of it produced in those high-energy collisions in the upper atmosphere (it’s basically a particle accelerator up there).  That anti-matter isn’t dangerous or anything.  When an individual, ultra-fast particle of radiation hits you it doesn’t make much of a difference if it’s made of anti-matter or not.

And there isn’t much of it; about 160 nanograms, which (combined with 160 nanograms of ordinary matter) yields about the same amount of energy as 7kg of TNT.  You wouldn’t want to run into it all in one place, but still: not a big worry.

Why is there a map on it?

A Van Allen radiation belt simulated in the lab.

In a totally unrelated opinion, this picture beautifully sums up the scientific process: build a thing, see what it does, tell folk about it.  Maybe give it some style (time permitting).

 

The right hand picture is from here.

Posted in -- By the Physicist, Astronomy, Particle Physics, Physics | 10 Comments