Black holes do two things that make them (potentially) dangerous: they eat and they pop. For the black holes we might reasonably create on Earth, neither of these is a problem.

The recipe for black holes is literally the simplest recipe possible; it’s “get a bunch of stuff and put it somewhere”. In practice, you need at least 3.8 Sun’s worth of stuff and the somewhere is anywhere smaller than a few dozen km across. That last bit is important: the defining characteristic of black holes isn’t their *mass*, it’s their *density*.

If you’re any given distance away from a conglomeration of matter, it doesn’t make much difference how that matter is arranged. For example, if the Sun were to collapse into a black hole (it won’t), all of the planets would continue to orbit around it in exactly the same way (just colder). The gravitational pull doesn’t start getting “black-hole-ish” until you’re well beyond where the surface of the Sun used to be. Conversely, if the Sun were to swell up and become huge (it probably will), then all of the planets will continue to orbit it in exactly the same way (just hotter).

To create a new black hole here on Earth, we’d probably use a particle accelerator to slam particles together and (fingers crossed) get the density of energy and matter in one *extremely* small region high enough to collapse. This is wildly unreasonable. But even if we managed to pull it off, the resulting black hole wouldn’t suddenly start pulling things in any more than the original matter and energy did.

For comparison, if you were to collapse Mt. Everest into a black hole it would be no more than a few atoms across. It’s gravity would be as strong as the gravity on Earth’s surface within around 10 meters. If you stood right next to it you’d be in trouble, but you wouldn’t fall in if you gave it a wide berth. In fact, that’s why mountain climbers aren’t particularly bothered by Everest’s mass; even if you’re literally standing on it, you can’t get within more than a few km of most of its mass (fundamentally, Mt. Everest is a big, spread out, pile of stuff).

But the amount of material used in particle accelerators (or any laboratory for that matter) is substantially less than the mass of Everest. They’re “particle accelerators” after all, not “really-big-piles-of-stuff accelerators”. The proton beams at the LHC have a mass of about 0.5 nanograms and when moving at full speed have a “relativistic mass” of about 4 micrograms (because they carry about 7500 times as much kinetic energy as mass). 4 micrograms doesn’t have a scary amount of gravity, and if you turn that into a black hole, it still doesn’t. A black hole that small probably wouldn’t even be able to eat individual atoms. “Probably” because we’ve never seen a black hole anywhere near that small.

The other thing that black holes do is “pop”. Black holes emit Hawking radiation. We haven’t measured it directly, but there a some good theoretical reasons to think that it’s a thing. Paradoxically, the smaller a black hole is, the more it radiates. “Natural” black holes in space (that are as massive as stars) radiate so little that they’re completely undetectable (hence the name: *black* hole). The itty-bitty black holes we might create would radiate so fast that they’d be exploding (explosion = energy released fast). The absolute worst case scenario at CERN (where all of the 115 billion protons in each of the at-most 2,808 groups moving at full speed are all piled up in the same tiny black hole) would be a “pop” with the energy of a few hundred sticks of dynamite.

That’s a good sized boom, but not world ending. More to the point; this is exactly the same amount of energy that was put *into* the beams in the first place. This boom isn’t the worst case scenario for black holes, it’s the worst case scenario for the LHC in general (cave-ins and eldritch horrors notwithstanding). It is this “pop” that would make a tiny black hole a hazard. The gravitational pull of a few micrograms of matter, regardless of how it is arranged, is never dangerous; you wouldn’t get pulled inside out if you ate it. However, you wouldn’t get the chance, since any black hole that we could reasonably create would already be mid-explosion.

A black hole with a mass of a few million tons would blaze with Hawking radiation so brightly that you wouldn’t want it on the ground or even in low orbit. It would be “stable” in that it wouldn’t just explode and disappear. This is one method that science fiction authors use for powering their amazing fictional scientific devices.

The kind of black holes that we might imagine, that are cold (colder than the Sun at least), stable, and happily absorbing material, have a mass comparable to a continent. Even then, it would be no more than a couple millimeters across. These wouldn’t be popping or burning things with Hawking radiation. But the real danger of a black hole of this size isn’t the black hole itself, so much as the process of creating them (listen, I’m making a black hole, so I need to crush all of Australia into a singularity real quick).

We have no way, even in theory, to compress a mountain of material into a volume the size of a virus. Nature compresses matter into black holes by sitting a star on it. That seems to be far and away the best option, so if we want to create black holes the “easiest” way may be to collect some stars and throw them in a pile. But by the time you’re running around grabbing stars, you may as well just find an unclaimed black hole in space and take credit for it.

]]>Same thing with magnets, if you have two bar magnets floating around, they’ll try to line up with their north side next to the other magnet’s south side.

As a result these “positive/negative” forces tend to balance out *really* fast. There are “dipole forces” (one charge might be a *little* closer, so it pulls just a *skosh* harder), but dipole forces are tiny and decrease much faster with distance (technically, all magnets are dipole). In your body right now you have somewhere in the neighborhood of 10^{28} or 10^{29} (ten to a hundred thousand trillion trillion) charged particles in the form of protons and electrons. The number of extra, unbalanced charges on a good Van de Graff generator that’s dangerous to approach is less than a billionth of a billionth of that.

Point is; with magnets and charges you always have a problem with things canceling themselves out almost perfectly. The strength of the electric force between (for example) two protons is just a hell of a lot stronger than the gravitational force (about 1,000,000,000,000,000,000,000,000,000,000,000,000 times bigger), but you’d never know it since those huge forces are all balanced and cancelled out by all of the negative charges around.

Gravity, on the other hand, has only one kind of “charge”: matter. All matter attracts all matter, so despite being far and away the weakest force, gravity is basically the last man standing on large scales. You might imagine that if gravity acted like magnetism there would be planets and stars pushing and pulling each other every which way, but in all likelihood we just wouldn’t have large structures in the universe like planets in the first place.

]]>**Physicist**: The question is about the fact that if you type a fraction into a calculator, the decimal that comes out repeats. But it repeats in a very particular way. For example,

7 is a prime number and (you can check this) all fractions with a denominator of 7 repeat every 7-1=6 digits (even if it does so trivially with “000000”). The trick to understanding why this happens in general is to look really hard at how division works. That is to say: just do long division and see what happens.

When we say that , what we mean is . With that in mind, here’s why .

and so on forever. You’ll notice that the same thing is done to the numerator over and over: multiply by 10, divide by 7, the quotient is the digit in the decimal and the remainder gets carried to the next step, multiply by 10, …. The remainder that gets carried from one step to the next is just .

*Quick aside*: If you’re not familiar with modular arithmetic, there’s an old post here that has lots of examples (and a shallower learning curve). The bracket notation I’m using here isn’t standard, just better. “[4]_{3}” should be read “4 mod 3”. And because the remainder of 4 divided by 3 and the remainder of 1 divided by 3 are both 1, we can say “[4]_{3}=[1]_{3}“.

These aren’t the numbers that end up in the decimal expansion, they’re the remainder left over when you stop calculating the decimal expansion at any point. What’s important about these numbers is that they each determine the next number in the decimal expansion, and they repeat every 6.

After this it repeats because, for example, . If you want to change the numerator to, say, 4, then very little changes:

So the important bit to look at is the remainder after each step. More generally, the question of why a decimal expansion repeats can now be seen as the question of why repeats every P-1, when P is prime. For example, for we’d be looking at and for we’d be looking at . The “10” comes from the fact that we use a base 10 number system, but that’s not written in stone either (much love to my base 20 Mayan brothers and sisters. Biix a beele’ex, y’all?).

It turns out that when the number in the denominator, M, is coprime to 10 (has no factors of 2 or 5), then the numbers generated by successive powers of ten (mod M) are always also coprime to M. In the examples above M=7 and the powers of 10 generated {1,2,3,4,5,6} (in a scrambled order). The number of numbers less than M that are coprime to M (have no factors in common with M) is denoted by ϕ(M), the “Euler phi of M”. For example, ϕ(9)=6, since {1,2,4,5,7,8} are all coprime to 9. For a prime number, P, every number less than that number is coprime to it, so ϕ(P)=P-1.

When you find the decimal expansion of a fraction, you’re calculating successive powers of ten and taking the mod. As long as 10 is coprime to the denominator, this generates numbers that are also coprime to the denominator. If the denominator is prime, there are P-1 of these. More generally, if the denominator is M, there are ϕ(M) of them. For example, , which repeats every 12 because ϕ(21)=12. It *also* repeats every 6, but that doesn’t change the “every 12” thing.

Why the powers of ten must either hit every one of the ϕ(M) coprime numbers, or some fraction of ϕ(M) (, or , or …), thus forcing the decimal to repeat every ϕ(M) will be in the answer gravy below.

**Answer Gravy**: Here’s where the number theory steps in. The best way to describe, in extreme generalization, what’s going on is to use “groups“. A group is a set of things and an operation, with four properties: closure, inverses, identity, and associativity.

In this case the set of numbers we’re looking at are the numbers coprime to M, mod M. If M=7, then our group is {1,2,3,4,5,6} with multiplication as the operator. This group is denoted ““.

The numbers coprime to M are “closed” under multiplication, because if you multiply two numbers with no factors in common with M, you’ll get a new number with no factors in common with M. For example, . No 7’s in sight (other than the mod, which is 7).

The numbers coprime to M have inverses. This means that if and , then . This is a consequence of Bézout’s lemma (proof in the link), which says that if a and M are coprime, then there are integers x and y such that xa+yM=1, with x coprime to M and y coprime to a. Writing that using modular math, if a and M are coprime, then there exists an x such that . For example, , , , and . Here we’d write , which means “the inverse of 3 is 5”.

The numbers coprime to M have an identity element. The identity element is the thing that doesn’t change any of the other elements. In this case the identity is 1, because in general. 1 is coprime to everything (it has no factors), so 1 is always in regardless of what M is.

Finally, the numbers coprime to M are associative, which means that (ab)c=a(bc). This is because multiplication is associative. No biggy.

So , the set of numbers (mod M) coprime to M, form a group under multiplication. Exciting stuff.

But what we’re really interested in are “cyclic subgroups”. “Cyclic groups” are generated by the same number raised to higher and higher powers. For example in mod 7, {3^{1},3^{2},3^{3},3^{4},3^{5},3^{6}}={3,2,6,4,5,1} is a cyclic group. In fact, this is . On the other hand, {2^{1},2^{2},2^{3}}={2,4,1} is a cyclic *subgroup* of . A subgroup has all of the properties of a group itself (closure, inverses, identity, and associativity), but it’s a subset of a larger group.

In general, {a^{1},a^{2},…,a^{r}} is always a group, and often is a subgroup. The “r” there is called the “order of the group”, and it is the smallest number such that .

Cyclic groups are closed because .

Cyclic groups contain the identity. There are only a finite number of elements in the full group, , so eventually different powers of a will be the same. Therefore,

That is to say, if you get the same value for different powers, then the difference between those powers is the identity. For example, and it’s no coincidence that .

Cyclic groups contain inverses. There is an r such that . It follows that . So, .

And cyclic subgroups have associativity. Yet again: no biggy, that’s just how multiplication works.

It turns out that the number of elements in a subgroup always divides the number of elements in the group as a whole. For example, ={1,2,3,4,5,6} is a group with 6 elements, and the cyclic subgroup generated by 2, {1,2,4}, has 3 elements. But check it: 3 divides 6. This is Lagrange’s Theorem. It comes about because cosets (which you get by multiplying every element in a subgroup by the same number) are always the same size and are always distinct. For example (again in mod 7),

The cosets here are {1,2,4} and {3,5,6}. They’re the same size, they’re distinct, and together they hit every element in . The cosets of *any* given subgroup are always the same size as the subgroup, always distinct (no shared elements), and always hit every element of the larger group. This means that if the subgroup has S elements, there are C cosets, and the group as a whole has G elements, then SD=G. Therefore, in general, the number of elements in a subgroup divides the number of elements in a whole group.

To sum up:

In order to calculate a decimal expansion (in base 10) you need to raise 10 to higher and higher powers and divide by the denominator, M. The quotient is the next digit in the decimal and the remainder is what’s carried on to the next step. The remainder is what the “mod” operation yields. This leads us to consider the group of which is the multiplication mod M group of numbers coprime to M (the not-coprime-case will be considered in a damn minute). has exactly ϕ(M) elements. The powers of 10 form a “cyclic subgroup”. The number of numbers in this cyclic subgroup must divide ϕ(M), by Lagrange’s theorem.

If P is prime, then ϕ(P)=P-1, and therefore if the denominator is prime the length of the cycle of digits in the decimal expansion (which is dictated by the cyclic subgroup generated by 10) must divide P-1. That is, the decimal repeats every P-1, but it might *also* repeat every or or whatever. You can also calculate ϕ(M) for M not prime, and the same idea holds.

**Deep Gravy**:

Finally, if the denominator is not coprime to 10 (e.g., 3/5, 1/2, 1/14, 71/15, etc.), then things get a little screwed up. If the denominator is nothing but factors of 10, then the decimal is always finite. For example, .

In general, if the denominator has powers of 2 or 5, then the resulting decimal will be a little messy for the first few digits (equal to the higher of the two powers, for example 8=2^{3}) and after that will follow the rules for the part of the denominator coprime to 10. For example, . So, we can expect that after two digits the decimal expansion will settle into a nice six-digit repetend (because ϕ(7)=6).

Fortunately, the system works:

This can be understood by looking at the powers of ten for each of the factors of the denominator independently. If A and B are coprime, then . This is an isomorphism that works because of the Chinese Remainder Theorem. So, a question about the powers of 10 mod 28 can be explored in terms of the powers of 10 mod 4 *and* mod 7.

Once the powers of 10 are a multiple of all of the of 2’s and 5’s in the denominator, they basically disappear and only the coprime component is important.

Numbers are a whole thing. If you can believe it, this was supposed to be a short post.

]]>This is exciting stuff. The reason we have big fancy pictures of the *eight* planets in our solar system is because we’ve sent cameras to them. Hubble (and future space telescopes) are great, but sometimes you’ve just gotta be there.

Tomorrow morning (July 14, 2015) New Horizons will pass Pluto at mach 48 (48 times the speed of sound, which is a misleading and entirely inappropriate way of measuring speed in space). It will furiously take pictures and measurements for a couple hours and then continue into interstellar space, where its docket will be pretty open for the next few million years.

Already New Horizons has sent us the clearest images of Pluto ever.

Unlike the last post in this vein, there’s nothing for you to personally do. But still: now we get to learn stuff about the planet-turned-dwarf-planet that’s been a bit of an asterisk for 85 years. Good times!

Update (July 15, 2015): Huzzah!

]]>When we picture an atom we usually picture the “Bohr model”: a nucleus made of a bunch of particles packed together (protons and neutrons) with other particles zipping around it (electrons). In this picture, if you make a guess about of the size of electrons and calculate how far they are from the nucleus, then you get that weird result about atoms being mostly empty. But that guess is surprisingly hard to make. The “classical electron radius” is an upper-limit guess based on the electron being nothing more than it’s own electric field, but it’s ultimately just a gross estimate.

However, electrons aren’t really particles (which is why it’s impossible to actually specify their size); they’re waves. Instead of being in a particular place, they’re kinda “smeared out”. If you ring a bell, you can say that there is a vibration in that bell but you can’t say where exactly that vibration is: it’s a wave that’s spread out all over the bell. Parts of the bell will be vibrating more, and parts may not be vibrating at all. Electrons are more like the “ringing” of the bell and less like a fly buzzing around the bell.

Just to be clear, this is a metaphor: atoms are not tiny bells. The math that describes the “quantum wave function” of electrons in atoms and the math that describes vibrations in a bell have some things in common.

So, the space in atoms isn’t empty. A more accurate thing to say is that the overwhelming majority of the matter in an atom is concentrated in the nucleus, which is tiny compared to the region where the electrons are found. However, even in the nucleus the same “problem” crops up; protons and neutrons are just “the ringing of bells” and aren’t simply particles either. The question “where exactly is this electron/proton/whatever?” isn’t merely difficult to answer, the question genuinely doesn’t have an answer. In quantum physics things tend spread out between a lot of states (in this case those different states include different positions).

The atom picture is from here.

]]>But there are a lot of physical phenomena that poke holes in it pretty quick. For example, Foucault pendulums (more commonly known as “big pendulums“) swing as though the Earth were turning under them and in a way that exactly corresponds to the way everything in the sky turns overhead (not a coincidence).

The classic way that heliocentrism (the idea that the Sun is at the center of the solar system) is demonstrated to be better that geocentrism (the idea that the Earth is at the center of the solar system) is by looking at the motion of the other planets. This was essentially what Copernicus did; point out that with the Earth at the center the motions of the other planets are crazy, but with the Sun at the center the motion of the planets (including Earth) are simple ellipses. His original argument was essentially just an application of Occam’s razor: simpler is better, so the Sun must be at the center.

Occam’s razor is a great red flag for detecting ad-hoc theories, but it’s *not* science. With that in mind, it’s impressive how much Copernicus got exactly right. Fortunately, about a century and a half after Copernicus, Newton came along and squared that circle. Newtonian physics says a lot more than “gee wiz, but ellipses are pretty”; it actually describes exactly why all of the orbits behave the way they do with a remarkably simple set of laws for gravity and movement in general. Newtonian physics goes even farther, describing not just the motion of the planets, but also why we don’t *directly* notice the motion of our own.

If we still assumed that the Earth was sitting still in the universe, physicists would have spent the last couple centuries desperately trying to explain what’s hauling the Sun (and the rest of the planets) around in such huge circles. We’d need a bunch of extra, mysterious forces to explain away why the center of mass in the solar system doesn’t sit still (or move at a uniform speed), but is instead whipping by overhead daily.

What follows is a bunch of Newtonian stuff.

Position and velocity are both entirely subjective, but acceleration is objective. What that physically means that there is absolutely no way, whatsoever, to determine where you are or how fast you’re moving by doing tests of any kind. Sure, you can look around and see other things passing by, but even then you’re only measuring your relative velocity (your velocity *relative* to whatever you’re looking at). So, hypothetically, if you’re on a big ball of stuff flying through space, you’d never be able to tell. Acceleration on the other hand is easy to measure.

At first blush it would seem as though there’s no way, from here on Earth, to tell the difference between the Earth moving or sitting still. If the Earth is sitting still, we wouldn’t be able to tell. If the Earth is moving, we also wouldn’t be able to tell. But we’re doing more than just moving; we’re moving in circles and as it happens traveling in a circle requires acceleration. The push you feel when you speed up or slow down comes from the exact same source as the push you feel when you turn a corner or spin around: acceleration.

If the Earth were “nailed to space” and never accelerated, then we’d only have one each of the lunar and solar tides. If the Earth never moved, then the Moon’s gravity would pull the oceans toward it and that’s it. But the Earth does move. The Moon is heavy so, even though the Earth doesn’t move nearly as much, the Earth does execute little circles to balance the Moon’s big circles. The Moon’s big circles generate enough centrifugal force to balance the Earth’s pull on it (that’s what an orbit is), and at the same time the Earth’s little circles balance the Moon’s pull on us.

The same basic thing happens between the Earth and the Sun. Things closer to the Sun orbit faster and things farther away orbit slower. But the Earth has to travel as one big block. The side facing the Sun is about 4,000 miles closer, and traveling slower than it would if it were orbiting at that slightly lower level. As a result, the Sun’s gravity “wins” a little in the “noon region” of the Earth and we get a high tide (pulled toward the Sun). The side facing away is moving a little bit faster than something at that distance from the Sun should, so it’s flung outward a little more than it should be and we get another high tide at midnight. These are called the “solar tide” and they’re harder to notice because they’re about half as strong as the lunar (regular) tides. That said, the solar tides are important and they exist because the Earth is traveling in a circle around the Sun.

Long story short: If the Earth were stationary (geocentrism) then we’d have to come up with lots of bizarre excuses to explain why Newton’s laws work perfectly here on the ground, but not at all in space, and we’d only have one solar and lunar tide a day. If the Earth is moving (specifically: around the Sun), then Newton’s simple laws can be applied universally without buckets of caveats and asterisks*, and we get two lunar and solar tides a day.

*or even †’s.

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