The “0.999… thing” has been done before, but here’s the idea. When we write 0.9, 0.99, 0.999, 0.9999, etc. we’re writing a sequence of numbers that gets closer and closer to 1. Specifically, if there are N 9’s, then . What this means is that no matter how close you want to get to 1, you can get closer than that with enough 9’s. If the 9’s never end, then the difference between 1 and 0.999… is zero. The way our number system is constructed, this means that “0.999…” and “1” (or even “1.000…”) are one and the same in every respect.

As a quick aside, if you think it’s weird that 1 = 0.999…, then you’re in good company. Literally everyone thinks it’s weird. But be cool. There are no grand truths handed down from on high. The rules of math are like the rules of Monopoly; if you don’t like them you can change them, but you risk the “game” becoming inconsistent or merely no fun.

The same philosophy applies to every base. A good way to understand bases is to first consider what it means to write down a number in a given base. For example:

372.51 = 300 + 70 + 2 + 0.5 + 0.01 = 3×10^{2} + 7×10^{1} + 2×10^{0} + 5×10^{-1} + 1×10^{-2}

As you step to the right along a number, each digit you see is multiplied by a lower power of ten. This is why our number system is called “base 10”. But beyond being convenient to use our fingers to count, there’s nothing special about the number ten. If we could start over (and why not?), base 12 would be a much better choice. For example, 1/3 in base 10 is “0.333…” and in base 12 it’s “0.4”; much nicer. More succinctly: 0.333…_{10} = 0.4_{12}

Because we work in base 10, if you tried to “build a tower to one” from below, you’d want to use the largest possible number each time. 0.9_{10} is the largest one-digit number, o.99_{10} is the largest two-digit number, 0.999_{10} is the largest three-digit number, etc. This is because “9_{10}” is the largest number in base 10.

In the exact same way, 0.8_{9} is the largest one-digit number in base 9, 0.88_{9} is the largest two-digit number, and so on. The same way that it works in base 10, in base 9: 1_{9} = 0.888…_{9} !

The easiest way to picture the number 1 as an infinite sum of parts is to picture 0.111…_{2} , “0.111…” in base 2.

If you cut take a stick and cut it in half, then cut one of those halves in half, then cut one of those quarters in half, and so on, the collected set of sticks would have the same length as the original stick. One half, 0.1_{2 }, plus 1 quarter, 0.11_{2 }, plus 1 eighth, 0.111_{2 }, add infinitum equals one. That is to say, 1_{2 }, = 0.111…_{2 }.

But things get tricky when you get to base 1. The largest value in a given base is always less than the base; 9 for base 10, 6 for base 7, 37 for base 38, 1 for base 2. So you’d expect that the largest number in base 1 is 0_{1 }. The problem is that the whole idea of a base system breaks down in “base 1”. In base ten, the number “abc.de_{10 }.” means “ax10^{2} + bx10^{1} + cx10^{0} + dx10^{-1} + ex10^{-2}” (where “a” through “e” are some digits, but who cares what they are). More generally, in base B we have abc.de_{B }= axB^{2} + bxB^{1} + cxB^{0} + dxB^{-1} + exB^{-2}.

But in base 1, abc.de_{1 }= ax1^{2} + bx1^{1} + cx1^{0} + dx1^{-1} + ex1^{-2} = a+b+c+d+e. That is to say, every digit has the same value. Rather than digits to the left being worth more, and digits to the right being worth less, in base 1 every position is the same as every other. So, base one is a number system where the position of the numbers don’t matter and *technically* the only number you get to work with is zero. Not useful.

If you’re gauche enough to allow the use of the number 1 in base 1, then you can count. But not fast.

In base 1, 1 = 10 = 0.000001 = 10000 = 0.01. Therefore, the infinitely repeating number 0.111…_{1} = ∞. That is, if you add up an infinite number string of 1’s, 1+1+1+1…, then naturally you get infinity.

In short: The “1 = 0.999… thing” is just a symptom of how the our number system is constructed, and has nothing in particular to do with 9’s or 10’s. The base 1 number system is kind of a mess and, outside of tallying, isn’t worth using. Base 1 is broken when we consider this particular problem, but that’s to be expected since it’s usually broken.

**Answer Gravy**: We can use the definition of the base system to show that 1 = 0.999…_{10} = 0.333…_{4} = 0.555…_{6} etc. For example, when we write the number 0.999… in base 10, what we explicitly mean is

The same idea is true in any base B, . Showing that this is equal to one is a matter of working this around until it looks like a geometric sum, , and using the fact that .

Notice that issues with base 1, B=1, crop up twice. First because you’re adding up nothing, 0=B-1, over and over. Second because when B=1. So don’t use base 1. There are better things to do.

The excellent pdf about constructing the real numbers was written by this guy.

]]>**Physicist**: It turns out that even if you really stare at how often each object shows up, your estimate for the size of the set never gets much better than a rough guess. It’s like describing where a cloud is; any exact number is silly. “Yonder” is about as accurate as you can expect. That said, there are some cute back-of-the-envelope rules for estimating the sizes of sets witnessed one piece at a time, that can’t be improved upon *too* much with extra analysis. The name of the game is “have I seen this before?”.

*Zero repeats*

It wouldn’t seem like seeing no repeats would give you information, but it does (a little).

The probability of seeing no repeats after randomly drawing K objects out of a set of N total objects is . This equation isn’t exact, but (for N bigger than ten or so) it’s way to close to matter.

The probability is one for K=0 (if you haven’t looked at any objects, you won’t see any repeats), it drops to about 50% for and about 10% for . This gives us a decent rule of thumb: in practice, if you’re drawing objects at random and you haven’t seen any repeats in the first K draws, then there are likely to be at least objects in the set. Or, to be slightly more precise, if there are N objects, then there’s only about a 50% chance of randomly drawing times without repeats.

Seeing only a handful of repeats allows you to very, very roughly estimate the size of the set (about the square of the number of times you’d drawn when you saw your first repeats, give or take *a lot*), but getting anywhere close to a good estimate requires seeing an appreciable fraction of the whole.

*Some repeats*

So, say you’ve seen an appreciable fraction of the whole. This is arguably the simplest scenario. If you’re making your way through a really big set and 60% (for example) of the time you see repeats, then you’ve seen about 60% of the things in the set. That sounds circular, but it’s not *quite*.

For example, we’re in a paranoia-fueled rush to catalog all of the dangerous space rocks that might hit the Earth. We’ve managed to find at least 90% of the Near Earth Objects that are over a km across and we can make that claim because whenever someone discovers a new one, it’s already old news at least 90% of the time. If you decide to join the effort (which is a thing you can do), then be sure to find at least ten or you probably won’t get to put your name on a new one.

*All repeats*

There’s no line in the sand where you can suddenly be sure that you’ve seen everything in the set. You’ll find new things less and less often, but it’s impossible to definitively say when you’ve seen *the last* new thing.

I turns out that the probability of having seen all N objects in a set after K draws is approximately , which is both admittedly weird looking and remarkably accurate. This can be solved for K.

When P is close to zero K is small and when P is close to one K is large. The question is: how big is K when the probability changes? Well, for reasonable values of P (e.g., 0.1<P<0.9) it turns out that is between -1 and 1. You’re likely to finally see every object at least once somewhere in . You’ll already know approximately how many objects there are (N), because you’ve already seen (almost) all of them.

So, if you’ve seen N objects and you’ve drawn appreciably more than times, then you’ve probably seen everything. Or in slightly more back-of-the-envelope-useful terms: when you’ve drawn more than “K = 2N times the number of digits in K” times.

**Answer Gravy**: Those approximations are a beautiful triumph of asymptotics. First:the probability of seeing every object.

When you draw from a set over-and-over you generate a sequence. For example, if your set is the alphabet (where N=26), then a typical sequence might be something like “XKXULFQLVDTZAC…”

If you want only the sequences the include every letter at least once, then you start with every sequence (of which there are ) and subtract all of the sequences that are missing one of the letters. The number of sequences missing a particular letter is and there are N letters, so the total number of sequences missing at least one letter is . But if you remove all the sequences without an A and all the sequences without a B, then you’ve twice removed all the sequences missing both A’s and B’s. So, those need to be added back. There are sequences missing any particular 2 letters and there are “N choose 2” ways to be lacking 2 of the N letter. We need to add back. But the same problem keeps cropping up with sequences lacking three or more letters. Luckily, this is not a new problem, so the solution isn’t new either.

By the inclusion-exclusion principle, the solution is to just keep flipping sign and ratcheting up the number of missing letters. The number of sequences of K draws that include every letter at least once is which is the total number of sequences, minus the number that are missing one letter, plus the number missing two, etc. A more compact way of writing this is . The probability of seeing every letter at least once is just this over the total number of possible sequences, , which is

The two approximations are asymptotic and both of the form . They’re asymptotic in the sense that they are perfect as n goes to infinity, but they’re also remarkably good for values of n as small as ten-ish. This approximation is actually how the number e is defined.

This form is simple enough that we can actually do some algebra and see where the action is.

Now: the probability of seeing no repeats.

The probability of seeing no repeats on the first draw is , in the first two it’s , in the first three it’s , and after K draws the probability is

The approximations here are , which is good for small values of x, and , which is good for large values of K. If K is bigger than ten or so and N is a hell of a lot bigger than that, then this approximation is remarkably good.

]]>There is a “symmetry” in physics implied by our most fundamental understanding of physical law, and is never violated by any known process, that’s called the “CPT symmetry“. It says that if you take the universe and everything in it and flip the electrical charge (C), invert everything as though through a mirror (P), and reverse the direction of time (T), then the base laws of physics all continue to work the same.

Together, the PT amount to putting a negative on the spacetime position, . In addition to time this reflects all three spacial directions, and since each of these reflections reverses parity (flips left and right), these three reflections amount to just one P. You find, when you do this (PT) in quantum field theory, that if you then flip the charge of the particles involved (C), then overall nothing *really* changes. In literally every known interaction and phenomena (on the particle level), flipping all of the coordinates (PT) and the charge (C) leaves the base laws of physics unchanged. It’s worth considering these flips one at a time.

*Charge Conjugation* Flip all the charges in the universe. Most important for us, protons become negatively charged and electrons become positively charged. Charge conjugation keeps all of the laws of electromagnetism unchanged. Basically, after reversing all of the charges, likes are still likes (and repel) and opposites are still opposites (and attract).

*Time Reversal* If you watch a movie in reverse a lot of *nearly* impossible things happen. Meals are uneaten, robots are unexploded, words are unsaid, and hearts are unbroken. The big difference between the before and after in each situation is entropy, which *almost* always increases with time. This is a “statistical law” which means that it only describes what “tends” to happen. On scales-big-enough-to-be-seen entropy “doesn’t tend” to decrease in the sense that fire “doesn’t tend” to change ash into paper; it is a law as absolute as any other. But on a very small scale entropy becomes more suggestion than law. Interactions between individual particles play forward just as well as they play backwards, including particle creation and annihilation.

*Parity* If you watch the world through a mirror, you’ll never notice anything amiss. If you build a car, for example, and then build another that is the exact mirror opposite, then both cars will function just as well as the other. It wasn’t until 1956 that we finally had an example of something that behaves differently from its mirror twin. By putting ultra-cold radioactive cobalt-60 in a strong magnetic field the nuclei, and the decaying neutrons, were more or less aligned and we found that the electrons shot out (β^{–} radiation) in one direction preferentially.

The way matter interacts through the weak force has handedness in the sense that you can genuinely tell the difference between left and right. During β^{–} (“beta minus”) decay a neutron turns into a proton while ejecting an electron, an anti-electron neutrino, and a photon or two (usually) out of the nucleus. Neutrons have spin, so defining a “north” and “south” in analogy to the way Earth rotates, it turns out that the electron emitted during β^{–} decay is always shot out of the neutron’s “south pole”. But mirror images spin in the *opposite* direction (try it!) so their “north-south-ness” is flipped. The mirror image of the way neutrons decay is impossible. Just flat out never seen in nature. Isn’t that weird? There doesn’t have to be a “parity violation” in the universe, but there is.

Parity and charge are how anti-matter is different from matter. All anti-matter particles have the opposite charge of their matter counterparts and their parity is flipped in the sense that when anti-particles interact using the weak force, they do so like matter’s image in a mirror. When an anti-neutron decays into an anti-proton, a positron, and an electron-neutrino, the positron pops out of its “north pole”.

CPT is why physicists will sometimes say crazy sounding things like “an anti-particle (CP) is like the normal particle traveling back in time (T)”. In physics, whenever you’re trying to figure out how an anti-particle will behave in a situation you can always reverse time and consider how a normal particle traveling into the past would act.

This isn’t as useful an insight as it might seem. Honestly, this is useful for understanding beta decay and neutrinos and the fundamental nature of reality or whatever, but as far as your own personal understanding of anti-matter and time, this is a remarkably useless fact. The “backward in time thing” is a useful way of describing individual particle interactions, but as you look at larger and larger scales entropy starts to play a more important role, and the usual milestones of passing time (e.g., ticking clocks, fading ink, growing trees) show up for both matter and anti-matter in exactly the same way. It would be a logical and sociological goldmine if anti-matter people living on an anti-matter world were all Benjamin Buttons, but at the end of the day if you had a friend made of anti-matter (never mind how), you’d age and experience time in exactly the same way. You just wouldn’t want to hang out in the same place.

The most important, defining characteristic of time is entropy and entropy treats matter and anti-matter in exactly the same way; the future is the future is the future for everything.

]]>How can we [theoretically] prove this? (I.e. without assumption that there’s equal probability to mate for any two individuals).

Also: can we prove that one of 3 billion women currently living on Earth will be the only ancestor of all human population some day in the future, and all other currently living women (except her mother and daughters) will have no descendants at that day?

**Physicist**: The fact that everyone on Earth has a common female ancestor if you go back far enough is a direct consequence of the theory of common descent. It looks like everything that lives is part of the same *very* extended family tree with a last universal common ancestor at its base. In order to have two familial lines that never combine in the past you’d need to have more than one starting point for life, and all the evidence to date implies that there’s just the one. Luckily, you don’t have to go all the way back to slime molds to find common ancestors for all humans; the most recent were standard, off-the-shelf people.

It turns out that animal and plant cells aren’t particularly good at producing usable energy, so before we could get around to the business of existing we needed to get past that problem. The solution: fill our cells with a couple thousand symbiotic bacteria. Literally, they’re not human; mitochondria reproduce on their own and have their own genetic code. There’s a hell of a lot of communication and exchange of material between them and our cells, and without them there wouldn’t be an us, but they are (arguably) separate organisms that we are absolutely dependent on and which are completely dependent on us.

Here comes the important bit: eggs cells have mitochondria but sperm cells don’t, so mitochondria are passed strictly from mother to child. There’s no implicit reason for your mitochondria and your father’s to be related at all. The nice thing about that is that it keeps the genetic lineage very simple: all of your mitochondria are essentially clones of those in the egg cell you started as and (for our female readers) any of your children’s mitochondria will essentially be clones of yours. The genes of sexually reproducing beings are a lot trickier to keep track of over time; every generation half of our genes are dumped and the other half are shuffled with someone else’s (which makes your DNA is unique). The one real advantage to talking about mtDNA (mitochondrial DNA) is that you only have a *single* chain of ancestors to worry about. By the way: you can do exactly the same thing with the Y-chromosome and direct male lines.

If you have a group of creatures with two types of mitochondria, two “haplogroups“, living under a population ceiling, then eventually one or the other will be bred out. The math behind this is essentially the Drunkard’s Walk. The number of folk in a haplogroup can increase or decrease forever, unless it gets to zero; given nough time and no where else to go (a population limit), eventually the drunkard’s walk will take him off a cliff (zero population). So, if you start with a small village and several haplogroups, then after a few generations you’ll probably have fewer.

That isn’t saying much. It boils down to the rather fatalistic statement that “in order to be the last thing standing, you just have to wait for everything else to die off”. We can’t *prove* that a woman today will eventually be declared, very post-mortem, the Mitochondrial Eve to everyone (that is; all but one haplogroup will die off). But *statistically*: that’ll definitely happen. To within less than a 1% error, every inherited line of every kind has died off; practically every species, sub-species, gene, haplogroup, whatever, has gone extinct leaving only the amazing scraps that remain. That’s evolution in a nutshell: you chip away all the life that isn’t an efficient, functioning organism (and then a hell of a lot more besides) and the inconceivably tiny fraction that remains is (some of the) efficient, functioning organisms. So, chances are that every living haplogroup presently around will go extinct eventually. When there’s one haplogroup left, then you can say that they all have a common Mitochondrial Eve and when there are zero haplogroups left, then all human issues become moot.

In order to definitely *not* get a modern Mitochondrial Eve, you’d need human populations that are absolutely independent (and viable) forever. Maybe if we colonized Mars and then completely forgot it?

So, if any given haplogroup eventually dies out, then why is there more than one? Well good news: over long time scales (millennia) mtDNA accrues tiny changes through random mutation, leading to a relatively few distinguishable lineages. We live in a kind of meta-family tree, where each branch is entire groups of female lines. Even though some branches stop, others will randomly sprout new branches (“new” meaning “with mtDNA that’s detectably different at all”).

In fact, by carefully looking at the differences in our mtDNA and theirs, we can show that Neanderthals are not a parent species of ours, but cousins, and the common “Eve” that we share with them lived around half a million years ago. We can do the same thing with regular genetics and damn near any living thing to see how and how closely we’re related.

Humanity’s (present) Mitochondrial Eve is not our *unique* common ancestor nor is she our most *recent* common ancestor. Mitochondrial Eve is merely the most recent ancestor of all living people by means of a direct female line alone. If you allow for the inclusion of both men and women, then our most recent common ancestor jumps from around 120,000 – 150,000 years ago, for direct female lines only, to as recent as 3,000 years ago, for any ol’ lines. There’s no way to even reasonably guess who or where any of these common ancestors were. Probably lived near big population centers? Maybe?

Looking at mitochondria is a solid, simple way of understanding evolution and inheritance, but it doesn’t paint an accurate picture of how genes move around populations. An important fact to keep in mind is that a huge fraction of the people alive today will eventually be a common ancestor to all of humanity. Even if your family doesn’t increase or decrease the population (every couple has two kids), your family is still going to grow exponentially (2 kids, 4 grandchildren, 8 great-grandchildren, …). It only takes about 30 generations to have a billion descendants (less if you really work at it), so if you have kids, and they have kids, and so on, then in less time than you’d expect (not forever anyway) your genes will be spread thinly throughout all of humanity. Many of your particular genetics won’t make it, but many of them will. For example, if your haplogroup dies out then your mtDNA won’t be around, but the genes that dictate, say, the shape of your earlobe might end up all over the place.

Arguably, that’s the reason for sex. Maybe not the *first* reason most folk would cite, but they weren’t at the meeting a billion years ago. By mixing our genes every generation we can prevent genetic lines from disappearing forever, which is good: more diversity means more combinations for evolution to try out in a pinch. Almost as good, useful mutations and combinations of genes can be distributed and used by (a random subset of) the entire species after a mere few thousand years! Sexual reproduction literally makes us much better at evolving. Huzzah for doing it!

When you put gas in a tube and pass electricity through it you get light. Electro-dynamically speaking, this is basically just beating the hell out of the atoms and letting the atoms ring like bells (only emitting light instead of sound). Individual atoms are like simply shaped bells; the “tones” they make (or absorb) are very specific. The colors emitted by atoms, their “spectra”, are different for different elements. This is tremendously useful because it allows us to look at the light coming from something and immediately know what that thing is made of.

Some colors fall into the gaps between the spectral lines of all elements (technically, almost all of them do). So you can be forgiven for thinking that there are some colors that just never show up in nature. Fortunately, there are a lot of effects that shift all those lines, blur them, or even split them.

So you can create any color by starting with a few distinct colors and then moving your light source either toward or away from an observer to Doppler shift one of your colors to the target color. That’s a little like using a piano to get some notes, and then driving it around to get all the notes in-between.

You can also just use a non-atomic source of light, like something that’s glowing hot, and then select out the color you want with a monochromator (the rest is chucked out). But, as with any process that involves throwing out almost everything, this is remarkably inefficient.

So, say you want to create a very specific color of light with as little “waste light” as possible. Well, a good place to start is lasers. For some slick quantum reasons, the photons in laser beams are all kinda “clones” of each other; inside of any kind of laser device, the presence of the right kind of photon encourages the creation of other identical photons. Pretty soon your laser is bubbling over with coherent, identical photons and not a lot else. These share, among other things, a common color.

It turns out that only a very small fraction of atomic spectral lines are good candidates for lasers. It is possible to create laser light at any frequency between microwaves and X-rays, but the technique is a *long* way from efficient. You can use the Doppler effect to change the color of your laser, but in order to make any significant change you’ll need to get it going a significant fraction of the speed of light.

If you want to efficiently create *any* very specific color of light, you just need to strap a laser to a starship. So… no need to be picky.

The new planet is called “Proxima B” because the naming convention says that’s what you should call the fist planet discovered around a star called “Proxima” (if/when another planet is discovered, it would be called “Proxima C”, then D, etc.). This is *hopefully* a place holder until someone comes up with a better name, like Krypton or Xena. Strictly because it’s a little cumbersome to refer to both the star, Proxima, and the planet, Proxima B, for the purposes of this post I do hereby declare that the newly discovered planet will be called “Bacchus”, after the god Bacchus. (Bacchus: when you’re too drunk for organized religion, welcome to the Bacchanalia.)

Just for a rough sense of scale, the Earth is 1 AU from the Sun (this is how the “Astronomical Unit” is defined), Proxima is a little over a quarter million AU away from us and less than a tenth that distance from Alpha Centauri (a pair of stars it may-be/probably-is orbiting). Bacchus orbits Proxima every 11 days in an orbit that’s a twentieth the size of ours (0.05AU).

Bacchus was discovered using the radial velocity technique. As Bacchus orbits, its host star, Proxima, wobbles and we can detect its tiny back-and-forth movement using the Doppler Effect (things moving toward us appear slightly bluer and things moving away appear slightly redder). This wobble method reveals how long the orbit is (11 days) and gives a rough idea of the mass of the planet (around 1.3 Earths, give or take), but unfortunately it tells us very little more. If we were lucky enough that Bacchus’ orbit brought it between its star and us, then we’d be able to (possibly) get a look at its atmosphere and determine its size. Unfortunately, like the vast majority of exoplanets, Bacchus’ orbital plane is pointing off in some random direction, so it never eclipses Proxima from our perspective.

So, beyond how much it weighs and where it is, we have no direct data on Bacchus. But take heart! We also have access to well-reasoned speculation! Gas giants tend to be gigantic, so with only 1.3 (give or take) Earth masses to work with, Bacchus is almost certainly a rocky planet. You’d expect that since Bacchus orbits so close to Proxima it would be hot, but Proxima is very dim (~1/10 the mass and ~1/10 the light of our Sun), so Bacchus should have more or less the temperature necessary for liquid water (the way Earth does).

Tight orbits also tend to lead to tidal locking. Like our Moon’s relationship with Earth, Bacchus may have one side always pointed at Proxima. It may literally have a dark side and a light side and if that’s the case, then there’s probably only a thin habitable region near the ring of twilight between the two.

We do know a lot about Proxima from telescopes. In particular, it’s a flare star which means that its magnetic field plays a central role in its behavior; energy stored in its twisted up magnetic field is occasionally released as bursts of x-rays and solar flares. That may mean that Bacchus’ atmosphere is already long gone. This is something stars do to their planets (see Mercury and Mars, for example), it’s just that flares and x-rays are especially good at it and Bacchus is especially close. That said, an already thick atmosphere and a healthy magnetic field go a long way toward preserving what air a planet has. Fingers crossed.

There is some good news: Proxima is *extremely* stable. Both our Sun and Proxima are about five billion years old, ours brightens over time and is due to burn out in another five billion years, but Proxima will still be doing what it does for the next few trillion (with a “t“) years. When Proxima runs out of fuel the universe will be several hundred times older than it is now, so even if there’s no life on Bacchus (or any other planets around Proxima), there’s still an incomprehensible amount of time to get it right. Things have a way of changing over a few trillion years (one would assume).

The bird/Venus/Sun picture is from here.

]]>