countability

Favourite jokes

  • Referring to any four-legged animal as a weird dog
  • Massively underestimating the number of nearly uncountable objects
  • Massively overestimating the number of clearly countable objects
  • Bad puns in TV episode titles
A solution

When I first heard the submarine puzzle in the post below, I wasn’t able to solve it until I got some hints.  The trick, as with many puzzles and even serious math problems, is to replace the problem at hand with one easy enough you can solve it, but which still captures the essential difficulty of the original problem.  This tends to be easier said than done, unless you have a great deal of experience and so-called “mathematical maturity”.  Sometimes you have to work your way through a series of gradually harder problems until you settle the whole matter.

Step 1: Assume the submarine starts at position 0 and is traveling with positive speed, i.e. from left to right.

Now it would be pretty useless to say “Just bomb 0 immediately!” in this situation, as this provides you no insight to the original problem.  Instead you have to decide what you would do if you missed on the first bombing attempt.  Well, suppose the submarine’s speed is as simple as possible: 1.  Then the strategy would be to bomb position 1 on the next possible occasion.  If this bombing fails, we can conclude that the submarine does not travel with unit speed.  Perhaps, then, its speed is 2?  If so it would have been at position 2 when you bombed position 1, so the next spot you should target is position 4.  If this bombing fails, we conclude the speed is different from 2.

Now you should have the idea of what to do in this simplified scenario: the bombing sequence goes (1,4,9,16,25,…,v^2,…).  This is guaranteed to eventually hit the submarine, specifically on the vth bombing, at location v^2, where v is the sub’s speed.

Step 2: Assume the submarine starts at position 0 without putting any constraints on the direction of travel.

This is handled with a small modification to the above strategy: we alternatively bomb to the right and the left of the origin.  The bombing sequence goes (1,-2,6,-8,15,-18,28,-32,…).  This is messier than the last sequence, but not harder to understand; the negative terms are where the sub would be assuming right-to-left travel.

Step 3: Assume the submarine starts at position m, where m is any integer.

Then we simply translate the last bombing sequence by m.  There was nothing special about 0 in the above argument.

Now for the truly tricky and interesting part.  For convenience, we can represent the situation of the sub starting at position m with velocity v as an ordered pair (m,v).  Such an ordered pair is often called a lattice point (or Gaussian integer, when the plane is thought of as a geometric manifestation of the complex numbers).

Step 4: Remember that the lattice points of the plane are countable.

Informally this just means the set of points (m,v) can be listed out in a systematic way that doesn’t miss or double-count a single member.  Here’s one way to do it helpful in our situation, using a spiral winding its way out from (1,1):

In this picture, I didn’t bother plotting any points of the form (m,0), because we know the submarine has non-zero velocity.*

So, what’s the bombing plan for the most general situation?  Employ the listing of lattice points illustrated above, and when you get to point (m,v) in the list, bomb the location where the submarine would currently be if it had started with velocity v at point m (a situation we fully understand thanks to Step 3 above).  This is guaranteed to hit the sub in finite time - though it may take a rather long time if (m,v) is very far from (1,1)!

By the way, the winding spiral above also proves that the set of all rational numbers is countable, if we merely observe that (m,v) may be identified with the fraction m/v, since v is not zero!

*Actually, if the original problem were modified to include the possibility of a motionless submarine, we would still be fine because of this argument.

“same-sized” infinities

The idea of “two sets are the same size iff there is a bijection between them” leads to the interesting property that you can extend the idea of size to infinite sets.

Here are illustrations of how two infinite size sets can have the same size, some of which might seem surprising if you are only used to finite sizes.

Some sets I will use as examples are integers Z={…,-2,-1,0,1,2,…}, positive integers Z+={1,2,3…}, non-negative integers Z*={0,1,2,3…}, the non-negative even integers 2Z*={0,2,4,6,…}, the positive rational numbers Q={p/q : p,q are positive integers and q is not 0}, and positive real numbers R+, which include the irrationals.

Z+ and Z* are the same size

The function is f:Z*->Z+, f(x)=x+1

Z* and 2Z* are the same size

The function is f:Z*->2Z*, f(x)=2x

Z* and Z are the same size

The function is one where we split Z* into evens and odds, and then take the evens to the negatives and the odds to the positives

Q on the interval [1,infinity) and Q on the interval (0,1] are the same size

The function is f:Q[1,infinity)->Q(0,1], f(x)=1/x. (We can also use a similar function to show any interval of real numbers has the same size as the entire real number line)

Z* and Q* are the same size

This one is a bit trickier. We will show that Z* is the same size as the cartesian product Z*×Z*, and notice that Q* is a subset of Z*×Z*, so cannot be larger than it, but Q* contains Z*, so it cannot be smaller.

First, we split the cartesian product into lines based on the sum of the two numbers in each pair

Let t(n) be the nth triangular number, or the sum of all non-negative integers below and including it. Notice that t(p+q) is the number of pairs with sum below p+q, because each row where the sum is m has m-1 elements.

We construct a function f:Z*×Z*->Z*, f(p,q)=t(p+q)+p

R and R×R have the same size

First, we restrict R to the interval [0,1), which has the same size as R. Then, we realize that we can represent each real number by its decimal expansion, which has a Z+ sized expansion. We can split every expansion into even and odd number places, and make a new number in R×R restricted to [0,1)×[0,1) using the two expansions to make its two numbers.

ardentsonata replied to your post: ardentsonata replied to your post: on…

you lost me at cardinality of the continuum but that does sound fascinating. The halting problem has something that has been fascinating me for a while though now. :x

what would we work without wikipedia? cantor’s diagonal argument is cool :3

there are as many real numbers on the interval [0, 1] as there are real numbers.

CoC is the cardinality of the reals and has been shown to be the next biggest number after countable infinity (aleph null). i find it quirky because of things wolfram lists.

I slept at night thinking and actually believing that I was over him.
His texts were right there, unreplied and unread and curiosity didn’t kill me this time.
But I woke up in the morning and suddenly remembered something he had once told me. It wasn’t anything in particular, just a normal conversation out of the countably finite ones we had.
But it made me laugh and it made me see him in the light he used to be as opposed to what he is now.
I had to fight the urge of going back and reading our old emails and letters. I had to fight the urge of letting warm tears wet my cheeks. I had to fight the urge of calling him up. I had to fight the urge of admitting to myself that I’m still not over him.
With all this fighting, I don’t know if I have the fight in me to breathe without pain anymore.

A remarkable thing happened while I was walking my dog tonight.

I am one of those countless people (okay, countable.  I’m sure Niantic has an exact count of just how many people have downloaded their app) who has downloaded and plays Pokemon Go.  I groaned as I did it, I was embarrassed by it, but the truth is I’ve come to enjoy it as I’ve progressed.  I’m not that invested, but the concept is cool, filed with potential, and motivates me to get off my ass for a little while longer than I normally would each night.

Which brings me to the remarkable thing.

There’s a point where I usually turn back when walking, but tonight I was running low on pokeballs, so I decided to walk to the closest Pokestop at a park a few blocks away (translation:  I didn’t have anymore magic containers to throw at wild creatures so I had to go to a nearby landmark on the map and collect more)  On my way, I witnessed the extremely unusual sight of roughly seven children, each under the age of ten, and about five adults walking toward the corner of the block.  As we crossed paths, I overheard their conversation.

“Did you get it?  Did you get it?”

“Maria thinks she just saw a Pikachu over there!”

“Mom, how many pidgeys do we have already?”

I’m pretty positive I was smiling while I was passing the last adult in the group, a woman who looked to be about old enough to have young grandchildren their age, and she looked at me and asked, “are you with the Pokemon, too?”  I sheepishly held up my phone screen, app open, and replied, “yep.”

Look, there are a lot of people where I live.  I see lots of them on walks, but rarely so many, rarely so late, and I pretty much never talk to them.  This family group was happily walking – outside, in 90 degree heat, at 11pm – and interacting with one another and passerby.

I walked by them again, on the other side of the block, and we exchanged intel.  Someone had seen a dratini in the direction I was headed, I’d noticed a jigglypuff on my radar.  I found the dratini, but he ran away.

For all its oddities, faults, and potential dangers, this game has managed, for whatever time frame, to get kids AND parents outside, walking, literally searching for virtual creatures like some global scavenger hunt.  It’s given these people who have nothing in common but that game a connection, even if it’s brief and fleeting and otherwise meaningless.  It’s a social game, even if you play it alone.

The world’s basically drenched itself in gasoline and lit a match in the last few weeks, and this country is more divided than ever, but I got to share a human moment with seven children and five adults tonight over a phone app.  A phone app.

How amazing is that?

The Rado graph

The Rado graph is the unique (up to isomorphism) countable graph R such that for every finite graph G and every vertex v of G, every embedding of G−v as an induced subgraph of R can be extended to an embedding of G into R. This implies R contains all finite and countable graphs as induced subgraphs.

Rado gave the following construction: identifiy the vertices of the graph with the natural numbers. For every x and y with x<y, an edge connects vertices x and y in the graph if the xth bit of y’s binary representation is nonzero.

Thus, for instance, the neighbors of vertex 0 consist of all odd-numbered vertices, while the neighbors of vertex 1 consist of vertex 0 (the only vertex whose bit in the binary representation of 1 is nonzero) and all vertices with numbers congruent to 2 or 3 modulo 4.

Via #RobertReich

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The Question That Could Unite Quantum Theory With General Relativity: Is Spacetime Countable?

Current thinking about quantum gravity assumes that spacetime exists in countable lumps, like grains of sand. That can’t be right, can it?

One of the big problems with quantum gravity is that it generates infinities that have no physical meaning. These come about because quantum mechanics implies that accurate measurements of the universe on the tiniest scales require high-energy. But when the scale becomes very small, the energy density associated with a measurement is so great that it should lead to the formation of a black hole, which would paradoxically ruin the measurement that created it.

These kinds of infinities are something of an annoyance. Their paradoxical nature makes them hard to deal with mathematically and difficult to reconcile with our knowledge of the universe, which as far as we can tell, avoids this kind of paradoxical behaviour.

Continue Reading

math.stackexchange.com
Predicting Real Numbers
Here is an astounding riddle that at first seems impossible to solve.

100 rooms each contain countably many boxes labeled with the natural numbers. Inside of each box is a real number. For any natural number n, all 100 boxes labeled n (one in each room) contain the same real number. In other words, the 100 rooms are identical with respect to the boxes and real numbers.

Knowing the rooms are identical, 100 mathematicians play a game. After a time for discussing strategy, the mathematicians will simultaneously be sent to different rooms, not to communicate with one another again. While in the rooms, each mathematician may open up boxes (perhaps countably many) to see the real numbers contained within. Then each mathematician must guess the real number that is contained in a particular unopened box of his choosing. Notice this requires that each leaves at least one box unopened.

99 out of 100 mathematicians must correctly guess their real number for them to (collectively) win the game. Do they have a winning strategy?

Yo, Grammar: What's up with "There is a lot of (x)" and "There are a lot of (x)"?

Good question, m-elinya.

Whether we need is or are before “a lot of” depends on the type of noun that follows it.

If a mass noun (an uncountable noun) follows “a lot of,” use “is”:

  • There IS a lot of FOOD in the refrigerator.
  • There IS a lot of WATER in the ocean.
  • There IS a lot of SUGAR in ice cream.

If a plural countable noun follows “a lot of,” use “are”:

  • There ARE a lot of PEOPLE at the mall.
  • There ARE a lot of FLAVORS to choose from.
  • There ARE too many ERRORS in your essay.

SHORTCUT:

  1. Look at the sentence you’re not sure about: “There are a lot of people.”
  2. Take out “a lot of”: “There are a̶ ̶l̶o̶t̶ ̶o̶f̶ ̶people.”
  3. Look at what’s left: “There are people.”
  4. “There are people” is grammatical, so “There are a lot of people” is also correct. → ✓.

Pretty neat, right?

External image

The last thing you need to know about “a lot of” is that it’s considered informal; therefore, do not use it in your essays. Use “many” or some other synonym instead.

(Adventure Time GIF source: This Is What You Get; Jack Black GIF source: REPLYGIF.NET)

Here’s prize pack #2 for Most People Planking in a Pic!

- BODYPOP Victorious Bra by @bodypopactive
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This package is worth over $232! Yup, you guys deserve to be showered with gifts! You’ve all been working so hard! There’s still prize pack 3 to be announced :)

Remember that with this category, you need to be present in the pic. Pretty much as long as all the people in your photo are countable, this one’s easy to monitor and win if you keep checking the #popstersinplank hashtag to beat the numbers you see! Contest ends Oct 8th! GOOOOOO!!!! Tag all your friends who should be in this plank pic!

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Re[arctan(z)] and Im[arctanh(z)]

The angle shown in the first photo corresponds with the front view of one branch of Im[arctanh(z)] while the angle in the last photo corresponds with the front view of one of Re[arctanh(z)]; in other words, the two functions only differ by a 90° rotation. The flat edges of the painted areas in the center of the top and bottom Plexiglass plates are the location of two branch cuts, where there is a discontinuity. We could make the functions continuous by stacking countably many more copies of the sculpture (or branches, mathematically speaking) on top of each other to construct their Riemann surfaces, as these branch cuts correspond with where one [-π, π] iteration of these multivalued functions ends and another begins, which is easy to see in the middle photo.

“Types of Nouns in English - The difference between Common, Proper, Countable, Uncountable, Concrete, Abstract, Compound and Collective Nouns. #English #Grammar #ESL #LearnEnglish #EnglishTeachers #Nouns #WoodwardEnglish #Inglés #Sustantivos #AprenderInglés #Gramática” by @woodwardenglish on Instagram http://ift.tt/1Kuar2U

Let’s play “autistic thing or weird thing”

I can’t judge numbers at all like. mum will say “how many people turned up to the concert” and unless it’s a small, countable number (ie oh there was four people only) i’m “uhhhh???” i can do “about half the room” but not “roughly 60 people”

also: can’t judge length. “how wide is this room” hold on let me get a tape measure “no just guess” i have no idea