Tag Archives: math in life

The Theoretical Minimum: Lecture 1, The Nature of Classical Physics

This is part of my series of commentary on the physics book The Theoretical Minimum.

Notable Quotes

The job of classical mechanics is to predict the future. (p. 1)

I love this.

The rule that dynamical laws must be deterministic and reversible is so central to classical physics that we sometimes forget to mention it when teaching the subject.  In fact, it doesn’t even have a name.  We could call it the first law, but unfortunately there are already two first laws — Newton’s and the first law of thermodynamics.  There is even a zeroth law of thermodynamics.  So we have to go back to a minus-first law to gain priority for what is undoubtedly the most fundamental of all physical laws . . . (p. 9)

As a number-lover, this sort of thing just makes me all kind of amused.

But there is another element that [Laplace] may have underestimated [when he said the laws of physics could theoretically predict the whole future]: the ability to know the initial conditions with almost perfect precision. […] The ability to distinguish the neighboring values of these numbers is called “resolving power” of any experiment, and for any real observer it is limited.  In principle we cannot know the initial conditions with infinite precision. […] Perfect predictability is not achievable, simply because we are limited in our resolving power. (p. 14)

This concept, I am keenly aware, is what makes my Russell’s Attic books science fiction.  My main character is only able to do the calculations she can on the world around her because I permit her to have indefinitely good resolving power.  It’s kind of a required secondary power for what she does.  And reading this section, it completely tickled me that it has a name!

Thinky Thoughts

I’ve said before that I reduce all physics to doing math.  I felt like I was cheating a bit in this section, because saying a system is deterministic and reversible is the same as saying you can model it with a one-to-one function.  So I bopped along just thinking of the functional invertibility of the the rules, most of which I knew off the top of my head.

Sigh.  You can take the mathematician out of mathematics . . .

Fun with Numbers: 1066

I went to get a mailbox last week for my self-publishing venture, and the employee let me choose my own number.  Math nerd that I am, I stood there for a solid ten minutes looking at the bank of mailboxes and figuring out which number I would prefer.

1729 would’ve been my first four-digit choice, but they didn’t go up that high.  The obvious choices like 1024 were taken.  I started scrolling through this site, looking at what interesting mathematical qualities each of the open mailbox numbers had.

And then I spotted 1066.  Damn.  That was the one.

The Historical

When I told my friends I chose 1066, they immediately said, “Normandy?  Why?”  After rolling my eyes at having such smarty-pants friends, I explained that it wasn’t the Battle of Hastings per se that I was into, it was the Bayeux Tapestry.  Ever since I studied it in art history, something has tickled me about the Bayeux Tapestry — as a piece that has been so remarkably preserved, as a piece of craftsmanship so grand in scope, as a piece of such extensive narrative storytelling — and I just get a kick out of it.  I’ve never seen mention of anything else like it in the art world.

And I always associate 1066 with the Bayeux Tapestry.  So there was that.

The Numerical

From the “What’s Special About This Number?” list, the following happens to be true about 1066:

2\phi(n) = \phi(n+1)

Since I’ve done some work with the totient function before, and since it gets a shout out in Zero Sum Game when my main character is hallucinating (yes, she hallucinates math, what else would she hallucinate!), that seemed rather perfect.

The Historical / Mathematical / Personally Significant

There’s this game called 24.

I first learned it with cards.  You set out four cards, and you try to use each of them exactly once and end with a result of 24 (with the face cards being worth 11, 12, and 13).  So, an ace, a three, a five, and a jack could form (11 – 5)(1 + 3) = 24.  If you’re the first person to come up with a working combination, you win that round.

Some people claim the rules say you can only use addition, subtraction, multiplication, and division.[1]  I think it’s FAR more fun — and more challenging! — to allow any operation.

Anyway, I’m addicted to this game.  I play it solitaire-style with anything that has four numbers.  Like license plates when I’m stuck in traffic (CA plates have four numbers and three letters).  Or dates.  And 1066 is one of my favorite 24 combinations ever:

(6 - (6^0 + 1))! = 24

Boom.

I liked that solution so much that I put it in a story I wrote when I was in high school.  In the story, the character couldn’t remember what happened in the year 1066 — only that it was “something important,” and that the combination of the digits to make 24 was a cool one.

So, there you have it.  1066 became my new mailbox number.

  1. I know some people claim this, because they’ve tried to disqualify my creative solutions!

How to Use Mathematical Expectation in Daily Life (A Demonstration)

Mathematical expectation is one of the single most useful pieces of mathematics ever.

The technical stuff: to calculate it, you multiply the probability of something happening by how much return that thing will give you.  This allows you to compare different paths.  Cool!

I’ll show how this works by example.

I had two possible jobs this week (well, three, but I got called for the third after I’d already committed to one of the others).  One of those jobs was a definite, but it was also a job I would essentially be doing as a favor.  The other job was at my full rate, but I wasn’t sure it was going to happen.  Which to accept?

I’m not going to put the actual amounts down here, because I feel weird talking about what I make online, but the orders of magnitude are right.  To keep the numbers super easy, let’s say it was this:

Favor job: $400

Real job: $4,000

The probability of the favor job happening was pretty much 100%, or close enough.  The real job I wasn’t sure.  But I could estimate based on where the production was in the process and how they were interacting with me.  Let’s say, in my experience, 9 out of 10 jobs that get this far actually happen.  Then I estimate the real job’s probability at 90%.  Thus, my expected return is:

Favor job: $400 x 1.00 = $400

Real job: $4,000 x .90 = $3600

$3600 is way better than $400!  So the clear choice is to take the real job and turn down the favor job.  Even if the real job doesn’t happen, I can be comforted in knowing I’ve made the mathematically correct choice, the choice that was most likely to work out best.  (Yeah, I do find that shit comforting.  So shoot me.)

Okay, let’s say you don’t know the probabilities terribly well.  This can still be helpful.  How low a probability would the real job have to have to give me an expected return as low as the favor job?

Favor job: $400 x 1.00 = $400

Real job: $4,000 x p = $400

It’s pretty easy to see that p would work out to .1, or 10%.  So the real job would have to be so improbable as to only have a 10% chance of happening for both jobs to have the same expected return.  Even if I don’t know the real probability, I might be able to say that I’m pretty sure it’s over 10%.  (And it would have to be below 10% to have a lower expected return than the favor job and finally make the favor job be the better choice.)

So, there’s a quick guide to using mathematical expectation in life.  If you can estimate the rough probabilities of two paths, multiply those by the return, and you’ll get the expected return; compare those and pick the bigger one.  Done!

Man, math makes life decisions so easy.[1]

  1. Not really.

My Very Favorite Math Joke

If you hang around with mathematicians long enough, you hear All the Math Jokes.

My very favorite one is as old as time and is so corny not even other mathematicians find it funny.  Most math people I know hate it.

Since I feel like a funny blog entry today (by which I mean “funny to me”), I’m going to inflict it on you.

Background: There’s this thing called the commutative property that says an operation can work both ways.  For example, 4+7 = 7+4, because addition is commutative.  Easy enough!  A group is called an abelian group if the the elements in the group commute—so, for example, the real numbers under addition would be an abelian group.

So now . . .

The Joke:

Q: What’s purple and commutes?

Continue reading

The Mathematics of Comic Con: Conversations

My friend is ziptying frames of PVC pipe together to pack up the set pieces for our masquerade act.

Friend: I’m trying to make sure these don’t come apart. It’s a topology problem. There. Will those stay together?

::I pick up the frame pieces and shake them::

Me: Yup. (beat) Notice how I did that like an experimentalist rather than a theoretician. Instead of proving it was solid, I tested it.

That night, we are verifying we’ve pinned the backdrops correctly for our sewing-capable friend[1] to sew them. We are momentarily concerned because we’ve pinned the Velcro parallel to the direction of sewing rather than perpendicular to it.

Sewing-Capable Friend: Parallel is fine, as long as they’re all pointed away from the direction I’m sewing in so I can pull them out as I go.

::other friend and I look at each other::

Me: Did we do that?

Other Friend: Well, it’s a binary choice. Either we did or we didn’t . . . fifty-fifty chance . . .

Me: But the chance we did it right on all of them is more like one-half to the power of how many pins we put in—

Other Friend: True, except the probabilities for the rest of the pins were probably conditioned on the direction of the first, because I think we kept going in the same direction.

::we look at sewing-capable friend::

Me: Uh, we’ll check.

Later, when finding our car in the parking garage, which is structured with a concrete pillar every three cars:

Friend #1: Where did we park?

Friend #2: Well, I remember it wasn’t next to a pillar.

::everyone laughs::

Friend #1: Technically, that does diminish the possibilities by 2/3.

Me: Yes, but that 2/3 is a sieve across the whole parking lot, so it doesn’t actually limit our search area!

Just before masquerade, a regular con-goer approaches my friend, who is also a regular con-goer.

Regular Con-Goer: I hear your group this year is like . . . you guys, plus like five MIT people.

My Friend: That’s not entirely inaccurate.

  1. By “sewing-capable” I mean “massively ridiculously talented, and generously deigning to direct sewing-incapable peons like myself in ways we can help.” Just to be clear.

Conversation of the Night: All Hail the Singularity

Friend #1: “You’re so unique you only have one eigenvector.”

Me: “Yes, I’m a very singular person.”

Friend #1: ::Groan::

Me: “Are you groaning because it’s such a bad pun or because it’s technically mathematically incorrect?”

Friend #1: “BOTH!”

Other friend: “Wait, a singular matrix doesn’t have any eigenvectors.”

Me: “YES ARTISTIC LICENSE I’M SORRY!”

Other friend: “And isn’t it impossible to have only one?”

Me: “DON’T EXPLAIN THE JOKE!”

Today’s Exercise in Self-Control

While getting patted down by an admittedly very polite TSA officer, I had to listen to the TSA officer patting down the guy next to me go on and on very sanctimoniously about how we passengers are unqualified to have opinions on airport security procedures because we’d be talking about something we “know nothing about.”

It took every ounce of self-control not to speak up and say that 1) I read experts like Bruce Schneier, who most certainly do know what they’re talking about, and their general opinion is that TSA policies are dumb as bricks, 2) part of this reading has included extensive evaluations about how ineffective the TSA is at stopping any potential terrorist attacks, and 3) I happen to have read statistics on things like the body scanners and how many people avoid the TSA in favor of ground transportation and also the numbers on, y’know, terrorism, and the TSA kills way more people than terrorists do, so this mathematician says in your face!

Ahem.

Anyway, it was really, really hard not to say all this, guys! In fact, if the guy had been talking to me, I don’t think I would have been able to resist. (And I’d probably be stuck in a small room right now being interrogated and missing my flight.)

Don’t I have spectacular self-control this morning?

Boston: When Math Doesn’t Help

I still have so many friends in Boston.  I’ve been haunting Facebook pages and emailing people all afternoon, trying to get confirmation that everyone is okay, terrified I’m forgetting someone, terrified that someone I used to know but didn’t keep in touch with was at the marathon.

My closest friend in Boston, someone I care about very deeply, still hasn’t gotten back to me.

I know the chance that someone I know was injured is tiny.  I did the math, even—despite the length of time I lived there, there’s less than a 1 percent chance anyone I know was affected, probably more like a .01 percent chance.

But that doesn’t matter.  The math doesn’t help.  I still have to know.  I still need them to tell me they weren’t hurt.

(And I feel selfish saying all this—because we know how many people died and how many people are injured, and nothing will change that number, and if they aren’t my loved ones they’re someone else’s loved ones.  But I can’t help it.)

I still think of Boston as my city.  Some cities are places you live in—Boston was home.  This attack feels more than tragic to me; it feels violating.

Boston’s a small city, too, impossibly small.  Even if my friends are all okay, will I hear next week about a friend of a friend?  About a former professor’s daughter, or the spouse of someone I used to train swords with?  This attack is too close, too intimate; it feels impossible that it won’t somehow resonate through my social circles, that someone I know won’t be personally affected by it.  Boston isn’t like LA; almost every time I meet someone from Boston we have something in common: some experience, some hang out, some mutual friend.  It’s too small a city.

I don’t know what I’ll find out later today, or later next week, or three weeks from now.  All I can do right now is wait.  Wait, and stalk Facebook with the shitty hope that they were all someone else’s loved ones.

And Now for a Nerdy Math Song! You Know You Wanted One!

I was working on some sort of Deep Insightful post today about institutional prejudices and society and the way people respond to shit and . . . well, I just kinda ran out of steam on it.  Sometimes blogging is cathartic, but other times I can only spend so long thinking about a subject that depresses me.

So, in Operation Brighten My Day, instead I’m sharing one of my favorite songs!  Possibly my favorite song ever ever!

“Finite Simple Group of Order Two,” by the Klein Four:

This song always makes me giggle.  So.  Much.

Happy Pi Day!

In celebration of one of our favorite transcendental numbers, today we enjoyed chicken pot pie, macaroni and cheese pie (seriously, it exists), and chocolate pie.  Mmmmm!

Pie.  Public domain.

Pi on pie.  Wikimedia commons/ public domain.