Tag Archives: mathematics

The Theoretical Minimum: Interlude 1, Spaces, Trigonometry, and Vectors + Lecture 2, Motion + Interlude 2, Integral Calculus

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

I’m lumping this all together because they’re all math background.

I read through these sections without skipping anything even though I didn’t have to, because, you know, math; I can literally do this stuff in my sleep (by which I mean I have successfully done calculus after being up for more than 40 hours).  But it was very interesting to read the authors’ crash course in single-variable calc — the angle they came from was informative, and I picked up a few pleasing pedagogical tricks that I may use with some of my own students at some point.

Yay calculus!

Notable Quotes

That’s basically all there is to differential calculus. (p. 36)

This made me howl in amusement.  Not because the book is wrong — it’s not!  And they did a great job on teaching calculus in ten pages! — but because I imagined telling my high school students that.  (To be clear, for a book on physics that is aimed at people who know calculus but may need a refresher, it was excellently done IMO.)

The fundamental theorem of calculus is one of the simplest and most beautiful results in mathematics. (p. 50)

Agreed!

There are some tricks to doing integrals.  One trick is to look them up in a table of integrals.  Another is to learn to use Mathematica.  But if you’re on your own and you don’t recognize the integral, the oldest trick in the book is integration by parts. (p. 55)

This perspective is fascinating to me, and I suspect it’s one of those mathematician/physicist divides that made me twitch in my physics lectures in undergrad (and my friends would all roll their eyes and laugh at me).  Because I would never consider either tables or Mathematica to qualify as actually “doing” integrals!

I was also surprised at the weight given to integration by parts as “the” basic integration tool.  To me substitution is the most basic (and is a backwards chain rule rather than a backwards product rule) and is the first go-to.  Integration by parts feels to me on the level of partial fractions or trig substitutions — useful for its own specific subset of problems, but hardly the broad skeleton key the book seems to imply.  I do wonder if this is another mathematics/physics difference — if the vast majority of integrals physicists deal with happen to fall into the type that are solvable by parts, it makes perfect sense that it would be considered the basic tool of the trade.

Thinky Thoughts

I liked the framing of an indefinite integral as a definite integral with a variable limit — I hadn’t seen it explained in quite that way before.

Also, I knew integration by parts came straight out of the product rule, but I’ve been doing it for so long that a layer of abstraction had built up and I’d forgotten!  Always good to be reminded of these connections. :)

Worked Problems

Exercise 4: Prove the product rule and the chain rule.

I know I’ve done these proofs before, but it’s been long enough that I didn’t remember how they went, so I figured why not.  My blog is being difficult about Latex formatting for some reason, so I’ll just give the gist.

Product rule: Add and subtract the appropriate quantity, and it all falls out very nicely.

Chain rule: Realize that g + Δg = g (t + Δt).  This is easy to intuit via a visualization of a t versus g graph, with points marked (t, g(t)) and (t + Δt, g(t + Δt)).  The gap between g(t) and g(t + Δt) becomes Δg, and voila.

Guns and Math: Does 1 MOA *Really* Equal 1 Inch at 100 Yards?

I’m doing rifle marksmanship training right now, and the rule of thumb for sight adjustment is that 1 minute of angle (MOA) equals one inch at 100 yards.  That means that if you adjust your sights to sweep in a certain direction (left, right, up, or down) by an angle measuring one minute, your point of impact on a target 100 yards away will move in that direction by 1 inch.  So, for example, if you adjust your rifle by 1 minute of arc left, your point of impact will shift left by 1 inch on the 100-yard target.

Of course, being a mathematician, my thought is automatically, “One MOA equals 1 inch at 100 yards?  How convenient!  I should do the math to see how close it actually is.”

It’s pretty easy to convert the angle measure of an arc to the arc length.  Take a look:

Here, theta is the angle, r is the radius (imagine the pie slice as a sector of a circle with radius r), and s is the length of the arc.

Here, theta is the angle, r is the radius (imagine the pie slice as a sector of a circle with radius r), and s is the length of the arc. (Wikimedia Commons, public domain.)

If you have the measure of the central angle in radians, the measure of the arc length — denoted s here — will be that number times the radius:

s = \theta r

Since the angles we’re looking at are so small, we can use a small angle approximation to say that our length of shift on the target equals the arc length —

. . . OKAY FINE, I shudder when physicists use the small angle approximation, but when I did it ABSOLUTELY EXACTLY without using the small angle approximation, the length of the chord matched the length of the arc out to 7 decimal places, which means our error here is about the width of an atom.  SO FINE.  We’ll use it and say the chord approximately equals the arc.

Arc / chord comparison

The red curve is the “s” we’re finding; the blue line is the actual straight-line distance on the target. For angles as small as we’re examining, the difference in length between them is about the width of an atom or two.

Let’s go back to the above formula.  We need one minute in radians.  One minute is 1/60 of a degree, so we get

\left(1/60 \text{ degrees}\right) \left(\frac{\pi}{180 \text{ degrees}}\right) = 0.000290888209 \text{ radians}

s = 0.000290888209 * \left(100 \text{ yards}\right)

s = 0.0290888209 \text{ yards}

Which equals about 1.0472 inches.

(Look at me, truncating before we get out to 7 decimal places so the small angle approximation holds!  To find the exact number, you’d have to do 2 times the sine of half the angle you’re looking at (so, 2 times the sine of half of 1 minute converted to radians, in this case), and then multiply by the radius (100 yards) and then convert to inches.  Again, for angles this small it’s equal to the above out to an absurd number of decimal places.)

So, yeah, 1 MOA is 1.0472 inches of distance at 100 yards — it’s pretty close to one inch!

The rifle marksmanship rule of thumb continues to say that we’re at 1 inch of point-of-impact change per 100 yards out, so at 200 yards 1 MOA would be equivalent to 2 inches on the target, at 300 yards 1 MOA would be equivalent to 3 inches on the target, etc. (and at 50 yards or 25 yards, 1 MOA would be equivalent to .5 inches or .25 inches on the target respectively).  Here’s what the actual numbers are:

Distance 1 MOA Equivalence (Rule of Thumb) 1 MOA Equivalence (True)
25 yards .25 inches  0.2618 inches
50 yards .5 inches  0.5236 inches
100 yards 1 inch  1.0472 inches
200 yards 2 inches  2.0944 inches
300 yards 3 inches  3.1416 inches
400 yards 4 inches  4.1888 inches
500 yards 5 inches  5.2360 inches
1000 yards 10 inches  10.4720 inches

So we’re edging up to half an inch difference at 1000 yards.  That’s a lot!

Now the question becomes — well, scopes often adjust as 1 click = 1/4 inch.  But are they adjusting 1/4 of an MOA (and thus 1/4 of 1.0472 inches), or 1/4 of a true inch?  Wikipedia had the answer:

One thing to be aware of is that some scopes, including some higher-end models, are calibrated such that an adjustment of 1 MOA corresponds to exactly 1 inch, rather than 1.047″. This is commonly known as the Shooter’s MOA (SMOA) or Inches Per Hundred Yards (IPHY). While the difference between one true MOA and one SMOA is less than half of an inch even at 1000 yards,[5] this error compounds significantly on longer range shots that may require adjustment upwards of 20-30 MOA to compensate for the bullet drop. If a shot requires an adjustment of 20 MOA or more, the difference between true MOA and SMOA will add up to 1 inch or more. In competitive target shooting, this might mean the difference between a hit and a miss.

Hey, look how useful math is!

Why New Math Fads Are Bad For Math Education

An insightful remark by one of my Twitter friends today sparked some thoughts about mathematics education — specifically, my thoughts about the new fads in math education that seem to crop up all the time in schools across the U.S. like ever-more creative varieties of fungus.  As someone who loudly and often condemns many of these new “methods” for teaching math, it turned out I had far too many thoughts for Twitter.

I’ve worked for many, many years as a private math tutor, at varying degrees of full/part time (right now I don’t do it for my main job, but maintain a few students because it’s fun and I like teaching and I like the company I work for).  My students have crossed a wide variety of middle and high schools, with a wide variety of curriculum tracks (from off-beat “hippie”-ish private schools to standard state curricula to college, and including a range of clients from students for whom math was not their forte to students who were brilliant and wanted more enrichment to students who were smart but lazy).  Pretty much the only selection bias my students have had in common is that they and their families tend to be in a financial bracket to hire me and care enough about academics to do so, which has slanted the demographic I work with to be much more likely to attend schools that purportedly each really try to have a good math curriculum.

And so many of them . . . well, don’t.

Education is a hard thing.  I realize that.  It’s really, really hard to figure out the optimal way of teaching a lot of disparate students with a lot of disparate skill levels some knowledge they don’t necessarily all want to learn.  Teachers are often underfunded, with too many students, and constrained by administrations or standardized tests that work against them.  I’m willing to cut teachers a lot of slack for not batting 1000 for all students at all times.  But what really gets my goat when it comes to mathematics education is what I call “fad math.”

Fad math is my students whose school thought the best way to teach them geometry was to put them in small groups and say, “figure it out” with no additional guidance (this is based on the textbook, by the way, which does not teach at all).  Fad math is my students whose curriculum jumps from topic to topic with no discernible connective tissue, and then assigns a mountain of problems . . . maybe three each on each of the wildly different topics.  Fad math is the private school that decided real-world math was more important than algebra and calculus, so taught taxes and mortgage amortization instead.  I could go on.  (And on.)

I get the motivations behind trying out these fads.  Mathematics teaching in this country is (in many ways correctly) perceived as broken, and people are looking for the magic formula, the Holy Grail, the thing that will work.  For example, in the first example above, I get that the idea was probably originally that more “figuring out” should happen in math teaching and less rote memorization; I support that in principle, but when we’ve reached a point at which any actual teaching has disappeared and students are basically being required to re-derive modern mathematics from scratch (with the end result that most of them just learn nothing), we’ve gone way, way, way, way too far.  The second curriculum I mentioned comes from the fear that students lose material when they study one topic in bulk rather than repeating the skills over a period of time, which, again, is a problem worth addressing — but gutting students’ ability to gain an in-depth understanding of any material is not the way to do that.  And while I applaud schools looking to teach students about real-world math like taxes and mortgages — again, a good idea! — lacking a more traditional math base completely derailed students who moved or transferred high schools and also snapped off the math foundation needed for any students who wanted to go on to a STEM field in college, including fields like pre-med and economics.

Fad math, in my view, seems more about people being proud of a shiny new idea — “this idea will work for pumping math knowledge into kids’ heads!” rather than being concerned with teaching, which is, in my opinion, where people should be concerned.  My best math teachers have never used any gimmicks, ever.  But they were really, really, really good at explaining things in a way that made sense.

(And that’s the kind of teacher I try to be, too: one who explains things in a way that makes the lightbulb go off and the student say, “Oh!  So then that’s why this happens!”  It’s amazing to be able to help someone reach that place.)

For what it’s worth, here are the main problems I see with math teaching in this country from working with my students:

  1. Math taught as a “how to” instead of a “why.”  If all you’re doing is memorizing that this number should go here and that one should go there when you see a certain symbol, then that’s . . . . well, almost useless.  If all students are doing is following a flow chart by rote memorization, there’s no mathematical understanding going on.  Students have to know why a thing makes sense in order for it to, well, make sense.  Teachers need to teach the why instead of just giving a recipe for the how.[1]
  2. No connection between mathematical ideas.  Math is ridiculously interconnected.  Every topic is related to every other.  And when you help a student relate a new topic back to an old one, it enhances understanding of the old one while giving the student an intuitive basis for the new one.  Trying to learn math concepts in isolation is a ridiculous proposition, and yet, that’s what students are so often asked to do.
  3. A horrifying number of high school math teachers don’t seem to have a deep understanding of the concepts they’re  teaching.  I can’t count the number of times my students have come to me confused about explanations their teachers gave them — explanations that were off-base, muddled, or just plain wrong.[2]  The cynic in me bets that this is because the teachers learned math by learning “how” as well, so when students ask the “why,” the teachers might genuinely not know.  (This is not, of course, true of all math teachers, and is perhaps not even true of most math teachers — I don’t know — but that it is true of a noticeable number concerns me.)
  4. A smaller number (but even more horrifying in its existence) of math teachers either just don’t care or are actively derogatory towards students.  This includes everything from being hostile toward giving extra help to sexism toward female students.
  5. Math teachers also have roadblocks thrown in their path from every conceivable corner.  Class sizes are too large, stripping away teachers’ abilities to tailor explanations to individual needs or to give a struggling student the extra help that might make the difference.  Standardized tests and state-imposed math standards often do more harm than good, as they pressure teachers into hammering the “how” into students hard enough that they’ll get the right answers without ever addressing the “why.”  Teachers are underpaid, overworked, and often struggle against turgidly bureaucratic administrations.  And in math specifically, the sort of “fad math” I’ve referred to here is often forced on teachers from the outside, hobbling their ability to actually teach.

I don’t mean to sound hostile toward teachers.  Like I said, I’ve had some brilliant math teachers in my time — in fact, I don’t think I’ve ever had a single math teacher who was bad at what she or he did.  And god, look, that’s probably a huge part of the reason I fell in love with math and went into it: because of my teachers.  Two of my high school teachers in particular (one in my high school and one in a summer program) are probably directly responsible for me going to MIT and majoring in math.

But this just reinforces my point: Good teachers are so freakin’ important.  Not a single one of my math teachers ever used a gimmick or a fad or some krazy new-fangled new idea for gettin’ math into the heads of them dumb-dumb math-hatin’ students.  They used blackboards, white boards, or transparencies.  And — and now that I’m thinking about it, this was true without exception — they pretty much spent the entirety of each class period writing and talking and explaining things.  And it worked.  I mean, I know it might seem like, okay, this is me talking, and I’m smart and good at math so it worked for me — but no, it worked in general.  I still use visualizations taught to me by my geometry teacher 17 years ago with my students and they find them incredibly helpful.  My calculus teacher was exceedingly proud of the fact that every single one of her AP students would consistently get 4’s and 5’s on the AP exams.  Every single one, in a public school.  (A good public school, but still.)  And she didn’t teach to the test; she just taught well (and was incredibly beloved by students, not just me).

Yeah, I was very lucky.  But I’d like all students to be able to be that lucky.  To be as well-taught and inspired as I was.  To feel that they’re not just passing tests, but that they truly understand what they’re learning.

There’s a lot that is hard about education reform.  A lot.  But for Pete’s sake, one thing we can do is stop it with the fad math.  Stop dropping shiny new assembly line algorithms across school curricula in the hope that they’ll press out perfect little cubes of students who know how to factor properly.  You can’t teach math by plugging a student into a flow chart.

You teach math by teaching it.  There are many, many excellent ways and methods of teaching, of course, and I’m not saying discussing those isn’t valuable — I could probably write a book on all the different ways I’ve discovered to explain calculus.  But so many of these math fads stop valuing teaching entirely.  And that makes our education system, one in which there are already so many things to fix, just that much more broken.

  1. There’s this commercial for an online tutoring service that drives me bonkers.  It’s meant to show a good math tutor.  The student calls up the tutor and says, “How do you find the area of a triangle?” The tutor says something like, “Well, [Student’s Name], the area of a triangle is one-half base times height!  So you take the base, and multiply it by one-half and by the height!” and she writes A = 1/2 bh.  And the two of them smile at each other like this is just peaches.  And I scream every time I see this commercial, because teaching a kid to memorize a formula, that’s not teaching math.  In fact, area of a triangle is one of the easiest things to explain — A = bh is quite intuitive for a rectangle (and if not can easily be demonstrated via a visualization of 1×1 boxes), and then you can show the area of a triangle as being half the area of a rectangle by drawing a rectangle and slicing it down the diagonal to make two triangles, so for a triangle A = 1/2 bh.  (Slightly more rigorously, you can teach area of a parallelogram in between those, but “triangle as half a rectangle” is actually easier for most students when intuiting the reason for the formula, and the other connections can be drawn later.)  In any case, this commercial is everything I hate about bad math education in one thirty-second soundbite.
  2. In most cases it’s pretty easy for me to tell when it’s just the student who’s confused versus when the teacher was actually confusing.

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 Much Bigger is a 4.7 Earthquake Than a 4.4? Twice As Big.

OMG it’s a math post!

Yesterday we had an earthquake here in SoCal.  It was the most dramatic one I’ve been here for, by a long shot, and my friends have all said the same — probably because we all live near the epicenter and it was quite a shallow quake.

It turned out the earthquake was only a 4.4 — a bit of a disappointing number after what we had felt!  Anyway, it was first reported in as a 4.7, and then revised to a 4.4, and someone retweeted this tweet:

Well, yeah.  But that doesn’t really tell us anything, does it?  It’s a difference of .3 on the Richter scale, but the Richter scale is logarithmic, and we silly humans aren’t used to thinking in logarithms unless we’re astronomers.  So how much bigger IS a 4.7 than a 4.4?

(cut for math)

Continue reading

Links to Analysis Regarding AuthorEarnings.com

Because I am, apparently, incapable of keeping my mouth shut when it comes to certain things.

Like math.

Like bad math.

Like people using bad math to support their pet Cause when the data do not support those conclusions.

If you don’t know what I’m talking about: self-publishing evangelist Hugh Howey and a silent partner went and scraped a bunch of Amazon data.  That’s fine.  That could be cool, even.  But then they made a bunch of pretty charts and used it to bang their pro-self-publishing / anti-trade publishing drum, and wrote a whole lot of paragraphs next to the pretty charts as if they were Conclusions, when, in fact, those paragraphs were not in any way implied by the data collected.

This pains me in my mathematician heart.  And it makes me angry when people misinform aspiring authors this way.  Mr. Howey touts himself as an author advocate, but that’s not what this is.  These data do not support his “conclusions.”  To be fair, they don’t disprove his ideas, either; they just don’t really say much of anything.  And when Howey pretends that they do support him, he’s giving authors bad information.

I’m not saying all this because I’m anti-self-publishing (I’m not!  I’m doing it myself, in fact!).  But science isn’t about “sides.”  When talking about science or math, there shouldn’t be sides; there’s no “teach the controversy” or “we’ll let the people who believe Earth is flat have equal air time.”  Or there shouldn’t be.  There’s just what the data imply, and what they don’t.  And there’s absolutely no shame in saying, “I firmly believe in XYZ.  And I just collected a lot of data in the field . . . but unfortunately those data don’t support XYZ.  They don’t contradict it, either, but there are just too many limitations here, and too much we don’t know.  That said, I still believe my ideas on XYZ are right and that the data will bear them out eventually!”

There’s no shame in that.

But that’s not what Howey did.  He used the numbers to pretty up a dog-and-pony show that pretends to support his preconceived notions with data, and he posted a piece that is actively detrimental to anyone trying to cut through the obfuscation and agendas and learn about publishing.

Now, who wants MATH?  Have some links![1]

How (Not) to Lie With Statistics.  “[The authors of the report] make claims that the data cannot possibly support […] they do a lot of inferring that is analytically indefensible.” (emphasis in the original)  I highly suggest reading the whole thing.  It’s a very detailed and well-written analysis by someone trained in research and sociological methods, and it concludes, as I did, that these data do not imply anything like what Howey claims.

Some Thoughts on Author Earnings. “The failure to compare the model’s results to actual measurements before making pronouncements is a huge problem.”  Courtney Milan is an extremely successful self-publisher, so obviously she’s pro-self-publishing.  She’s also clearly incredibly knowledgeable about data analysis, and she points out a myriad of problems with the way these findings are presented, as well as also some possible discrepancies in the raw data.

The Missionary Impulse. “Sorry, Hugh.  There is absolutely nothing in your blog post that justifies that conclusion.  This is not the same as saying that your conclusion is wrong.  Maybe it’s right.  But if it’s right, it’s not because of anything — anything! — in your blog post.”  This makes many excellent points and comes with a context of a lot of details of the publishing industry (the author is a literary agent).  Once again, the conclusion is that the data do not actually allow Howey to make any of the extreme claims he’s making.

Digital Book World: Analyzing the Author Earnings Data Using Basic Analytics.  “For myself and others, I wish I had more optimistic findings that showed we could all share in an incredible gold rush, but the data are the data.”  This article makes a case that the data are actually entirely consistent with the site’s own (far more pessimistic) prior survey, and can’t be used to prove anything more extreme.  (Obviously it’s possible there’s a bias there, and I can’t comment on the DBW survey as I haven’t seen the full thing, but I think what’s said here is valuable and knowledgeable regardless, and I note that the author is exceptionally qualified at data analysis.)

Some Quick Thoughts On That Report on Author Earnings. “[W]hile the report gives the illusion of providing hard data, it appears to be as built on guesswork as anything else we’ve had.”  Steve Mosby also makes excellent points about the unique path a published book takes, and that this can’t be repeated with hindsight.

Edited to add: Comparing self-publishing to being published is tricky and most of the data you need to do it right is not available “Unfortunately, Hugh’s latest business inspiration — a call to arms suggesting to independent authors that they should just eschew traditional publishing or demand it pay them like indie publishing — is potentially much more toxic to consume.” Mike Shatzkin weighs in with a long list of other variables Howey’s report does not take into account.

 

Look, you can’t list a lot of numbers and a lot of pretty charts and then list “conclusions” next to them and say one follows from the other because they happen to be next to each other on the page.  Science doesn’t work that way.

The poor way these data have been presented only serves to feed the adversarial “us vs. them” mentality that (some) self-publishers and (some) trade published writers are for some strange reason so invested in.  Personally, I want to see that attitude go away forever.  It’s not productive.  It’s not helpful.  I wish to all that is holy that Howey had come out with this spreadsheet in a more professional way, an invitation to other people in the writing/publishing world to analyze the data and see what we might be able to learn.  That might’ve been nifty, a positive addition to the knowledge base.  Instead, by presenting it as part of such a massive load of bad math and misinformation, he’s only clouded the discussion even more.

That’s not good for anyone.  And speaking as a self-publisher, it embarrasses me.  False conclusions that are unsupported by data, written up in something that pretends to be a study but is anything but—it just looks desperate.  Self-publishing is all grown up now, and the people most responsible for stigmatizing us in the eyes of other writers and publishers are the self-publishers themselves who pull stunts like this one.

 

Comments are closed, as I don’t have time to babysit the blog right now and from what I’m seeing elsewhere this subject can be rather contentious.  I may reopen them later.  If you have something you feel would be a valuable addition to this post, feel free to send me the comment through the Contact page and I will post it here.  Be warned that I am only going to be prone to posting contributions of the dry academic variety on this one.

  1. Note that this list is, in order, a researcher who doesn’t write fiction, a successful self-publisher, a literary agent, a data analytics professional whose research is in digitization, and a trade published writer.  And I’m a math nerd who is self-publishing my fiction books.  The biases we’d be expected to have are all over the map, but like I said, science doesn’t take sides.

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.

The Mathematics of Walking and Running

So one of my betas gave me the feedback that she wants EVEN MORE MATH in my already-excessively-mathy book.

My reaction: “MORE math?  I CAN DO THAT!”

In particular, my beta (who is not a mathematician, by the way) wanted a few more technical specifics at some points.  I’d consciously tried to keep a balance between where I mentioned technical words and where I handwaved and basically said “because MATH,” and she thought some of the handwaving could stand a little more detail.  (Incidentally, I was quite chuffed the technical bits were interesting enough that she wanted more of them!)

Anyway, one of the bits my beta flagged was a spot where the MC is drawing conclusions about a person from the way he walks.  Here’s what’s in the book now by way of explanation:

It came to me in numbers, of course, the subtle angles and lines of stride and posture.

So, having been given the note of adding a touch of the specific to this part, I found myself researching the mathematics of walking.

OMG SO COOL.

This post is basically a ramp up to tell everyone to go to this website:

Modelling, Step by Step

which models walking and running mathematically, and can I say again, OMG SO COOL.  There’s mathematics behind the maximum speed we can walk (without breaking into a run), why running is more efficient, and why people who are trained to speed-walk can actually walk faster than people who aren’t.  HOLY CRAP THIS IS COOL.

On a side note, I’m constantly excited by how much I learn doing research for this book series.  I’m a theoretical mathematician, which basically means that the only mathy parts I’ve been able to write without research are the ones using high school-level math or physics (e.g. projectile motion) or when my MC was hallucinating.  I know very little applied math at all.  Writing this series has taught me all sorts of useful things, like whether a bullet can knock a grenade off course, and that blood spatter involves trigonometry, and now about the circular motion of walking!

If You’ve Never Played with a Möbius Strip Before . . .

. . . or if you don’t know what one is, then drop what you are doing and go do this:

  1. Cut a strip of paper (about an inch or so wide is good).  It should be long enough for you to join the ends to make a ring—but don’t do that yet!
  2. Put the ends together as if you were going to make a ring for a paper chain or something, but first put a half twist in the strip of paper.  You can do this easily by putting the ends together and then turning one side over so the “back” of the paper on one side is meeting the “front” on the other.
  3. Tape the ends together so you have a paper ring with a half-twist.  Now . . . what does it do?
  4. Take a pencil and draw down the center of the strip of paper, like you’re drawing the line in the middle of a highway—well, a funny half-twisted highway . . .
  5. Keep drawing until you’ve met up with where you’ve started.
  6. Keep drawing . . .
  7. MIND = BLOWN, right???
  8. Now, for extra fun, take a pair of scissors and cut along the line you just drew, like you’re trying to cut your one paper ring into two paper rings.  (Make sure you don’t cut through the strip as you do this—snip or push the scissors through the paper in the middle of the strip to start it off, rather than starting from the edge.  Again, think as if you’re trying to make two separate rings out of the first one . . .)
  9. AMAZEBALLS!

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?

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