The Higgs Field Fallacy

A little while ago, I did a comic in which the joke was a lady saying something like “if you don’t agree that math is gender neutral, you will be!” while brandishing a knife. Interestingly, this got some trans advocacy groups AND some mens’ rights groups mad. For the sake of this argument, I’m more concerned with the latter.

The claim being made was that the joke mocked genital mutilation. I suppose this is, strictly speaking, the truth. The joke is that this lady will cut your dick off for reasons that are amusing because of a pun on “neutral.” I can’t deny that it’s a joke that involves the idea of cutting someone’s dick off, but I don’t agree that this is necessarily a serious matter. I was trying to figure out the exact reasoning, when I happened to come across a passage in a book of funny etymology stories.

The story was about military K rations. More interesting to me than the etymology was a joke from the author, who presumably spent some time in the military. The joke went something like this: “K rations served the dual purpose of feeding soldiers and making them angry enough to kill.”

That is, of course, a joke about killing a fellow human being. I think I can safely say I’d rather have my dick cut off than be killed, which suggests to me that one is more serious than the other. Yet this joke is considered so tame that it appears in a cute illustrated book with the unintimidating title “Who Put the Butter in Butterfly?” So, I’m assuming that most people would not take it as advocating or trivializing murder.

And then I realized what the problem was. We’re really talking about a particular straw man argument, in which a statement intended as a joke is taken to be serious. Because something that is light is made out to be massive, I thought it might be funny to call it “the Higgs field fallacy,” since the Higgs field gives rise to mass. It’s possible it already has a name, in which case forgive me.

The essence is this: Sometimes jokes are meant to be serious and sometimes they are not. This is the advantage and disadvantage of being a comedian – you can choose whether your past statements are serious or not (and so can other people). We sometimes assume that all statements carry weight, when in fact many really, truly, seriously don’t. When your parents taught you to play Hangman, for example, I’m guessing neither of you felt like you were mocking capital punishment.

It is perhaps unfortunate that communication is imprecise, resulting in light things seeming heavy to one person and vice versa. However, this is a condition of signal transmission in this universe, so there’s not much sense in fretting over it.

Now that I put a name to the fallacy, I can think of lots of instances where I’ve come across it. It’s a great debate tactic, because in casual language we often laugh at the dreadful. But, if you’re in a debate, and someone does this, all you have to do is keep a serious face and say “Well, I’m glad you feel you can laugh about it.”

What’s particularly strange is how often we invoke this fallacy on public officials. When a public official uses a non-PC term we’ve used or heard a thousand times in private company, we get mad. Or if a public figure makes an offensive joke we’ve told or laughed at a thousand times, we call him an asshole. Why? Well, our jokes are massless. To theirs, we give a Higgs field.

Posted in Pondering | 27 Comments

A Favorite Chesterton Quote

Nicely threads the needle, as he often does. This is from “What I Saw in America”

Travel ought to combine amusement with instruction; but most travellers are so much amused that they refuse to be instructed. I do not blame them for being amused; it is perfectly natural to be amused at a Dutchman for being Dutch or a Chinaman for being Chinese. Where they are wrong is that they take their own amusement seriously.

PS: Yes, I’m well aware he was an anti-evolutionist! But, also a great writer. C’est la vie.

Posted in Quote | 1 Comment

Three Original Senryu

(ref)

“Monogamy gene?
How’s junior’s genome have it
If yours and mine don’t?”

“Kilt your son,” He says.
“Are you completely insane?”
He glowers at Abe.

“Why do Bobby’s pants
Smell like old lady?” asks Mom.
“And where’s my lotion?”

 

Posted in Poems | 1 Comment

An Old Joke

Dork that I am, I was enjoying an excellent book on senryu today, called “Light Verse from the Floating World.” The short version of senryu is that they’re haikus as we in the west think of them, but are closer to observational humor than lofty poetry.

Anyway, I happened on an interesting one just now. A few of you may know this modern joke: “How does every racist joke start? With a look over your shoulder.”

Well, here I am reading a book of 18th century Japanese poetry, when I find this similar one. First, I’ll put it in Japanese for those who read Japanese (I do not), then in English.

subete onna
to iu mono to
sokora wo mi

“All women…”
he begins, and then
glances around.

Ha! It’s the exact same joke, though shifted from race to gender.

 

Posted in Book, Poems | 2 Comments

End of the Science Posts

After giving it a lot of thought, I’ve decided to indefinitely stop with the sciencing blogs. There are a number of small reasons, but the main one is this: I just don’t have the time.

The big appeal to me of studying science is “horizontal knowledge.” That is, there is some knowledge that is vast, but doesn’t build upon itself (literature, for example, generally speaking), which you might call “vertical knowledge.” Horizontal knowledge is when you must have one floor of the edifice to climb to the next. Logic, math, and most of the sciences work this way.

I like to think it’s good to have both. As you may guess from the last post, I feel I’m doing relatively well on the vertical end. But, it’s been slow going on the horizontal end, largely because, well… teaching is hard. Understanding is hard enough. Trying to find the right words, the right analogies, the right thought modes, is even harder. It’s especially hard when I understand a concept very well. That to me is the real art of teaching – understanding something well while remembering why it was that you didn’t always understand it well.

But, I want to move faster. Some rough estimation I did tells me I could be learning at least three or four times more quickly if I weren’t obligated to transcribe what I’ve learned. Although I’ve been really happy to have people say that I helped them learn, I believe it’s probably a better use of my time to give myself the best possible education I can, then use my larger channels (comics, books, etc.) to distribute the information.

Additionally, hell, a lot of people are doing this a lot better. Khan Academy, Code Academy, Coursera, OpenCourseWare, BetterExplained, and many more. These people are dedicated teachers and damn good at it. Although I enjoy explaining things to people, I don’t consider teaching a calling. It’s fun, but I don’t get out of bed in the morning for it. Other people do. There might be a few people who want to learn from me, but there is nobody who needs to learn from me.

I want to thank everyone who read along with me, especially those of you who offered corrections and insight. Some of you follow me on twitter, and I will no doubt be abusing your kindness as I continue my self-education.

So, with some regret, I’ll be stopping with the formal regular science blogs as of now. I’ll only be posting new stuff when I have some particular insight or excitement about a topic. I hope this’ll free me up to spend more time on science and more time on my other backburner projects. I’ll make sure to keep you all in the loop.

<3

Zach

Posted in Personal | 3 Comments

Weinersmith’s Book Club Part 5

Wow. It’s been over a year since the last one of these. I got an email request to do another, so here we are. I’ve read a good number of books since then, so I don’t have time to properly review all. So, I’ll be using the following system:

1/5: Avoid

2/5: Not good

3/5: Read if you’re into the subject matter or genre

4/5: Definitely read

5/5: Cherish that shit

NC: No comment because it was written by a friend (or acquaintance or friend of a friend or relative or anyone else who’d make it awkward) and I don’t want to go down that road.

NR: I don’t feel I remember this book well enough to judge. That said, if that’s the case it may be a reflection on book quality.

NW: No way, I refuse to rate a classic like this. You should read it.

July 7 – Malcolm X: A Life of Reinvention (Marable): NR

July 8 – Just Six Numbers (Rees) 3/5

July 9 – Pale Blue Dot (Sagan) 5/5

July 10 – Marie (Haggard) 3/5

July 12 – The Disappearing Spoon (Kean) 4/5

July 13 – Politics (Aristotle) NW

July 17 – Fairy and Folktales of the Irish Peasanty (Yeats) 5/5

July 18 – Eiffel’s Tower 4/5

July 27 – The Rational Optimist (Ridley) 4/5

July 29 – Polio: An American Story (Oshinksy) 4/5

July 30 – Child of Storm (Haggard) 3/5

Aug 1 – Mooonwalking with Einstein (Foer) 3/5

Aug 2 – Preferred Risk (Pohl/Del Ray) 2/5

Aug 3 – History of Western Philosophy (Russell) NW

Aug 4 – 1864: Lincoln at the Gates of History (Flood) 3/5

Aug 8 – Fiasco (Lem) 4/5

Aug 9 – A Plum in Your Mouth (Taylor) 4/5

Aug 10 – The Matisse Stories (Byatt) NR

Aug 12 – Ghengis Khan (Lamb) NR

Aug 15 – Game of Thrones 1 (Martin) NC

Aug 17 – Buddha (Armstrong) 3/5

Aug 19 – Life and Times of the Thunderbolt Kid (Bryson) 5/5

Aug 19 – Not Forgotten (Oliver) 3/5

Aug 20 – The Opposite of Fate (Tan) 1/5

Aug 20 – The Midas Plague (Pohl) 1/5

Aug 21 – Tarzan of the Apes (Burroughs) 2/5

Aug 23 – Wolfbane (Pohl) 2/5

Aug 30 – Rudyard Kipling to Rider Haggard (Ed. Cohen) 3/5 (Personal 4/5, but it’s pretty fucking obscure)

Aug 31 – Winner Take Nothing (Hemingway) 3/5

Sep 1 – Kabloona (de Poncins) 5/5

Sept 2 – Chitty Chitty Bang Bang (Fleming) 5/5

Sept 5 – Growing Up Bin Laden (Sasson) 2/5

Sept 8 – Crime and Punishment (Dostoevsky) NW

Sept 9 – Everything is Obvious Once You Know the Answer (Watts) 3/5

Sept 12 – Nichomachean Ethics (Aristotle) NW

Sept 14 – The Sky is Not the Limit (Tyson) NC

Sept 17 – 1920: The Year of 6 Presidents (Petrusa) 3/5

Sept 18 – America 1908 (Rasenberger) NR

Sept 18 – Pythias (Pohl) – Short Story 2/5

Sept 23 – Leviathan Wakes (Corey) NC

Sept 24 – The Professor and the Madman (Winchester) 4/5

Sept 26 – Moll Flanders (Defoe) NW

Sept 27 – Ghost in the Wires (Mitnick) 3/5

Sept 27 – The Hated (Pohl) – Short Story 2/5

Sept 27 – The Tunnel Under the World (Pohl) – Short Story 2/5

Sept 28 – The Garden of Eden (Hemingway) 4/5

Oct 3 – On China (Kissinger) 3/5

Oct 10 – Acting White: The Ironic Legacy of Desegregation 4/5

Oct 11 – J Robert Oppenheimer: Shatterer of Worlds (Blythe) 3/5

Oct 25 – Nature via Nurture (Ridley) 3/5

Oct 26 – The Futures (Lambert) 3/5

Oct 26 – A Plague of Pythons (Pohl) 2/5

Oct 28 – Newton and the Counterfeiter (Levenson) 2/5

Nov 2 – The Confessions of St. Augustine NW

Nov 5 – The Other Brain (Fields) 5/5

Nov 6 – The Space Trilogy 2 (CS Lewis) 2/5

Nov 8 – The Day of the Boomer Dukes (Pohl) 1/5

Nov 8 – Beyond Good and Evil (Nietzche) NW

Nov 10 – Snowcrash (Stephenson) 3/5

Nov 12 – Nonsense on Stilts (Pigliucci) NC

Nov 15 – Aeneid (Virgil) NW

Nov 16 – Plain Tales from the Hills (Kipling) 4/5

Nov 18 – American Gods (Gaiman) NC

Nov 21 – The Bell Jar (Plath) 5/5

Nov 22 – Hiroshima (Hersey) 5/5

Nov 30 – 1493 (Man) NR

Nov 30 – Farmer in the Sky (Heinlein) 4/5

Dec 7 – Caesar – Life of a Colossus (Goldsworthy) 4/5

Dec 9 – She and Allan (Haggard) 3/5

Dec 11 – Steve Jobs (Isaacson) 4/5

Dec 11 – White Man’s Burden (Easterly) 5/5

Dec 12 – Sex at Dawn (Ryan and Jetha) 3/5

Dec 16 – The Black Arrow (Stevenson) 2/5

Dec 19 – Truman (McCullough) 4/5

Dec 22 – Keynes-Hayek (Wapshott) 3/5

Dec 23 – Nothing to Envy (Demick) 3/5

Dec 24 – Uranium (Zoellner) 3/5

Dec 28 – The Rise of Theodore Roosevelt (Morris) 5/5

Dec 31 – Theodore Rex (Morris) 5/5

Dec 31 – Consider the Lobster (Wallace) 4/5

Jan 4 – Colonel Roosevelt (Morris) 5/5

Jan 7 – Thus Spoke Zarathustra (Nietzche) NW

Jan 10 – Foundation and Empire (Asimov) 2/5

Jan 10 – Autobiography of Charles Darwin NW

Jan 12 – Dracula (Stoker) NW

Jan 15 – The Long Dark Teatime of the Soul (Adams) 3/5

(Move to Alabama)

The Quest for Cosmic Justice (Sowell) 3/5

The Most Powerful Idea in the World (Rosen) 5/5

Cleopatra (Grant) 4/5

Jan 31 The Iliad (Homer) NW

Feb 7 – The Everlasting Man (Chesterton) NW

Feb 9 – Coolidge – An American Enigma (Sobel) 3/5

Feb 10 – Eternity Soup (Critser) NC

Feb 13 – A Universe from Nothing (Krauss) 3/5

Feb 15 – Broom of the System (Wallace) 3/5

Feb 15 – Gateway (Pohl) 5/5

Feb 17 – Age of Wonder (Holmes) 4/5

Feb 17 – John Carter of Mars 1 (Burroughs) 2/5

Feb 17 – Tom Sawyer Abroad (Twain) 5/5

Feb 17 – Bad Astronomy (Plait) NC

Feb 19 – Shadow of the Hegemon (Card) 3/5

Feb 20 – The Sword in the Stone (White) 5/5

Feb 26 – Storm of Steel (Junger) 4/5

Feb 27 – Intellectuals and Society (Sowell) 2/5

Feb 28 – Fatal Journey (Mancall) NR

March 1 – Dialogues (Plato) NW

March 1 – The Sixth Column (Heinlein) 3/5

March 7 – The Witch in the Wood (White) 5/5

March 7 – The Poison Belt (Doyle) 3/5

March 9 – The Ego Tunnel (Metzinger) 4/5

March 9 – The Color Purple (Walker) 5/5

March 11 – Polk (Borneman) 4/5

March 13 – Madison’s Metronome (Weiner, Greg) NC

March 13 – Man Plus (Pohl) 3/5

March 23 – Heart of the World (Haggard) 3/5

 

(ECCC Prep, forgot to record dates…)

 

Armageddon Science (Clegg) 2/5

Hedy’s Folly (Rhodes) 4/5

The Ill-Made Knight (White) 6/5 (JESUS, what a good book)

 

April 4 – The Last Days of Ptolemy Grey (Mosley) 3/5

April 5 – One Flew Over the Cuckoo’s Nest (Kesey) 4/5

April 7 – Faster (Gleick) 3/5

April 8 – The Prince and the Pauper (Twain) 4/5

April 10 – Incognito (Eagleman) 3/5

April 13 – The One Percent Doctrine (Susskind) 3/5

April 14 – Last Chance to See (Adams) 2/5 (though charming!)

April 15 – God and the State (Bakunin) NC

April 16 – The Oxford Book of Ballads (1910) 5/5 (wonderful)

April 16 – Empress Orchid (Min) 3/5

April 18 – Endless Forms Most Beautiful (Carroll) 4/5

April 18 – Notes from a Small Island (Bryson) 4/5

April 19 – Sweet Thursday (Steinbeck) 5/5

April 20 – The meaning of Everything (Winchester) 4/5

April 21 – Hitchiker’s Guide to the Galaxy 5 (Adams) 2/5

April 24 – Madame Bovary (Flaubert) 5/5

April 26 – Glory (Nabokov) 4/5

April 28 – The Candle in the Wind (White) 5/5

April 29 – Money (Amis) 3/5 (though I have the strange suspicion that I just didn’t “get” this book?)

April 29 – Letter to my Daughter (Angelou) 2/5

 

Away for ROFLCon

 

May 9 – Distrust that particular flavor (Gibson) 3/5 (book of essays, some better than others)

May 9 – Judging Edward Teller (Hargittai) 4/5

May 11 – Alexander Hamilton (Chernow) 5/5

May 12 – Scourge (Tucker) 3/5

May 14 – The Fountains of Paradise (Clarke) 3/5

May 18 – The Greater Journey (McCullough) 4/5

May 20 – The Time Traveler’s Wife 2/5

May 20 – War As I knew It (Patton) NW

May 25 – The Better Angels of Our Nature (Pinker) 5/5

May 25 – Bhagavad Gita NC

May 28 – The Prisoner of Zenda (Hope) 2/5

June 2 – Wizard (Seifer) 3/5

June 3 – The Lucifer Effect (Zimbardo) 2/5

June 3 – Physarum Machines (Adamatzky) NC

June 4 – My Life and Hard Times (Thurber) 5/5

June 6 – The Reformation (Collinson) 3/5

June 6 – Burmese Days (Orwell) 3/5

June 9 – The Illustrated Man (Bradbury) 3/5

June 9 – Brief Interviews with Horrible Men (Wallace) 5/5

June 10 – The Golden Apples of the Sun (Bradbury) 3/5

June 10 – Night (Wiesel) 5/5

June 11 – Economics and Politics of Race (Sowell) 3/5

June 15 – The Beginning of Infinity (Deutsch) 3/5

 

Trip to Luling

 

June 24 – Dandelion Wine (Bradbury) 5/5

June 27 – Decoding Reality (Vedral) 2/5

June 31 – A Planet Called Treason (Card) 2/5

July 1 – Science and Religion (Russell) NW

July 2 – Eisenhower: The White House Years (Newton) 5/5

July 4 – Explorers of the Nile (Teal) 3/5

July 9 – Beyond the Blue Event Horizon (Pohl) 3/5

July 11 – The New Geography of Jobs (Moretti) 5/5

 

Comicon

 

July 20 – Cro-Magnon (Fagan) 3/5

July 20 – The Breast (Colacci) 4/5

July 21 – Economic Facts and Fallacies (Sowell) 3/5

July 22 – Charlie and the Chocolate Factory (Dahl) 5/5

July 26 – The Panic Virus (Mnookin) 3/5

July 27 – Alvin Maker (Card) 4/5

July 28 – The State of Jones (Stauffer)4/5

July 30 – Rabbit, Run (Updike) 4/5

July 31 – Abundance (Diamandis) 3/5

Aug 4 – Islands in the Stream (Hemingway) 6/5 (DAMN WHAT A BOOK!)

Aug 4 – The Forever War (Haldeman) 4/5

Aug 7 – American Lion (Meecham) 4/5

Aug 8 – The Particle at the End of the Universe (Carroll) NC

Aug 10 – Stranger in a Strange Land (Heinlein) 4/5

Aug 11 – In the Company of the Courtesan (Dunant) 3/5 (but fun!)

Aug 14 – Moscow, December 25: 1991 (O’Clery) 4/5

Aug 14 – Tom Swift and the Electric Hydrolung (Lawrence) NW

 

Lost to the West (4/5

Rabbit Redux (4/5)

Old Man’s War NC

The Unbearable Lightness of Being 4/5

 

Away for Worldcon

 

Sept 7 – The Darkest Jungle (Brick) 4/5

Sept 9 – Pale Blue Dot (Sagan) 5/5

Sept 11 – The Future of Power (Nye) 3/5

Sept 14 – The Cool War (Pohl) 3/5

Sept 15 – The Snakehead (Keefe) 4/5

Sept 19 – The Life and Times of Michael K (Coetzee) 4/5

Sept 25 – Dark Sun (Rhodes) 4/5

Sept 27 – Birdsong (Faulkes) 4/5

Oct 3 – The Dictator’s Handbook (Bueno de Mesquita) 5/5

 

Last SMBC-T Shoot

 

Oct 9 – Gods Graves and Scholars (Knopf) 5/5

Oct 9 – War is a Racket (Butler) NW

 

NYCC

 

Oct 17 – Heaven’s Command (Morris) 5/5

Oct 19 – On Monsters (Asma) 4/5

Oct 21 – Something Wicked This Way Comes (Bradbury) 4/5

Oct 23 – The Drunkard’s Walk (Mlodinow) 3/5

Oct 24 – The Price of Politics (Woodward) 3/5

Oct 26 – The Acts of King Arthur and His Noble Knights (Steinbeck) 4/5

Oct 28 – The Napoleon of Notting Hill (Chesterton) 3/5

Oct 29 – Render Unto Rome (Berry) 4/5

Oct 31 – What Technology Wants (Kelly) 2/5

Nov 5 – The Information (Gleick) 4/5

Nov 6 – I Sing the Body Electric (Bradbury) 3/5 (collection, some better than others)

Nov 7 – The Star Diaries (Lem) 5/5

Nov 8 – Lincoln at Gettysburg (Wills) 4/5

Nov 8 – Thumbelina and Other Stories (Andersen) NW

Nov 9 – A Heartbreaking Work of Staggering Genius (Eggers) 5/5

Posted in Book Club | 5 Comments

Discrete! #33: Discrete Mathematics and Its Applications 2.4 Part A

2.4: Sequences and Summations

My general sense on this sections is that the real action is in problem-solving, since summations and sequences are conceptually pretty easy. So, I’m gonna be a bit breezy here and then we can get into more depth with the problem sets. If you’ve been following along with the calc posts, you’re already pretty up on this stuff.

Introduction

BOOK DEFINITION:

“A sequence is a function from a subset of the set of integers (usually te set {0,1,2,…} pr the set {1,2,3,…} to a set S. We use the notation a_n to denote the image of the integer n. We call a_n a term of the sequence.”

I think that’s pretty clear. If your sequence is a_n=2n then the output is just the list “2,4,6,8,10,…”

So, a_n=2n is basically saying “Make a list such at a at each position 1 to infinity is given by 2*(that position).”

Special Integer Sequences

Just because you recognize a pattern in a sequence doesn’t mean you know how to denote it with a pretty function like the one above. If you’re a dork, you’ve probably come across riddles about predicting the next element of a sequence.

The book gives a slightly tricky one. Conjecture a formula for {1,7,25,79,241,727,2185,6559,19681,59047}

Now, although can’t instantly say what it is, you can try some things. Is there the same difference between consecutive digits? Nope. How about a constant multiplier between digits? Nope. You step back a second, and you notice it contains prime numbers like 7 and 29. So, at the least, there’s probably some addition or subtraction. You also see that the numbers grow quickly, so we’re probably talking about some sort of exponentiation here.

After some fiddling, you’ll see that you square n then subtract 2.

Of course, there’s no one right way to do this. But, you want to get some methods of attack to be intuitive.

Summation

A summation is just a sum of the elements of a sequence over a certain interval.

\sum_{i=1}^na_i is a pretty typical way to write out a summation.

The i is just the variable. The =1 just means that for the series we’re about to sum, we use {1,2,3,…}. If it’d been i=0, we’d have used {0,1,2,3…}. You could set some other starting point, but it’s fairly rare. The n at the top is where you end. So, if i=1 and n is 5, you care about the values of at {1,2,3,4,5}.

I have to say, it always seemed to me like it’d be more clear to write something like \sum_{1 to 5}a(i), but perhaps there’s a good reason not to.

In any case, the idea is pretty simple. A summation is the sum of a sequence over a given interval.

The book goes on to give some examples, including a proof of the geometric series. It’s probably worth your time to go over, and I don’t really have anything to add here.

My general feeling on these things is it doesn’t help to try to memorize these things outright. You remember them if you use them regularly. In general, what’s good is to have intuition about these things. Seeing a function or list of numbers, it’s useful to be able to say “That feels exponential.” or “That feels sinusoid.”

Last, they get to infinite series. As you may know from calc, an infinite series is just the sum of an infinite sequence. If this sounds weird, remember that a lot of sequences start to get to really small numbers really quickly. For example, if the function you’re worried about is 1/x!, as the sequence gets bigger and bigger, the function’s output gets smaller than smaller. Pretty quickly, you’ve got some number right around 2.71812 or so. Often, you can get a simple formula for the sum.

Next Stop: Cardinality

 

Posted in Autodidaction, Discrete Math | 1 Comment

Calculus! #65: Early Transcendentals 5.2

5.2: The Definite Integral

Remember last time when we talked about a special kind of limit? The limit of a sum of rectangles where you get the area under a curve by using smaller and smaller rectangles? That special limit turns out to be useful all over the place. Seriously, really useful. Like, useful in the way addition is useful.

So useful, it gets a special symbol. A stretched out S: ∫

Now, just as some functions are differentiable and some are not, some functions are integrable and some are not. Understanding the integral can give you a sense of when it’s integrable and when it isn’t.

For example, if you’re trying to figure the area of a function, but it divides by 0 somewhere. Well, you can’t do that. But, you CAN define the same function over a certain interval such that it DOES work.

When we define an integral over a certain interval, it’s called a “definite integral.” This isn’t always done just for integrability (is that a word?). There might just a certain area you’re interested in. Like, for example, if you want to know how far a runner went during a race, you just integrate velocity with respect to time, giving you position. However, if your function for velocity describes the runner’s entire life, your numbers are gonna get screw when he goes home after the race. You’re only interested in the time interval from the start of the race to the end.

Now some housekeeping.

To denote the area of interest, you put the starting point at the bottom of the ∫ and the end point at the top. This is arbitrary. So it goes. After that, there’s the function you care about, usually denoted f(x). Then, there’s dx.

As the book notes, “dx” doesn’t have a lot of meaning. It’s really just a placeholder for the infinitesimal change in width of the rectangles in your limit. However, we’ll see later that, much like in derivatives, having the right notation is valuable, and the dx is quite handy.

Now that you have the gist of integration, we can start getting more in depth.

The book notes that you can have “negative area” under the curve when your rectangles fall below the x axis. If you integrate over a positive area, you’ll get a positive value. Over a negative area, you get a negative value. Over both and it depends on which area is larger.

For a real world example, image you have a function that describes food eaten over time and you want to integrate it from today to the same time tomorrow to determine total food eaten during that 24 hour period. Now, say you eat 3 pounds of uncooked shrimp in the first hour. The integral over that hour will be positive. But then say you feel terrible and spend 15 hours throwing up. Then the first 16 hours will total 0. In fact, if you vomit stuff you ate before this all started, it’ll be negative. If after that, you start eating again, you’ll get back to positive territory.

More simply: When you integrate over a function, the total area is going to be the difference between positive areas (areas below the curve and above the x axis) and negative areas (those below the x axis and above the curve).

Now then, let’s talk integrability. The book says:

“If is continuous on [a,b], or if f has only a finite number of jump discontinuities, then is integrable on [a,b].”

This makes sense, right? Another way to say it might be: If you have a function where you can always get a better and better sense of the area by drawing little rectangles under it, it’s integrable. You may note that jump discontinuities are bad for derivatives, but apparently fine for integrals. Why?

Remember a derivative is a rate of change. Rates of change make no sense when there are sudden discontinuous jumps. An integral is about how much of something happened in total, so jump discontinuities don’t matter. In fact, they’re quite easy to put rectangles under.

Then the book gives some rules for “evaluating integrals,” which will actually be superseded pretty soon. So, I’m gonna skip’em.

The Midpoint Rule

I’m not sure why, but her we’re introduced to the midpoint rule. The basic idea is that you take a sum, of rectangles, but instead of having the curve always touch the left or right corner of the rectangles, you have the curve go through the middle of the top side of the rectangle. This is accomplished by taking the midpoint between two x points and using that as the reference for your sum.

At it turns out, for most functions, midpoints are a much better way of doing business. For “nice” functions, midpoints basically split the loss of always putting rectangles under functions with the gain of always putting rectangles over functions. The result is a more accurate approximation.

I’m not really sure what it’s doing in the middle of this chapter, but so it goes!

Properties of the Integral

You’ve got the math written out for you. I want to give you a verbal sense of the rules:

1) The integral of a constant over an interval is just that constant times the difference between the start and finish of the interval.

Think about it like this: Say you’re running at constant velocity. If you want to know change in position and you integrate velocity with respect to time, you’re basically just say “how long did I go and how fast did I go?” and then multiply those two things. How fast you went is the constant. How long you went is just the difference between your start and stop time.

2) The integral of the sum of two functions over an interval is the sum of the integrals of each of those functions over the same interval.

That is, if you have an integral of two functions (like, ∫(x+1/x)dx) you can split it into two separate integrals. This makes sense physically. Suppose you have two light bulbs and they each emit a certain amount of light per unit of energy put into them. The total amount of light for a given amount of energy (aka, the integral with respect to energy) is going to be the same regardless of whether you look at the bulbs together or look at them separately and add things up.

3) Integrating a function times a constant is the same as that constant times the integral of the function.

That is, if you see a constant times a function, you can yank the constant outside of the integral. Remember, an integral is the limit of a sum. It doesn’t matter if you have 3 times every member of the sum or you sum it and just multiply by 3. The total is the same.

4) Rule three also works if the two functions are subtracted from each other. This should be pretty obvious if you believe rule 2.

5) Say you have three points on a line in this order: a, c, b. In that case, the integral from a to b is the same as the sum of integrals from a to c and from c to b.

That is, if you have an integral between two points, you can always subdivide it into smaller chunks. This is often useful. For example, if you have a function from -10 to +10, you can divide it into the part that goes from -10 to 0 and then from 0 to +10. Having those zeroes can make the math a little easier.

Additionally, it makes physical sense. If you’re figuring the amount of sweat that comes out of your body over a certain distance running, and you have a function for sweat vs. distance, calculating total sweat from 0 to 10 miles is the same as calculating sweat from 0 to 5 and 5 to 10, then adding it all up.

6) If a function is never negative on a certain interval, then its integral is non-negative.

This certainly makes sense graphically. Plus, physically. Say you have a function for velocity and you want to know displacement by integrating over time. Well, if you know you were always going forward or standing still, you know displacement can’t be negative. Unless maybe you bump into a wormhole or something.

7) If a function is never positive on a certain interval, then its integral is non-positive.

This is just the negative analog of (6), obviously. You could also state that if a function is always 0 on a certain interval, its integral is always zero. But, that might be too obvious…

8) If a function is always between two constants (call them m and M) on a certain interval) then the integral of that function on that interval is less than or equal to the integral of M alone and more than or equal to the integral of m alone.

This may be confusing both in the math language and in mine. Here’s a more human friendly version. If you draw two horizontal bars and a function is always between them, then its integral is at least as big as the lower bar’s integral and no bigger than the upper bar’s integral.

So, for example, if you know a car can’t go faster than 10 m/s and can’t go slower than 1 m/s, then you know that over 10 seconds its max distance is 100 meters and its min distance is 10 meters. Thus, no matter what it does in that interval, the integral for displacement is between 10 and 100, inclusive.

Next Stop: Interesting Exercises in 5.2

Posted in Autodidaction, calculus | 1 Comment

Physics! #48: University Physics Chapter 7 Exercises

Chapter 7 Exercises

I just finished working all the odds here. That is, I did a homework assignment from hell. That said, it’s always an excellent refresher. I’m (generally) selecting one problem from each subsection that was clever or difficult.

7.5

“A baseball is thrown from the roof of a 22.0 m tall building with an initial velocity of magnitude 12.0 m/s and directed at an angle of 53.1 degrees above the horizon. (a) What is the speed of the ball just before it strikes the ground? Use energy methods and ignore air resistance. (b) What is the answer for part (a) if the initial velocity is at an angle of 53.1 degrees  below the horizontal. (c) If the effects of air resistance are included, will part (a) or (b) give a higher speed?”

This one’s not too hard, but gives you a (perhaps) counterintuitive result.

For (a), you can calculate max height using the kinematic equations. Using that (Added to the starting height) will give you max potential energy when there is no kinetic energy. Then, you just have to convert that value to kinetic energy and determine the velocity.

For (b) You start with a certain kinetic energy, then just pile the U on top.

The cute part is that both problems produce the same result! How does this make sense? Well, remember that energy is a scalar quantity, not a vector. So, energy is a description of something about the baseball in this system. In both cases, it started with a certain amount of potential energy and then got an energy bump. Then, in both cases, it converted all that energy into motion. Thus, there was not point at which it was able to lose or gain total energy. Thus, in both cases, it has the same final energy.

That leads us to (c). It should be clear by now that throwing down will produce the highest speed. You might think it’s because you’re throwing “into” gravity. However, in a vacuum, that doesn’t matter. Throwing the ball upwards in a vacuum will get to the same velocity eventually. The reason going up results in a lower final velocity if you’re talking about air drag is that the baseball is constantly losing energy to the air. The ball that goes up simply travels through more air en route to its final destination.

7.13

“A 10.0 kg microwave oven is pushed 8.00 meters up the sloping surface of the sloping surface of a loading ramp inclined at an angle of 36.9 degrees above the horizontal, by a constant force with a magnitude 110 N and acting parallel to the ramp. The coefficient of kinetic friction between the oven and the ramp is 0.250. (a) What is the work done on the oven by the force. F? (b) What is the work done on the oven by the friction force? (c) Compute the increase in potential energy for the oven. (d) Use your answers to parts (a), (b), and (c) to calculate the increase in the oven’s kinetic energy. (e) Use F=mto calculate the acceleration of the oven. Assuming that the oven is initially at rest, use the acceleration to calculate the oven’s speed after traveling 8.00m. From this, compute the increase in the oven’s kinetic energy and compare it to the answer you got in part (d).”

Woof. As far as I can tell, in basic physics it’s hard because all the questions are 200 words long, and then in advanced physics it’s hard because all the questions are 10 letters long.

(a) is pretty straightforward. You know there is a constant force up the hill and it’s 110 N. W=Fs, and you know s=8. And, you know the force is in the same direction as the displacement. Thus W=880 joules.

(b) Now, reach back a few chapters and recall how to calculate the normal force, which gives you the force of friction. Then, get its component along the incline. This force acts directly opposite the force in (a), which I called “push force.” Since the incline and mass are constant, the friction force is constant. So, once again you just do FS. In this case, I got about 157 joules.

(c) This is the simplest part yet. Remember, gravitational potential energy is all about height. So, you just run mgh where h=8sin(36.9). For mgh, I got about 470 joules.

(d) Now, you need to calculate increase in K. You know K increases because the work from pushing is greater than the opposing forces of gravity and friction. In this example, the only possible way for the oven to express the leftover is in motion. So, it’s just the push work energy minus the energy lost to friction minus the energy lost to potential energy change.

(e) Here, you sum the forces (push, gravity, and friction), which is a slightly annoying amount of arithmetic. Then you say F=ma and solve for a. From there, you can use equations of motion to calculate final velocity. You can use that to calculate final K, which should agree with your answer in d.

Neat, huh?

7.23

“A 2.50 kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 J of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur.”

First, whenever you see cm, convert it to meters immediately. SI is meters, and if you don’t use meters all your units will get screwy.

Now, since you’re dealing with the system, when you find out that 11.5 J are stored, you know that is all the energy on that mass. So, to calculate velocity, just assume it all gets converted to kinetic energy. From there, max speed is easy to compute, and of course it happens after the spring has released the mass.

For (b), we want to know greatest acceleration. Well, we know that force is always ma, and in the case of a spring, force is also given by kx. So, we say ma=kx. m and k are given and constant, so let’s just say k/m=c. That leaves a=cx.

Well, that’s convenient. More compression means more acceleration on release, and the relation is simply linear. We could go more in depth, but all you care about is this: more compression means more acceleration. SO, we’re looking for the maximum compression.

Therefore, all we have to do is solve for compression using the spring potential equation, knowing K and knowing that the total energy gotten is 11.5. Then solve for x and substitute it back into our acceleration equation. BAM!

Sections 7.3 and 7.4

I decided these were all too easy bother with here. For 7.3, you’re just multiplying friction through distance to show over and over that friction is non-conservative. If you don’t have this intuitively, it’s worth your time to solve the problems.

7.4 and 7.5 might be a little trickier since they involve calc, but if you can do a derivative, these are very simple and rather tedious. If you can’t do a derivative, check out the calc blogs.

Next Stop: Problems in Chapter 7

Posted in Autodidaction, physics | 1 Comment

Discrete! #32: Interesting Problems in 2.3 B

2.3: Interesting Problems in 2.3 Part B

Oh man, this one tried to kill me. I’m not sure if I’ve noted it explicitly, but I never actually got past chapter 3 in the book way back when I was self-teaching. So, we’re getting close to the edge of my knowledge here. Wish me luck.

39

For this problem set, we’re introduced to the horrific notation for inverse images. Horrific because it doesn’t quite mean “inverse function.” The book says “We define the inverse image of S to be the subset of A whose elements are precisely all pre-images of all elements in S. We denote the inverse image of S by f^{-1}(S).”

So, basically, suppose you have two sets, A and B. There’s a chunk of B called S (S for subset). Then, there’s a subset of A called the “inverse image” denoted f^{-1}(S). Or, in a pretty way, we say that f^{-1}(S)={ a∈A| f^{-1}(a)∈S }.

Simply put, we’re mapping between subsets in A and B.

NOW THEN, to the actual problem:

“Let g(x) = ⌊x⌋. Find the inverse image of

a) {0}

b) {-1, 0, 1}

c) {x| 0<x<1}”

For (a), we’re saying “You took the floor of x and got 0. What are the possible pre-images of g(x)? Well, if it floors to 0, it must be bigger than 0 and less than 1. So, that’s your set.

For (b), It floored to a couple numbers. So, the possibilities are a bit larger. It must be greater than -1 and less than 2.

For (c), you’re only talking about non-integers, and you can’t floor to a non-integer. So there are no images. The set of inverse images is just the empty set: ∅

41

“Let f be a function from A to B. Let S be a subset of B. Show that the inverse image of the complement of S equals the complement of the inverse image of S.”

You can actually prove this to yourself really easily by drawing it. Draw a circle on the left called A, with a subset that is the inverse image of S. Draw a circle on the right with a subset that is S. If you’re talking about the inverse function of B – S, you have to be mapping only to things in A-(inverse image of S), since by definition, the images in S are the images that map to the inverse image of S.

Once that makes sense, the proof is pretty straightforward.

51

“The function INT is found on some calculators, where INT(x)=⌊x⌋ when x is a nonnegative real number and INT(x)=⌈x⌉ when x is a negative real number. Show that this INT function satisfies the identity INT(-x)=-INT(x).”

So, you can basically think of INT(x) as “locate x on a number line, go toward 0, and stop at the first integer you pass.”

That idea helped me believe this was true. Either way you do it, you’re moving toward 0. So, you’re either starting in negative territory then moving to zer0 or starting in positive territory, then moving to zero, then flipping.

For the proof, with these ceiling and floor deals, it seems to be best to start by splitting your variable into its integer and decimal part. You can denote these as n and ϵ (epsilon).

So, now we can rephrase to: INT(-(n+ϵ))=-INT(n+ϵ)

For the left hand side, we can simplify to

INT(-n-ϵ)

This simplifies down to

⌈(-n-ϵ)⌉

Which is simply

-n.

For the right hand side, we have

-INT(n+ϵ)

Which is the same as

 

-⌊n+ϵ⌋

Which is the same as

-n.

BAM. You may at this point argue “HEY! You assumed x was positive.” This is true. But, if you follow the above logic, if x is negative you simply switch the floor and ceiling and switch the sign. And, it still all works out

 

There are a number of other good problems here, but in the interest of saving space and time I’m not going to work them all. A bunch of the 60s are just graphmaking busywork, but 67 was very interesting, as was 77. I recommend mentally chewing on them a bit when you work the problems yourself.

Next stop: Sequences and Summations

 

Posted in Autodidaction, Discrete Math | 3 Comments