Limits (Part 2)

SEPTEMBER 10TH, 2017

S: Last time didn’t go so well.

J: It wasn’t perfect.

S: Let’s face it: Nobody understood what Georgie got from his York Notes.  So, today, I’m going to continue the subject of limits by teaching it mostly myself – but I’ll still need your help.

…

…

S: It’s weird not having you say something.

J: You’re teaching it to me, right?

S: Yes, but that’s only because you didn’t do a good job of explaining it, and I need your help teaching it to you.  Your job is to come up with a story problem.

J: For limits?

S: Yes.  But I have since done some research, and “limits” is a broad subject.  Today, I guess we’ll be doing…things…on a graph.

J: So, if John was simply to play middle C, then we can say, “He’s playing middle C,” right?

S: Yes.

J: But what if John’s starting from Cb, using a pitch-bender control on his keyboard, and he’s moving it ever so slowly, ever so much more slowly-

S: I’m getting an image of this in my mind… He’s doing his evil grin as the pitch-bender moves slower and slower…

J: Ever so much more slowly…

S: Alright, get to the point!

J: Ever so much more slowly toward middle C…but without ever actually getting to middle C.

S: But we already understand infinite series.

J: Are you jumping ahead?  Because I’m not done talking.

S: Fine.  Continue.

J: So, we can’t just say he is playing middle C, can we?

S: No.

J: We could say that the difference between what he’s playing and middle C is the variable epsilon (ε)-

S: “Difference” as in subtraction?

J: Yes.

S: This isn’t Beatle-y or funny enough.

J: BAM!  Dad’s pasta dish falls to the patio with a smash and Dad freaks out as Mimi the dog leaps upon the mess to lick the pasta sauce from the scattered shards.

S: That’s not about the Beatles.  But I know something else about the Beatles – actually, I know an infinite number of things about them, but that’s not the point.  The point is that I wrote “I ♥ THE BAND OF THE FLAMING PIE” on my foot.  Look!

J: I don’t get the reference.

S: – –

J: Is that from one of your blogs?

S: No!  DAAAAAAAD, when John was 12, he had a vision where a flaming pie came to him and said, “You are Beatles with an ‘a’.”

J: He made that up!

S: Did you actually know that or did you just guess?

J: I guessed that he made it up being flipped to the inquisitive press.

S: Actually, it’s part of an article he wrote for Mersey Beat sometime around 1960 or ’61.  He did retell the story during a Capitol Records promotional interview for the release of Meet the Beatles!, the first of the American albums, so I guess that counts as “being flipped to the inquisitive press.”  So, try to tie my foot with the flaming pie into the story problem.

J: We shall see.  I’m not inspired at the moment.

S: So, the difference between what John is playing-

J: -and middle C… At any given moment, we can say that the difference between what he’s currently playing and middle C is ε.  So, as ε gets smaller and smaller as John gets closer and closer to middle C, but never reaching middle C, we might say that the limit of what he is playing approaches middle C.

S: And the reason that he never reaches middle C is because…?

J: He doesn’t care to.

S: Paul sticks his hand under John’s hand at the last moment and plays treble D, so it would slowly go up to Eb.

J: That is confusing it!  If you really want this to have educational value, we can’t jump all over the place.

S: Then put the Beatles in more!  John, when he’s just by himself and not talking or doing anything but slowly moving his hand, is not interesting enough for our purposes!

J: He sticks his tongue out at you.  The fact that he’s performing for you should be enough.  You sound pretty ungrateful for a Beatlemaniac!

S: I’M enjoying it more than anything else in the world!  I’m talking about the readers!

J: The paying readers?

S: What do you mean?

J: Where can people go to donate money to you to support the Beatles math blog?

S: Nowhere Land!  I don’t need support!…except for my music.

J: Well, I bought an amp, and it cost $25, so it costs money to feed your Beatlemania!

S: The amp has nothing to do with the Beatles.

J: Guitar?

S: Guitar and the Beatles aren’t the same thing!

J: So you’ll never play a Beatles song on guitar?

S: I play Beatles songs on piano and you know it!  And I have an acoustic guitar!  And we have never been this off-topic!

J: Sure we have!

S: No, we haven’t!

J: How about that time when we segued to a flaming pie tattoo on your foot?

S: THAT had something to do with the Beatles, and I was going to tie it into the story problem until YOU interrupted.

J: That was billions of nanoseconds ago!  Why bring up the past?

S: I’m a Beatlemaniac; I LIVE in the past!  And anyway, YOU were the one who brought up the past in regards to this lesson!  Okay, I’m going to officially take over teaching this.

J: Thank goodness!

S: But you still have to say things.

J: Yes.  So, the limit of the note as ε approaches 0-

S: What does zero have to do with anything?

J: It’s the difference between the note he’s playing and middle C.

S: Makes no sense.

J: As John moves the pitch-bender up, up, up, closer and closer and closer, we would say that the limit as he does so approaches middle C.

S: Yes, we know that, and on a graph, middle C, in this case, would be zero.

J/S: *begin arguing about why Starli’s statement is or isn’t true and put off finishing the lesson for a month and a half*

OCTOBER 22ND, 2017

S: Hello again!  We didn’t finish this lesson on September 10th, and, since then, the flaming pie “tattoo” has washed off of my foot, and we have come to a better understanding of how to teach limits.  So, John plays the key of middle C-

J: -but he holds down the pitch-bender one semitone, and uh, plays and holds down middle C.

S: And here’s the part we left out, the absence of which previously confused me.

J: And every second, John eases up on the pitch-bender stick half of the remaining amount to get back to neutral.

S: By “neutral”, do you mean middle C?

J: Yes.

S: During the first second, he moves the pitch-bender halfway to C; during the second second, he moves it half of halfway to C (or a quarter); during the third second, he moves it half of a half of a half (or an eighth).

J: But he never gets to C.

S: As you have said 618,000,000 times.

S: If we say one more thing in relation to the math component of this lesson, we will have progressed beyond what we did on September 10th.

…

…

S: So, what are you going to say?

J: So, middle C-

S: Make it Beatle-y!

J: So, middle C is the limit of what John is playing…

S: I know I asked before, but please put the other Beatles in, or at least one of them.

J: …and George watches the distance left to go on the pitch-bender as it gets closer and closer to zero but never gets to neutral.

S: So, Geo is watching epsilon (ε)?

J: Yes…as John approaches the limit and never gets there even though he plays on and on and on.

S: But this is the problem I had when Paulie was playing the piano in Limits Part 1 (before I knew that it only took four minutes) because whatever is left of the Beatles’ timeline will never happen!

J: I suppose we have forked off a parallel universe for this to happen in, probably so that there can be enough Pauls to go around for everyone.

S: I don’t understand.

J: Oh… I suppose that’s mixing things up a little too much.

S: And I have more to say.  First of all, do you have any mercy for the fans in that parallel universe?  With John Lennon stuck at a keyboard for all eternity, the Beatles cannot go on!

J: Maybe everyone in the world tunes in to listen – that’s all they ever listen to.

S: Alright, but wait – what would Paul and Ringo do being prematurely ripped away from their two bandmates?

J: Ripped away?  They’re watching it, too.  Paul gave John the idea!

S: Well, forgive me for saying so, but he should have thought it out better!

J: Hey, it’s the creative process!

S: Okay, fine; I can’t argue with that.  We’re almost ready to get back to the math, but first, explain to me how there would be enough Pauls to go around.

J: Well, if the universe splits into two copies every day, and those split into two every day, then it doesn’t take long before there’s one John, Paul, George, and Ringo for every Beatles fangirl in at least one of the universes.

S: Well, I knew that, but I thought you meant for all the fangirls in all the universes.

J: Well, I can’t speak to that seriously because I don’t subscribe to the parallel universes theory.

S: Well, I do.  Ah, to be in the chosen universe!  The only problem is that people can’t travel from one universe to another (according to my knowledge from Doctor Who).

J: That sounds saner.

S: Yes, I suppose.  BUT IT WOULD BE BETTER THE OTHER WAY!!!  *LOTS OF TEARS FOR LOTS OF IMPOSSIBILITIES*

…

…

S: Back to the math: We still have the problem that the remainder of the Beatles’ music couldn’t be written or recorded if John plays the keyboard forever.  It couldn’t be after Abbey Road (thereby being after the last Beatles music had been recorded) because they were never together in the same room again after staying up until 1:30 on the final recording day to decide the track order, except for that last Apple business meeting in September ’69.

J: But maybe they could do an extra track for Abbey Road so that it’s never over.

S: Like the Sgt. Pepper loop on the original vinyl…  The only problem with that is that if John can’t leave the keyboard, it can’t be released because the recording session won’t end until he dies…and Abbey Road will not be released in 1969, that’s for sure.

J: Maybe George just cuts it right there.

S: And now we don’t have a limits problem.

J: NOOOOOOOOO!!!  That universe explodes – the one in which he cuts it right there, and we continue in the one in which he doesn’t.  And there’s no mercy for those waiting for Abbey Road.

S: It’s not necessarily 1969; it could be when John and George first met in 1958!

J: Okay, well let’s have it happen then in another parallel universe!

S: I still feel as if I’m betraying my fellow fangirls, but it’s okay because we have to get on with the math.

J: The next step is graphing this so that everyone can see what’s going on.

S: So, ε is-

J: -how far the pitch-bender is pulled down away from the neutral position, and this value gets smaller as John moves it closer to the neutral position over time.

S: We will label ε on our graph and put the graph at the end of this post.  So, readers: I challenge you to confirm for yourselves that you understand by paying attention when we say what the variables are and then looking at the graph.

J: Or John can play it again for you.

S: Let’s move on to what the other variables on the graph represent.  L is middle C, right?

J: Yes.

S: And middle C is the limit and it is also an asymptote, right?

J: Right.

S: What is f(x)?

J: f(x) is the frequency of the note that John is playing over time.

S: Is f frequency?

J: Yes, but f generically means “function,” but it just so happens that function and frequency start with the same letter.

S: So f(x) is the frequency of the note over time, which is shown as the function (drawn in blue on our graph).  Notice the x and y axes on our graph.  x is defined to be time, and y is defined to be frequency.  Our last variable is N.

J: N is the time at which John happens to be at any given moment.

S: How does this all come together?

J: So, how we’re defining “limit” in this case is that for any pitch-bender distance (ε) greater than zero, there’s a time (N) for which all time (x) after that the following is true.

S: And “the following” of which you speak is the equation, which is L – ε < f(x) < L + ε.  Can you tell me what that means?

J: The frequency of the note middle C is L, and we’re subtracting the amount it gets pulled down in frequency from middle C.

S: Except that the frequency is going up.

J: As John goes higher, ε gets smaller as it gets closer to middle C, or L.  For any pitch-bender position that is not neutral (any pitch-bender position up or down), there’s a time (N)  past which all other times (x) the note being played f(x) is between middle C minus ε and middle C plus ε.

S: So, L – ε  is Cb or something between Cb and C.

J: Right, because ε is getting close to zero, which is small, and L is big – it’s the frequency of middle C, which is 261.6 hertz, and ε, which started at the difference between C and Cb-

S: What do you mean by “the difference between C and Cb”?

J: Well, let me show you.  *opens Google voice search*  What is the frequency of middle B?  No, Google, not “metal B”!  So, ε starts out at 14.69, the difference between middle C and middle B.  So, when John pulls down the pitch-bender and plays the key for middle C, the note that comes out is the difference, the frequency of middle C (which is 261.63 hertz) minus ε, which comes out to be 261.63 hertz – 14.69 hertz = 246.94 hertz, which, at time (x) zero, is parenthetically equal to middle B, or B3, I should say.

S: (For people who don’t know much about music theory, it helps to know that Cb and B are the same note.)  Anyway, all of this can be summarized in standard limit form: 

J: The limit of the frequency at a certain time as time approaches infinity is equal to middle C.

S: Here is the graph, which I made myself using Windows Paint.

S: And that was Limits Part 2!  I certainly understand limits better now.

J: *barely audible singing* Do-do-do!  Yes you do!

S: *rolling eyes* What a nice song… I suppose you’re too tired for a Beatle-y ending.

J:

J:

J:

J: So, after you say that you understand it better, have me saying, “So you’ve grabbed the brass Ringo!”

S: ???????

J: That’s a reference to a brass ring that kids would grab on a carousel – so you’re riding around the carousel on your little horsey and you reach out and try to grab the brass ring, and if you did, you won a prize!

S:

S:

S:

S: You quite honestly sound like Ringo.

Limits (Part 1)

JULY 16TH, 2017

S: Hi!  Sorry we didn’t have time for math last week.  July 7th was Ringo’s birthday, and I forgot to post something then, too.  So, happy belated birthday, Ringsy!

J: Speaking of Ringo, we could use some help playing an infinite series of notes.

S: Oh!  I remember infinite series!

J: Paul says-

S: Where are the Beatles?  We need a scene.

J: You make it up.

S: I can’t until I know what’s happening.

J: They’re at an undiscovered location, hiding out from fans.

S: And yet, we can hear them.  And possibly see them.  And I’m a fan.  So they’re doing a really ­bad job of hiding.

J: They took video of themselves, and we found it thanks to Google.

S: So it’s not live?  *weeping inside*

J: So, we should rewind.  No, we shouldn’t.  Forget what I just said.

S: No!  Rewinding is cute! 😊

J: *facepalm*

S: 😉

J: What did Paul say?

S: I don’t know – he never started talking!

J: He said, “We have a half note and a quarter note and an eighth note and a sixteenth note and a thirty-second note and a sixty-fourth note and a hundred-twenty-eighth note and a two-hundred-fifty-sixth note – and we’re still in the same measure.”

S: Oh!  NOW I understand infinite series!!!

J: “That looks like an infinite sum,” says George.  Ringo starts to play it.

S: Why is it ALWAYS Ringo who plays the notes in math?  Except for that time we had Paul play my keyboard… no, he didn’t play it; he was just thinking about playing it.  So, it can’t be Ringo again (though he does a fabulous job) because he was the star last time.  I already have two posts with featured images of him playing the drums – I don’t need a third!

J: So, you want to replace Ringo?

S: Yes, but only because it’s always him, and even though I have a favorite order, I still strive for Beatle equality.

J: We’ll replace him with Paul.

S: Playing what instrument?

J: Keyboard.  Wait, does John play keyboard?

S: Well, he can play it with his foot like he did at Shea Stadium… or was that his elbow?  I can’t remember.

J: Let’s have Paul playing the notes on keyboard.

S: Oh, and by the way… DOES JOHN PLAY KEYBOARD?  “IMAGINE” IF HE COULD! *COUGH COUGH COUGH*  Stupid questions are stupid.  Why is it keyboard as opposed to piano?

J: It’s piano.

S: So, what are the notes that Paul is playing?

J: Middle C for the half note, high C for the quarter note, middle C for the eighth note, high C for the sixteenth note and so on.

S: But it’s much easier to control two notes when they get really, really fast when you use one hand instead of two.

J: George Martin says, “Because with one hand, you’re using one side of your brain, but with two hands, the two hemispheres of the brain have to coordinate.”

S: So can we have those two notes be middle C and treble C so that Paul can reach them both with one hand?  Because past the hundred-twenty-eighth note, anyone’s hands would start to… malfunction – even a professional pianists’.

J: Sure!

S: Actually, all this depends on the tempo.  If we set the BPM at 1, then we can play the maximum amount of notes the easiest.  A half note lasts two minutes… Paul’s going to be bored out of his mind.  Well, maybe he has more patience than I do.  If I had to hold down middle C for two minutes straight, I might just die.  Anyway, back to the story problem: George says that it looks like an infinite sum; Paul starts to play it…

J: John says, “What’s an infinite sum?”

George says, “Well, if you have an infinite sum like 1/2 + 1/4 +1/8 + 1/16+···, it’s defined to mean the limit of the sum of the first n terms as n approaches infinity.

s_n = 1/2 + 1/4 + 1/8 + 1/16 +···+ 1 ÷ 2^n-1 + 1 ÷ 2ⁿ

s is the sum of the length of the first n number of notes.  So, with an algebraic interlude, multiply both sides by 2 to get 2s_n = 2/2 + 2/4 + 2/8 + 2/16 +···+ 2/2 n = 1 + [1/2 + 1/4 + 1/8 +···] = 1 + [s_n – 1/2 n].  And then subtract s_n from both sides to get s_n = 1 – 1 ÷ 2ⁿ.  You can see then that as the number of notes in the measure-”

S: -gets bigger and bigger going faster and faster… Wait, how can Paul know exactly when to play the sixty-fourth note exactly 3.75 seconds after the thirty-second note that was played exactly 7.5 seconds after the sixteenth note?

J: He just can… because he’s Macca.

S: *really un-called-for fangasm*  So, does he have a clock or a metronome ticking so that he knows approximately when to play the notes?

J: Maybe he just has a natural sense of it.

S: *another really un-called-for fangasm*  Why did that make me so ridiculously happy?  Anyway, let’s get back-

J: -to where you once belonged!

S: You’re turning into me!  ANYWAY, let’s get back to what George was saying.  I never knew that George knew so much about calculus!

J: He’s got the York Notes right there in his pocket.

“As I was saying, as the number of notes played in the measure (n) gets larger, 1 ÷ 2ⁿ gets smaller, so the sum of the lengths of the notes (s_n) gets closer to 1.”

S: And 1 represents the end of the measure, right?

J: Ringo says, “Well, the measure takes one whole note, so it must be 1, even though Paulie plays an infinite number of notes.”

S: So that means that he’ll just sit there forever, playing notes, and whatever is left of the Beatles’ timeline will never unfold???!!! 😱

J: Well, he’ll be done in… how many seconds?

S: It’s 1 BPM, so four minutes.  Thank goodness.  Still though, it’s just not possible for the human hand to play what he would have to be playing near the end of that time.

J: That’s another consideration of the turtle in one of Zeno’s paradoxes.

S: The TURTLE???

J: That’s for another time.  Now, we could say that the limit of the sum of the length of these notes is 1.

S: So, if we graphed that big equation that George was talking about originally, would it produce something that resembled a limit?

J: Yes.

S: Then let’s graph it.  And then we’ll do a practice problem. *starts graphing*

n	sn	FRACTION
1	0.5	1/2
2	0.75	3/4
3	0.875	7/8
4	0.9375	15/16
5	0.96875	31/32
6	0.984375	63/64
7	0.9921875	127/128
8	0.99609375	255/256
9	0.998046875	511/512
10	0.9990234375	1,023/1,024

So, we have graphed s_n = 1 – 1 ÷ 2ⁿ.  Now for a practice problem.  This time, it’s George playing notes on his guitar.

J:   This can be read as “the limit of x to the power of 1 over x as x approaches infinity”.  Now guess the limit based on a table of numbers substituted for x.

S:

x	x1/2	DECIMAL
1	1	1
2	√2	1.41
3	3√3	1.44
4	4√4	1.41
5	5√5	1.37
6	6√6	1.34
7	7√7	1.32

WOAH.  The quartic root of 4 is the same as the square root of 2!!!

J: Pretty cool, huh?  So, how about we jump ahead.

100^1/100 = 1.047

101^1/101 = 1.046

1,000^1/1,000 = 1.006

987,654,321^{1/987,654,321} = 1.00000002

S: You know I can’t resist 987,654,321…

J: Yes.  That’s why I used it.  Do you have any guess as to the limit?

S: How should I know out of all those numbers!?

J: What do the numbers seem to be approaching?

S: 1.

J: Correct.

S: And that was Limits Part 1.  But we need a Beatle-y ending because John was almost completely left out, and we didn’t even talk about George playing the notes.  So, what did it sound like?  Oh!  I just realized that the numbers have too many digits.  You can’t divide 1 beat by undecillion.

J: Sure you could!

S: In theory, you can, but it’s not what musicians do.

J: Thank goodness.

S: We need an ending!

J: No, no, no!  What’s George doing?  We didn’t say anything about what he was doing.

S: He’s sitting in the secret location, playing his guitar.  So, the ending…?

J: We did math.

Polar Functions

JULY 3RD, 2017

J/S:

J: Ringo says, “What’s this?  Dartboard paper?”

S: I don’t know, Ringsy… I’ve never seen anything like this before, though it’s probably some type of graph, since we’re doing polar functions.

J: That’s right.  And a polar function is a function expressed in terms of a radius r and an angle Θ (theta), as opposed to x and y.

S: So, ONLY r and Θ?  No numbers?

J: Oh!  You can have anything you want, but the point is that x and y are rectangular coordinates-

S: And this is a circular graph!

J: -whereas r and Θ are polar coordinates.

S: Polar, in this case, means that the graph is circular.

J: Ringo says, “Hmm… So, if I lay this graph on a snare, then I can see, in terms of r (the radius) and Θ (the angle), where I’m hitting the drum and note how the sound varies.”

S: But then, if anyone discovers the equations, they would know how to play in the elusive, top-secret Beatles percussion style.

J: I suppose you could write it in secret code, or, as da Vinci did, by writing things backwards in different languages.

S: I don’t think Ringo is bilingual.

J: Well, they all knew Liverpudlian, right?

S: *rolls eyes* WELLLLLLL, I guess Liverpudlian does count because of all the words they use that we don’t use in America (or in the rest of Britain, for that matter)… or they could use Paul and John and Ivan’s secret language!  All I know of the secret language is “Cranlock naval, Cranlock pie”.  No one knows what it means, but it’s from the poem that Paul wrote for Ivan when he died… in 1993… of Parkinson’s disease… I know way too much…

J: Hm!  You know a lot.

S: So, Ringo, we will attach this polar graph to your snare drum and see the angle and radius at which you hit it.

J: John is writing down something.  Hey!  It’s a polar function!  Looks like r = Θ/60°.

S: Ahhhhh… That’s a nice, simple function.  But where ARE the Beatles, just so that I can picture this more clearly?

J: They could be in a secret lab at Apple.

S: So, it’s 1968?  ‘Cause it must be.

J: Why must it be?

S: In ’67, they didn’t spend much time at Apple and it hadn’t really got going, and by ’69, it was falling apart…

J: “Hey, Ringo!  Play THIS!” says John, handing him the paper with the equation on it.

S: Have we already put the graph on the drum?

J: You can see it there, on the drum!

S: Oh, now I see.  Ringo thinks that it probably has something to do with the 60° mark on the circle.

J: Θ represents all the points going around the circle – it’s an angle.  So, if the point that Ringo hits has radius 0.25 and an angle of 15°, how would we graph that?

S: We would graph it by first finding 15° on the Θ line and then starting from the origin (the center of the circle) and going up the 15° line (which is a radial, not to be confused with a radian) 0.25 units.

J: And we get there because of John’s equation: 15 ÷ 60 = 0.25.

S: Oh!  So, should I graph it?

J: Absolutely!

S: *graphs it in red*

J: And now Ringo has hit a point on the 30° radial!

S: What IS the point?  Oh!  I know!  Paul and George are the point!  PUT THEM IN HINT HINT

J: “We’re trying to find out where the sweet sounds are on a snare,” George says.

“And whether it’s worth it to compose for snare in terms of r and Θ,” says Paul.

S: So, they’re all in their secret lab in the basement of Apple with a sterile table and-

J: Why does it have to be sterile?

S: It’s a lab!

J: It could be a loose R&D lab!

S: Whatever.  I just meant that it’s cleaned regularly.  Anyway, as I was saying, there are four plastic chairs in their favorite colors, and Ringo’s drum kit (the biggest one, of the three he had), and a tiny window near the ceiling that no one can see through so that their experiments are kept secret.  The sterile table is made out of marble.  They bought it back when they didn’t know that Apple was bankrupt and broke and in-debt and moneyless.

J: Goodness!

S: No, really – it was.  Also, John, Paul, and George are sitting at the table and Ringo is at his drum kit.  So, you said that Ringo hit a spot on the 30° radial?

J: Mm-hmm.  Can you keep up with him?

S: Are we still following John’s equation?

J: Yes.

S: 30 ÷ 60 = 0.5, so I’ll graph that… *graphs that in blue*

J: BAM!!!  He hit the 45° radial!  Looks like he’s hitting one radial every second, going around the paper from one radial to the next!

S: Snare drums don’t go “BAM”; they go “TSCHHHHHH”.  But, anyway, 45 ÷ 60 = 0.75, so… *graphs 0.75 on the 45° radial in purple*  Hey!  It’s going to make a circle, isn’t it?

J: It’s going to do something, but I don’t think it’s a circle.

S: Okay.  What point on the snare does Ringo’s drumstick hit next?

J: Well, Ringo’s hitting the next radial on the graph paper every second.

S: So (assuming we’re in 4/4), he’s playing quarter notes at 60BPM or half notes at 120BPM or whole notes at 240BPM or eighth notes at 30BPM or sixteenth notes at 15BPM or triplets at… what would that be?

J: Use a calculator.

S: *uses calculator* 20BPM.  So, Ringo’s next point would be on the 60° radial and at 1, since the pattern increases by 0.25 every time he hits the drum, and 60 ÷ 60 = 1.  *graphs that in green*  Now I’m just going to graph all the points for this equation that are on the radials.  *graphs up to radial 225°*  Oh, you’re right!  It’s not a circle… Maybe it’s a heart!  No, that would be a badly-shaped heart…

J: A cardioid.

S: *finishes graphing*  It’s a spiral!

J: Zzzzzzzhooooop!  Nice spiral!

S: Ritchie is an artist.  Actually, John is an artist because the spiral was his idea.  I really feel like Paul and Geo are being left out, even though they’re sitting there in the lab.

J: Paul says, “Okay, I like the sound of that spiral, but can we try a circle that is off to the side?”

S: Okay, Paulie!  But first, Dad, remember that the point of this is to see if it’s worth it to to them to do all the math, so keep that in mind as we continue… but it depends on their definition of “worth it”…

J: Well, if it’s a drag, they won’t do it again!

S: Yeah, because, since most bands don’t use precalculus to figure out drumbeats, it’s not a well-known drag.

J: *grimaces* Do you want to go to a drag show?

S: No!  It’s from A Hard Day’s Night!  You know – ‘she’s a well-known drag’ – Susan, the TV lady trendsetter!

J: Well, that’s what ‘a well-known drag’ means!

S:

J:

S:

J:

S: OH!  NOW I know why Geo was drawn in drag in that weird comic that Strabius on DeviantArt did!  Because of ‘a well-known drag’!!!

J:

S:

J:

S:

J: Paul says, “John, what’s the equation for a circle?”

S: I don’t understand why John is always the one with the equations!  He skipped class, never did homework, hated school, and thought math was stupid.

J: John (looking up from a math reference book) says to you, “I’m just a newborn genius!”

S: I still think that if a Beatle had to do the equations, it would be Paul!  He was a model student… until he met John.

J: Paul has trained John to look things up in his old CRC handbook.

“CRC?” George asks.

Ringo reads from the spine of John’s book, “Chemical Rubber Company.”

S: Is that really what it stands for?

J: Yes.  *gets copy of CRC Standard Mathematical Tables from 1961*

S: *looks in book*  WOW, this is complicated.

J: If you didn’t have a calculator, this is where you’d look.

S: But then how did the authors of the book figure out all this stuff?

J: By hand.

S: Ewww… Gruesome…

J: John speaks the function: “r = 4 cos Θ + 6 sin Θ.”  And now we graph it as Ringo starts to play it, still tapping one radial per second.

S: Starting at 0°… r = 4 cos 0 + 6 sin 0.  The cosine of 0 is 1, and the sine of 0 is 0.  4 · 1 = 4 and 6 · 0 = 0, so we’re left with r = 4.  So, I’m counting four units from the origin on the 0° radial… *graphs that* Now I’ll do the others until I get back to 0°.  *finishes graphing*

J: “Nice!” says Paul.  “That circle sounds nice.”

S: I’m so glad you think so! 🤣 But Georgie’s still left out…

J:

J:

J:

S: …because his mouth is full of gummy bears!

J: Really?  I was going to have him ask for a clover leaf or something with heart.

S: He tries to through the gummy bears.

J: George says, “How about something with heart: r = 1 + cos Θ.”  And I have marked 1 where 4 would normally be, since the equation fits nicely that way.  Now, this graph is in radians, so make sure that your calculator is set to radians instead of degrees when you do cosine.

S: *graphs equation*

It’s a sideways heart!  We should note also that the black lines that connect the colored points are just approximations of the function – if we were to plot EVERY point in the function, it would be more curved.

OMG, Dad, I sound like you!

J: George Martin pops in and says, “Say, is that a spec for a cardioid microphone?”

S: So, what happens then?

J: I guess it’s just left there, hanging…

S: …on the clothesline stretched across the lab for no apparent reason that Ringo crashes into every day.

J:

J:

J:

S: Okay, that was random.  Bye!

Rational Functions

JUNE 25TH, 2017

S: Just as last week was a noteworthy Beatles-related day, so is this week.  Today is World Beatles Day.  The reason that it is on June 25th is that it was on this day in 1960 that Stu Sutcliffe thought of the name “The Beetles” (as a sort of tribute to Buddy Holly and the Crickets).  John changed it to “The Beatals”, and then, finally, “The Beatles”, with “The Silver Beetles”, “The Silver Beats”, and “The Silver Beatles” all mixed in there somewhere.

Also, coincidentally, it was on June 25th in 1967 that the Beatles performed “All You Need Is Love” in the first ever satellite TV broadcast.

OMG – I sound like a history book!  Although, people have told me that I am a history book when it comes to the Beatles.

Anyway, Friday was Stu’s birthday.  He would have been 77.  And now, after those formalities: RATIONAL FUNCTIONS.

Great.  Now I sound like a sports announcer.

Anyway, what are rational functions?  BEATLES HINT HINT

J: “Well, what an insane party that was!” Paul exclaimed.

“Yeah,” George said.  “In the future, let’s attend more rational functions.”

S: HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA!!!  Did you just think of that?

J: Yeah.  And the Fool on the Hill thus intones, “A rational function is” *looks up definition on phone* “a fraction of two polynomials.”

S: So, Fool on the Hill, by “a fraction of two polynomials”, do you mean that the numerator and the denominator are both polynomials?

J: “Precisely!”

S: Aren’t you supposed to be a fool?

J: I’m not a fool; the Fool on the Hill said that!

S: Exactly!  He said that, not you!  So, how does he know about rational functions if he’s a fool?!  It is precalculus, after all!

J: He has Wikipedia.

S: Oh, well that TOTALLY clears things up.

J: Here’s an example of a rational function:

The equation for this function is y = [x³ – 2x]/[2(x² – 5)].

S: Okay, so it is not a continuous function.  That’s the main thing I can see.  So, do you want to explain what else makes it recognizable as a rational function?

J: No, because the formal definition is full of more than one hour’s worth of side topics.

S: You just don’t understand my question.  It’s one image.  Can we tell from one image whether or not it’s a rational function?

J: No, because we can’t see what’s off the graph.  So, looking at the equation, can you see where the denominator would be 0?

S: You mean, what values of x and y would make it 0?

J: No, what values of x would make the bottom of the equation 0?

S: I still don’t understand!

J: Try to make it so that you’re dividing by 0.  What values of x would make that equation undefined?  In other words, find the discontinuities.

S: Oh!  I know about discontinuities.  Hmm…  It’s not 0, because 0² = 0, and 0 – 5 is -5, and -5 · 2 is not 0.

J: What minus 5 is 0?

S: 5!  So, x² = 5.  Um… um… What’s the square root of 5?

J: √5.  That’s all you need write.

S: But I don’t like representations of numbers that aren’t really numbers!

J: But √5 is a real number!  It’s accurate!

S: But I can wrap my head around the concept much better when it’s in decimal form (which is about 2.2, for the square root of five)!

J: But it’s much better than when it’s in decimal form because it’s precise.  It equals the exact square root of 5.

S: Whatever.  Go on.  So, we found that ±√5 is a value of x that produces a discontinuity.

J: Yes.  Now, look at the graph above and find where x = ±√5.

S: The vertical dotted lines are discontinuities.  But what about the other dotted line?

J: Let’s look at the example equation.

y = [x³ – 2x]/[2(x² – 5)]

So, if x gets really, really big-

S: I need you to put the Beatles in right now.  It’s been too long.

J: “As x approaches infinity?” Ringo says.

S: It’s kind of a cheap way to include the Beatles, making them say things you would say.

J: Would you rather have the Fool on the Hill speak it?  Because I can do that accent.

S: The Fool on the Hill has never spoken, so you don’t know what he sounds like!

J: You don’t either, so you can’t say it’s wrong!

S: 😠

J: Would you rather a minor Beatle say it?

S: You mean Stu, Pete, Ivan, Pete Shotton, Nigel Walley, Eric Griffiths, Colin Hanton, John “Duff” Lowe, Rod Davis, Len Garry, Ken Brown, Chas Newby, Norman Chapman, Tommy Moore, Andy White, Jimmie Nicol, or Billy Preston?

J: Yes!  How did you know?

S: Because it couldn’t possibly be anyone else!  Those were ALL the people EVER in the Beatles!

J: As far as we know.  There may have been secret Beatles.

S: If you count Yoko, George Martin, and the session musicians as “secret Beatles”… So, we were deciding who would say it.

J: I don’t care who says it!  The faceless recording engineer says it over the headphones!

S: He’s not faceless!  His name is Geoff Emerick!

J: His face was melted off in a fire started by fangirls in the studio kitchen.

S: What would YOU know about his life?!

J: He is faceless, therefore, he wishes to remain anonymous, and I will respect that anonymity.  “Let’s simplify the equation,” he says.

y = [x³ – 2x]/[2(x² – 5)]

We distribute the 2, and we get:

y = [x³ – 2x]/[2x² – 10]

As x gets huge, the top is approaching x³, and the -2x is insignificant.  And the bottom approaches 2x².  And the -10 is insignificant, parenthetically – this is a gross simplification.  So, y approaches…?

S: Infinity?

J: No, x³/2x², which is equal to…?

S: *tries and fails to find the answer*

J: So, let’s expand the equation.

S: Since when are equations expanded when they need to be simplified?!

J: Since now.  So, write it in terms of x · x · x, and so on.

S: x · x · x / 2 · x · x

J: Now, x/x is equal to what?

S: 1, otherwise known as GET BACK TO THE BEATLES.

J: John says, “Can we get back to the EQUATION, please, and SIMPLIFY it?”

J/S: x · x · x / 2 · x · x

So, we can cancel x‘s, and now we have:

y ⇒ x/2   x ⇒ ∞

S: That reads as “y approaches x/2 as x approaches infinity”.

J: The dotted diagonal line you asked about a while ago is y = x/2.  It is an asymptote.  Okay, let’s graph x/2.

S:

J: It’s a dotted line because it’s an asymptote.

S: But I thought that we were graphing a rational function, but this is continuous!

J: We are just drawing the guidelines for the function right now – asymptotes are guidelines.

S: Oh!  So we’re basically just copying the first graph without looking at it.

J: Yes.  So, the next guideline is x = √5, regardless of the value of y.  So, it’s a vertical line.  That’s one of the two zeros of this function.

S: And the other is x = -√5.  The Beatles haven’t come up in forever!

J: Paul’s granddad appears through a portal in the floor.

S:

J:

S:

J:

S: Well, what happens?!

J: Nothing.

S:

J:

S:

J:

S: *finishes graphing x = √5 and x = -√5*

J: So, we often are interested in only one part of the curve.  For example, the curve in the domain of x > √5 (in Quadrant I).

S: Dad, if we were in a class on rational functions, what would the main point be?  What is the point of them?

J: If we have an equation, we’d like to understand it, and a good way to do that is A, graph it, and B, find out about discontinuities and asymptotes and such.  And for rational functions, we end up with an equation that looks like polynomials-

S: I thought you were about to say “an equation that looks like Paul McCartney”!

J: Oh!  HA HA HA

S: Wait, can we graph that???!!!  And use the graph to find out what the equation is and have the equation embroidered on my forehead with a needle and thread???!!!

J: Sure!  But first, you need to graph the neck of the guitar…

S: Alright.  Let me guess – the neck of the guitar is supposedly the guidelines that we’ve already drawn.  But whatever; let’s just get on with this so that we can be done.  So, now, the last step is to graph y = [x³ – 2x]/[2(x² – 5)], right?

J: Yes.  But let’s only do it in the domain of x > √5, as opposed to in all the Quadrants.

S: Okay.  Let’s do that…

Now we’re done.  Goodbye!

Oh, and, once again, happy World Beatles Day!!!

J: Not to be confused with Krimble.

 

Rational function graph from Wikipedia: Link

Conic Sections

JUNE 18TH, 2017

S: Today is, as all Beatlemaniacs know, Sir Paul McCartney’s 75th birthday.  So, some of this lesson must be about that.

J: It’s all about the birthday hats for his birthday celebration.

S: Is it a previous birthday so that the other three are involved?

J: Sure!  Pick a birthday.

S: Okay… Hmm… Maybe his 16th birthday?  John and Geo would probably be there… or his 21st; that was a nice one… or his 22nd; that was in my favorite year… You choose out of those three!

J: The 22nd.

S: Actually, I just got a really good idea for the 16th.

J: What is it?

S: Aunt Mimi.  She crashes the party because John did something or forgot to do something.

J: John forgot to do his conic sections math homework.

S: He never did ANY homework, though.  Aunt Mimi just about gave up.  But it’s supposed to be a story, so she comes in later.  Any ideas for how the story problem will go?

J: Hmm… Do they have any fans yet?

S: No!  This is 1958!  By late ’59, they probably did, but I don’t think they did in ’58… Wait – what kind of fans do you mean?  Rabid fans, or friends who just said, “Hey, I like your music!”?

J: Fans who want parts of their hats.

S: Oh!  I know!  Fans going back in time!  And if they’re really going back in time, it must be at least year 2300.  Okay, begin!

J: So, we’ve gone back in time-

S: Since when are YOU a Beatles fan?

J: The FANS have gone back in time to bring back samples of the Beatles’ birthday hats, but without disturbing the timeline.

S: Oooh.  That’ll be hard!

J: You’d think so.  But we have the ability to mathematically specify specific parts of these conical hats – hopefully by taking infinitely thin conic sections of the hats – and there will be no disturbance of the Beatles’ timeline.

S: Even though one of them is not there.

J: Here is the picture we’re basing this off of:

S: Okay, so, taking sections of cones that produce different shapes!

J: Suitable for bracelets or tiaras or other Beatlemaniac accessories.

S: But how would the fans wear them?  They are 2D shapes, not strings, and they are infinitely thin!

J: They’re easy enough to put on – but ONLY YOU will know that you wear part of a Beatles’ birthday hat.

S: The poor Ringo fans… Don’t you feel sorry for them?

J: Oh, well, they’ll just have to go back in time to a different birthday for themselves.

S: Okay, start the math component.  We’ve already seen the picture.

J: We’re taking a circle from Paulie’s hat.

S: Thanks for calling him Paulie!

J: So, we’re taking these sections by intersecting infinite planes with the cones, or party hats.

S: Are “we” using lasers?

J: No.  We’re using the time machine computer.  We give it the equation for the conic section, and it automagically extracts the conic section from the physical object in question, using an infinite plane to intersect it.

S: But doesn’t that mean that, since the section, no matter how thin, goes all the way through the hat, the top of the hat would fall off into Paul’s cake and disrupt time?

J: Well, we can rely on physics to hold it together.  So, first, the circle.  Do you see it?

S: Yeah.

J: It’s where the infinite plane is parallel to the bottom of the party hat (the so-called “cutting plane”).

S: Of laserness.

J: The equation for the circle is x² + y² = a².  We should also note that, parenthetically, conic sections have other useful, descriptive features, including eccentricity.

S: Here is the table with those descriptive features:

Could you please tell me what all those columns actually mean?  Like, what is the eccentricity of a given shape?

J: The eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.  So, if you look at the table, we’ve got an eccentricity e… So, you can see the eccentricities for different shapes with a fixed focus f and directrix-

S: Well, that doesn’t make sense because not all shapes are ellipses, so how could they all have eccentricities!?

J: So, if the eccentricity is between 0 and 1, there’ll be an ellipse.

S: Oh, I see!  Because the only eccentricity in the table between 0 and 1 is for the ellipse.  Okay, let’s move on to the next shape.  What’s the difference between a parabola and a hyperbola?

J: There are many different ways of looking at it, but the hyperbola intersects the double cone in two places, whereas the parabola intersects in one.

S: Oh!  That makes sense.  Here’s another image to show that:

J: That plane that creates the hyperbola doesn’t have to be vertical… But the plane for the parabola – that’s not going to hit the mirror image party hat.  The hyperbola is going to hit it in two places…

S: I can see that.  So, is there any sort of problem to solve?

J: Well, I thought we’d just describe the conic sections.

S: Okay, so, we already described the circle… next, the ellipse!  The ellipse as a conic section… is… at an angle – how do I explain it?!

J: It is an eccentricity between 0 and 1.

S: Yes, that is the crazy math term.  What is it in human terms?

J: Oh, well, it’s from a plane that’s NOT parallel to-

S: -the bottom of the cone, OR to Paul’s head!  Or to George’s or John’s, or the other people there.  But I don’t know who the other people there are.  Ooh!  I bet Ivan’s there!  I’m SURE Ivan’s there.  They have the same birth- wait.  They have the same birthday.  It’s probably not a joint party, but maybe it is!  Or maybe they just never go to each other’s birthday parties because they’re always on the same day… I don’t know.  So, Ivan MIGHT be there, or he might not, and if he is, then it’s also his birthday, or maybe they had birthday parties at different times on the same day, or maybe, in 1958, June 18th was a weekday, so they would have had birthday parties on separate weekends, or maybe it wasn’t…  What do you think?

J: *sweeps hand over head* Sheeeeeeew!

S: Okay, so, we were talking about ellipses, right?

J: Right.  Here’s a diagram of the parts of an ellipse:

S: So, the point (if we’re just talking about conic sections) is that ellipses are not parallel to the top of the head.  But no!  Human heads are not flat!  So this is ALL wrong!!!  DUN, DUN, DUUUUUUUUN!!!

J: The party hats may be just fine.

S: Well, they’re on heads that are not flat-

J: Well, perhaps they are very stiff, or else have been placed on the heads very gingerly, and so are not distorted.

S: But the point is that the heads aren’t flat, so the bottom of the cone isn’t flat, so you can’t measure this stuff!  Oh, whatever.

J: So, the equation for the ellipse?

S: *looks at table shown previously* x²/a² + y²/b² = 1

J: Right.

S: What ARE x, a, y, and b?

J: Well, we’ve got a and b, and this diagram- where is it? *takes a long time looking for the diagram*

S: *is waiting*

S: *gets bored*

S: Oh, and that other guy… whose name was Ian, I think?  Once upon a time, I got him and Ivan confused all the time, and by the way, they called Ivan “Ivy”.

J: *makes uninterested sound*

J: *still looking for diagram*

S: “Jive with Ive the Ace on Bass” was his catchphrase…

J: What now?

S: “Jive with Ive the Ace on Bass” was Ivan Vaughan’s catchphrase when he played tea-chest bass in the Quarrymen!

J: Hmm.

S: Okay, so, GET TO THE MATH…

J: *doesn’t get to the math*

S: What IS a tea-chest?  Is it a chest with tea in it?

J: I think so.

S: Jive with Ive the Ace on Bass, Jive with Ive the Ace on Bass, Jive-

J: So, you know what a radius is, right?

S: Of COURSE I know what a RADIUS is!!!

J: But an ellipse doesn’t have one radius, right?  Instead, we’ve got a and b.

S: Oh! Okay.

J: a is in the x direction; kind of the like the x-radius, if you will.  And you can think of b as the radius in the y direction.

S: So, y is left and x is right!  I mean…  That reminded me of that time in A Hard Day’s Night when Paul says, “I’m his grandfather!  I mean…”  Anyway, y is vertical and x is horizontal.

J: So, you could figure out what the effective radius would be, depending on the angle as we went around the ellipse from 0 to 360.

S: Yeah.

J: With this equation-

S: -the equation in the bottom left of the previous image-

J: r = b² / a – c cos θ

S: Yes.  r is the radius, by the way.

J: c is the distance from the center of the ellipse to the intersection with a-

S: Cosine?  Does it say cosine?

J: Yes – cosine theta.  Next, the parabola!

S: Yes, we know ALL about parabolas… ParaboliCAT *cough cough*

J: Well, we’re not cutting through, say, both sides of a cone in the way that we would get a circle or an ellipse; we’re just cutting through one side of it.  And the equation for the parabola is…?

S: y² = 4ax

J: And the eccentricity is 1.

S: So… that means… that it is an ellipse?  Or something?  Or does it mean that it’s a circle?

J: No.

S: Well, then-

J: It means that it’s a parabola.

S: Oh… But I thought you said- OH!  Because any eccentricity between 0 and 1 is a measure of how circle-like an ellipse is, right?

J: Right.

S: And a parabola is not a circle or an ellipse, so it doesn’t- I said “circur” instead of “circle”!

J: For the eccentricity of an ellipse to be 1, b²/a² would have to be 0.

S: Oh.  Do you know what we should never do?  Talk in Manchester accents.

J: Oh, I couldn’t pull anything off there.

S: *in a Manchester accent* MANCHESTA!!! *goes back to Liverpudlian/London/Cockney/Yorkshire/other British stuff accent that she was using before* No!  I’m so sorry, Beatles, forgive me!  Because I can’t remember if it was just John or if it was all of Liverpool who hated Manchester people and were their mortal enemies.  I just can’t remember if it was John or the entire city!!!

J: Oh no.

S: But either way, I can’t talk in a Manchester accent, because *switches to dramatic Italian-ish accent* then I would be betraying them!

J: And, of course, for the parabola, the semi-latus rectum and focal parameter are both 2a.  On to the hyperbola!

S: *switches back to British accent from Italian accent* Yes.

J: So, we have the hyperbola…

S: Which is an infinite plane that the computer shoots through two… party hats that don’t exist in that oppositely stacked position (please see third diagram from top of post).  Okay, here’s how the arrangement for the hyperbola does exist: George has an idea that he thinks won’t work because he’s tried it before, but he stacks two of the hats – John’s and his own – on the table, but he stacks them so that the tips are touching (again, please see third diagram from top if you haven’t already).  And everyone is so amazed that he actually succeeds and the hats don’t fall over!

J: And the bottoms of the hats are parallel.

S: Yes, they are.  But no one knows that it’s just the time travel computer sending messages to his brain that make his fingers set the hats up perfectly, and making sure the air current around the hats is favorable.

J: But this is without disturbing the Beatles’ timeline.

S: Yes, but it still works, somehow.

J: Very good.  Good computer.

S: No, wait a second – that would disrupt time because when George originally tried it in real life, they fell over, and now they’re staying up.

J: Perhaps for that moment when they were in perfect balance before they fell over.

S: Ah, yes.

J: That would be useful.

S: The computer is lightning fast!  It doesn’t need the hats to stay balanced for more than a decisecond!

J: So, we could have a section of an infinite plane that was going through both party hats, and that would give us a hyperbola.

S: Yeah.  Because both of the hyperbolas are the same, right?  Are they congruent?

J: They’re part of the same curve, but the two different parts don’t touch.

S: I know!  I was asking if those “two different parts” were the same shape!

J: It really depends.

S: On the angle of the curve, or does it have to be straight and parallel with the center line of the two hats to be the same on both sides?

J: It depends.

S: FOR THE DEFINITION OF A HYPERBOLA, DOES THE PLANE HAVE TO POSITIONED PERFECTLY PARALLEL TO THE CENTER LINE OF THE TWO HATS?

J: Well, the rectangular hyperbola – that’s a special form, there.  But just a general hyperbola, the top and bottom may look different.

S: Oh!  So the plane could be at any angle as long as it’s still intersecting both hats, but the rectangular hyperbola is different.

J: Yes.  So, the equation for the, uh…

S: I just winked at myself in my computer screen’s reflection like Paul does during “You’ve Got To Hide Your Love Away” in Help!

J: …hyperbola-

S: But, obviously, there weren’t computers then and he didn’t wink at himself; I just copied his winking style, that’s all.

J: -is-

S: He moves his jaw and opens his mouth sideways sometimes when he winks.  It looks weird when I try to do that.

J: -similar to-

S: But it somehow doesn’t look weird when he does it.  Maybe that’s why everyone loves his wink – because no one else in the world winks using their mouth.

J: -that of the ellipse.  So, we’ve got x²/a² – y²/b² = 1.

S: And the difference between that and the equation for the ellipse-

J: -is that where this uses subtraction, the ellipse equation uses addition.

S: *starts humming “Santa Claus Is Coming To Town”* Why am I humming “Santa Claus Is Coming To Town”?

J: And those are the conic sections that we take from party hats.

S: The fans are so happy!

J: “WAIT!” says Aunt Mimi, bursting into the room.  “I know you didn’t do your homework, Johnny, but there are the degenerate conic sections.”

S: Would Aunt Mimi really know about degenerate conic sections?  Maybe she would.

J: Well, some authors do not consider them conics at all!

S: Ugh!  How DARE they!!!

J: So, the degenerate conic sections are… a point-

S: A point!  One dimension.  1D.  Oh no!  1D!  One Direction!  HOW DARE YOU MENTION THEM, MATHEMATICIANS!!!

J: You can imagine a plane that was between those two party hats that George stacked on the table.

S: A plane that’s between… OH!  A plane that is between them, and is perpendicular to the bottoms of the hats; therefore, it doesn’t touch the hats at all, except for the point you were talking about, which is the only place where the two hats touch each other!

J: Right.

S: And that’s just one little dot – it’s not even considered a conic section!  But I think it is; I think it’s like a circle, because a dot is a circle.  And there is no space between the two hats!  But wait – there must be space between the particles that make up the hats, or they would… meld together as one… a statue of the party hats of two Beatles… preserved forever!  Actually, probably thrown in the recycling bin, as the Beatles weren’t famous yet…  Did they even have recycling bins in 1958?  Or did they just put everything in the garbage?  Yes, I think they just put everything in the garbage.  Recycling bins are a fairly recent development.

J: Well, if you had someone who had a shop-

S: That was 58 years ago!

J: -they would probably use metal recycling in some form.

S: No, wait, that was 59 years ago.  I was like, ’58 was 58 years ago!  But then I remembered that it’s 2017, not 2016.  I keep thinking it’s 2016!  Can you say that again?  I couldn’t hear you because I was yelling too loud.

J:

J:

J: Um, what was I saying?

S: I don’t know!  I couldn’t hear you over my yelling!  Never mind.  Okay, so, is that the end of this unordinarily long lesson?

J: No.

S: Ohhhhhhhhhhhhhhhh nnnnnnnnnnnnnnnnnnnnnnnnnoooooooooooooooooooooo.  My life is dead.  But life can’t die.  Life is the opposite of death.  That’s like saying, “My hot is cold, my cold is hot, my low is high, my high is low, my Beatles are One Direction, my One Direction is the Beatles.”  In other words, I sound like Dr. Seuss.

J: So, another degenerate conic section would be one that intersects the cone-

S: I JUST DID IT!!!

J: -in a line, or-

S: I JUST DID THE PAUL McCARTNEY-STYLE WINK PROPERLY!!!

J: Do it again!

S: I can’t – I don’t know how I did it!!!

J: Or it could intersect it in a pair of intersecting lines.  So, you could imagine a plane that chopped the party hats in half.

S: Oh, I see, okay.  Wait – is that not considered a conic section?

J: That is a degenerate conic section.  You end up with an X in space.  You can picture it, right?  The X where it cuts them?

S: Yeah.

J: And you can imagine a plane that just barely touched the top hat and the bottom hat.

S: But through the very middle?  Or through just one side?

J: Um, it goes through the middle, but it’s not flat; it’s just tipped until it touches them.

S: OOOOOOOOOOOOOHHHHHHHHHHHHHHHHHHHHHHHHHH, I SEE, I SEE!!!  It’s like you have two hats, and the plane goes like THIS:

J: If it’s tilted a little bit one way, then it intersects it in a point, and that’s the other degenerate case.

S:

J:

S:

J:

S:

J:

S: Quiz: Which Beatle has the most nicknames?

J:

S:

J: Paul.

S: No.  Geo.  Because, I mean, he’s called George, Georgie, Geo, Jorj, Jorg, Jorge, and Joj.  At least, those are all the names I’ve seen the fans call him in their blogs, fan art, fan comics, comments, memes, fan fictions, and the like.  But I shouldn’t say “they”, since I’m one of them.  But the sentence sounded confusing with “our”.  Anyway, Paul just has Paul, Paulie, Pol, and Macca.  And John has the least, because, really, the only thing you can do with John is Johnny.  And then Ringo has Ringo, Ringsy, Rings, Ritch, and Ritchie.  They probably all have more nicknames; those are just the mainstream ones.

J: So, for review, what are the seven conic sections?

S: Um… they are the circle… the ellipse… the parabola, the hyperbola… the, um… the dot…

J: The point, you mean.

S: Dot Rhone!  Sorry – whenever I say or hear the word “dot”, I can’t help thinking of her.

J: The line and the intersecting lines.

S: The line and the intersecting lines.

J: Yes, and those last three are…?

S: The dot, the line, and the intersecting lines?  THAT SOUNDS LIKE THE NAME OF A BOOK!!!  “The Dot, the Line, and the Intersecting Lines.”

J: Some authors do not consider those last three to be proper conic sections at all.

S: Yes, it’s terrible!

J: So, they’re called…?

S: Degenerate.

J: Okay, I think today’s lesson is over.

S: Yes.  BBBBBBBBBBBBBYYYYYYYYYYYYYYYYYYYEEEEEEEEEEEEE!!!  HAPPY BIRTHDAY, PAUL McCARTNEY!!!  I LOVE YOU!!!

J: HAPPY BIRTHDAY, PAUL!!!

S: Your 16th and your 75th, and all the ones before and in between.

Above fan art: I think I found this on DeviantArt a long time ago, but I can’t remember where, and I just spent an hour searching the entire internet to try to credit it here.  The only place I could I find it was on Zerochan, and I don’t think the person who posted it there drew it because there were other things they had posted that I knew were by PepperMoonFlakes or Wakonaga or WingsOverDA (all three on DeviantArt).  So, to whoever drew this, I’m sorry that I can’t find you.  If anyone reading this knows who drew this, please tell me so that I can credit it.  The only reason I searched for an hour and didn’t use something else that I COULD credit is that this is my favorite Paul McCartney birthday fan art EVER.

 

Walrus in party hat painting link to original: Link

 

All conic sections diagrams from Wikipedia: Link

 

NOTE: Yes, I know that this was not posted on Paul’s actual birthday, but at least we did the lesson on his birthday.

Logarithms

June 4th, 2017

J: I’m really stuck because I have some math I want to lay out for you, but I have no idea how to Beatle-ify it.

S: I’m sure there’s a way.  *sings* It’s ee-ee-asy!

J: George Martin says, “Okay boys, sit down.  We need to review some basic logarithmic facts.”

S: The boys want to know why he’s suddenly teaching them math.

J: He can’t tell them to sit down and shut up, can he?

S: 😱😡

J: Well, they may be required to pass a mathematics exam to enter India.

S: Okay.  But be warned: John thinks math is stupid.  Or, he did at one point.

J: He may say that math is stupid, but he flew on planes and rode in cars and slept in buildings that could not have existed without math.  Oh, and what about that counting?!  *sings*  One, two, three, four…

S: Well, it was Paul who wrote “All Together Now”.  But they all found it useful to use numbers to count-in on songs.  Actually, that isn’t always true.  It seems to me that they had some sort of telepathic connection with each other, because sometimes, they all just started playing a song perfectly in sync without even counting in!  Okay.  You were going to teach me about logarithms, right?  No, George Martin was going to teach the Beatles about logarithms.

J: Say we had x².  Then it’s true that the base x logarithm of x² is 2.

S: Because 2 is the exponent, right?

J: Right.  We’d write this as log_x(x²) = 2.

S: Am I supposed to be ALL four Beatles and you’re George Martin?  ‘Cause that’s hard.

J: Is it too hard for you?

S: You weren’t even doing it!  You were pretending that we weren’t doing it!

J: Weren’t doing what?

S: 😤

J: Said John.

S: Never mind, Johnny boy.  He’s just being annoying.  On with the lesson!

J: So, what’s log_x of x³?

S: 3.

J: How about log₁₀ of 100?

S: Well, that’s 10¹⁰⁰, so 100.

J: No.  No.  So, think of the 10 as x, so, what is 100?

S: The exponent!

J: No.  No.  It would be log₁₀(100).  What is that equal to?

S: I don’t understand what “base” is, anyway!

J: What is 100 in terms of 10 to a power?

S: 10².

J: Very good.  So, the log₁₀ of 100 is the same as the log₁₀ of 10².

S: Yes.  I knew that.  So, what’s the point?  I still have to solve that thing up there.

J: Yes, so the log₁₀(10²) is equal to…?

S: 2.  I still don’t understand why people need logarithms.

J: Well, we use them to make a slide rule so that we can use addition to do multiplication.

S: And humanity does not need slide rules anymore.

J: Well, logarithms are useful in solving equations, and also at looking at big numbers in simpler ways.

S: Please give me the dictionary definition of “logarithm”, as I STILL don’t know what it is as a thing – a concept – a noun!

J: *looks it up on phone* “A quantity representing the power to which a fixed number (the base) must be raised to produce a given number”.  Do those words mean anything to you?

S: No.  Use the Beatles to explain.

J: …

J: …

J: …

*dramatic opera music actually plays in background in real life*

J: Mr. Martin wishes to explain record sales-

S: He’s not the SALES expert!  He’s the PRODUCER!  BRIAN’S the one who should be talking to them about sales.

J: BRIAN has had a chart made showing record sales.  Record sales skyrocketed so quickly that a logarithmic chart had to be used.  Ringo says, crazily out of Nowhere Land, that saying y = bx is the same thing as saying log_b(y) = x.

S: Oh.  So, logarithms are just a way to write things differently.

J: Sure.

S: Guess who hasn’t been mentioned at all in this lesson?

J: Paul?

S: No, I mentioned that Paul wrote “All Together Now”.

J: Hmm… must be George.  “Wot?”

S: Was that in quotes?

J: Yes.  George says that.

S: Now he has been mentioned- he has mentioned himself by- wait, what?  He has been mentioned by himself, or he has mentioned himself, or what?!!  HA HA HA HA!!!  What should I write?!  HA HA HA HA!!!

J: I don’t know.

S: HA HA HA HA HA HA HA HA HA HA!!!  He has been mentioned by himself!  Or should I word it differently?  What does it mean?  HA HA HA HA HA HA HA HA HA!!!

J: I was just going to have him say something, and now we’re off on this blah-blah-blah tangent that has nothing to do with anything!

S: Well, wouldn’t it make more sense if he SUNG “Something”?

J: He has completely lost track of this train of thought.

S: Oh, I’m sure he’s been following it!

J: So, George says, “If we have ten record sales today, do we put a mark at 10 on the chart?”

S: Just a note to the readers: We will put the finished chart for the Beatles’ record sales in at the end of this post.

J: Yes.

S: NOW we’re getting somewhere!  We should look up some actual data on record sales for this.

J: *looks up data*

J: *can’t find anything except a list of the best-selling music artists of all time, which the Beatles are at the very top of*

S: Oh look!  Paul gets his very own spot at #42!  And I bet One Direction isn’t even on there!  YAY!!!  Oh… we should check, just in case… If they are, the Directioners will be MAAAAAAAAD when they read this.  Not that I care.  Wait, why would Directioners be reading a BEATLES math blog?  Unless they are an alien who likes both bands?  In which case, I celebrate them for bridging the gap – no, HUMONGOUS CHASM – between those two fandoms, even if I can’t understand… their life.  Wait a second!  Paul McCartney said that he likes One Direction!  Which means that I can’t understand HIS life!  I like to think that I can… but I probably can’t.  Wait – why am I arguing with myself?

J: *sighs*

J: *picks up phone*

J: Now this is about One Direction.  Do you know how long they’ve been a band?

S: No, and I’m glad I don’t.

J: Well, I don’t think that this list is updated enough to have New Direction on it.

S: ONE Direction, dad.  Do you even know who they are?

J: Yes.

S: Then why did you call them “New” Direction?

J: Becaaaaaaaaaaaaaause I want to go in a new direction with this lesson.

S: Good, because I don’t like them.

J: Who?

S: One Direction!

J: Oh, I wanted to go in a NEW direction!

S: I just can’t win.  Actually, with you, I always win.  I don’t think I’ve ever said “I just can’t win” before.  Anyway, ONWARD!  Didn’t George say something about base 10 or something a billion years ago?

J: Yes.  On this chart, if we sold ten albums-

S: WOAH.  When is this?  By album #10, it was 1968.  And they’re learning math to GO to India, where they wrote most of their tenth album (the White Album), and wait – Brian is dead, so how can he be showing them this chart?

J: George says, “I meant One Direction.”

S: You know, Geo, they don’t exist yet, and they won’t for a long time (thank goodness).  Don’t listen to us talking about the terrible future.

J: Ringo says, “Can we go in a new direction?  My head hurts.”

S: Okay!  We’re DONE with One Direction; we’re going to do more math… and John and Paul need to be in here more, especially Paulie.

J: “Here are some hypothetical record sales by day,” says Paul.  “1, 4, 9, 16, 25, 36, 29, 64-”

S: *sings* WHEN I’M SIXTY-FOUR!!!

S: Sorry.

J: “-81, 100, 121, 144, 169, 196, 225.”

S: I can already tell that the equation is x².

J: Great!  Let’s plot the equation on the logarithmic chart!

S: (To see this, look at the end of this post.  This equation is graphed in black.)

S: *starts to plot points*

S: Oh, I see!  It shows how fast the record sales are climbing.  Only 1 on the first day!  That must have been before even Please Please Me, because they had more than 1 fan who would want to buy it the second it came out when Please Please Me was released.

J: These are hypothetical.

S: *continues plotting points for x² *

J: John says, “So, we’re clearly graphing x², but, instead of flying up off the page as on a regularly-scaled graph, it stays close to home, well-behaved.”

S: That does not sound like anything he would say.

J: Oh, well, Ringo said that last bit, completing his sentence.

S: But it IS making a parabola, right? (ParaboliCAT *cough cough*)

J: Eh… It’s different.  It’s clearly a curve.

S: Oh!  Because their record sales never go down, so the parabola can never be completed!

J: Let’s go on to the next equation and graph 10^x.

S: Okay.  So, how is that done?

J: Well, let’s graph values of 10^x.  Here are the numbers-

S: Still record sales?

J: Yes.  The numbers: 1, 10, 100, 1000, 10000.

S: WOOOOOOO!!!!!!!!  YAY!!!!!!!!  And that was just over a period of five days!!!

J: Right.  Where the first day is day 0, where 10⁰ = 1.

S: *graphs 10^x*  (This is in blue ink.)

J: Paul says, “It’s a straight line!”  Ringo says, “I know!  Do 2^x!  Because peace is two fingers, baby!”

S: One finger stands for peace and the other stands for love.  PEACE AND LOVE, RINGO!!!

J: Awww… ☺️  2^x.  Shall I give you the numbers?  Because you know them.

S: I THINK I know them.  1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192…  That’s as much as will fit on our graph.  Now, I need to graph it in a different color…  But do I have a thin enough pen in a different color?  Oh!  I know!  *Gets a four-colored pen*

J: “A FOUR-colored pen!  How clever!” says George.

S: Awww… 😊  It’s just too bad the four colors aren’t the Beatles’ favorite colors…  Wait – can I have a pen like that?

J: Sure!

S: *graphs 2^x*  (This one is in red.)

J: Is it another straight line?

S: Yes!

J: We’ve come to our conclusion for this lesson.

S: Make it Beatle-y!

J: So, logarithms make big curves into straight lines.  The end!

S: Dad!  What did I just say?!

J: *thinks for 30 long, long, long seconds*

J: George Martin says, “Oi!  This is what we use for the decibels scale!”

S: Even though George Martin was from London and spoke the Queen’s English, which did not involve “oi”.  But I’ll let it slip just this once, since- hey!  Where did you go?!

Exponents

May 27th, 2017

S: Okay.  Hello!  I saw an adorable fan art where the Beatles were babies (above; © arisuyoshioko on DeviantArt; link to original: Link), so that is the Beatle-y theme for this lesson.  My dad wants to start with a table, but he can’t for the life of him think of a way to relate it to the baby Beatles.

J:

EQUATION	EXPONENT	CANONICAL FORM OF EQUATION
x	1	x
x · x	2	x2
x · x · x	3	x3
x · x · x…x }x	x	xx
√ x	1/2	x1/2
1/x	-1	x-1
1/x2	-2	x-2

S: That was boring.  *shows Jeff the baby fan art*  Now, come up with an exponential story problem based off of this.

J: *thinking*

S: Wait – are they in a Japanese house?

J: Yes.  They are on a tatami mat surrounded by rice paper walls.

S: Why?  They weren’t born in Japan.

J: I don’t know.

S: Maybe the artist is from Japan and thinks that all babies play on tatami mats, or maybe the artist just wanted to do that for an unknown reason, or maybe this is an alternate universe where the Beatles were Japanese.

J: Anyway, the complexity of any one Beatle baby activity is x.  So, with two Beatle babies, their complexity multiplies.  So, you have a complexity of x².  Then, with a third Beatle baby-

S: This is not fun at all!  Do something specific to the art!  Like, about the rice paper walls or the tatami mat or the pajamas or the blanket or Geo’s bottle or the blocks or the duck or Ringsy’s octopus or the other octopus or John’s pacifier or the submarine toy or the car or the paper or the stacking rings or the teddy bear!

J: The probability of peace and quiet with baby John is x.  When baby Paul hits the tatami mat, their probability of peace and quiet is multiplied.

S: But the more babies there are, the probability of peace and quiet goes down!

J: I’ll explain how it works out.  So, for example, if x is 1/2-

S: OHHHHHH!!!  It would go down!

J: Yes.  So, when baby George wanders onto the tatami mat, looking for his bottle, the probability of peace and quiet multiplies by the other two, and so we have x · x · x, or x³.

S: But baby John is asleep, so he won’t affect the peace and quiet level unless he’s woken up.

J: So, when baby Ringo crawls onto the mat, looking for something to drum on-

S: -but ends up just gnawing on an octopus-

J: -the probability of peace and quiet is x · x · x · x, or x⁴.

S: But it’s really x³ because baby John is asleep.

J: Well, if we’re going to take that into account, we’re going to model it off whether a baby is asleep or awake and active.  We’re going to model it in a crude way – we’re not going to model it on the whole human nervous system, for example.

S: I barely understood that.  But we must take into account the fact that if the others are too loud, John will wake up.  But we don’t know what that noise level is.

J: Right.  We’re modeling it more crudely than that.

S: Says who?

J: Gesundheit!

S: I did not sneeze.  You know very well what I was talking about.

J: So, let’s say that maybe x is 1/4 in this situation.  So, if the probability of peace and quiet is x⁴, then what is the probability of peace and quiet as opposed to there being a Beatle baby jam session?

S: Awww… a baby jam session with tiny plastic toy instruments…  adorableness overload!!!  Okay, fine; I’ll solve it.  Is it x · x · x · x?

J: Yes.  In canonical form, that is x⁴.  Let’s make a table to figure out the probability of peace and quiet for different values of x.

S: But you said that x is 1/4!

J: That’s just one value of x because baby John is asleep.  We’re looking at an equation from which we might get many different values of x – maybe you want to erase something because the point is – DON’T GET HUNG UP ON 1/4 FOR X!  BECAUSE BABY JOHN IS GOING TO WAKE UP!!!

S: Oooooookay.

J: So, let’s make a table.

J/S:

BABY CONDITION	X	X4	THE PROBABILITY OF PEACE AND QUIET
Asleep, tired, and with full bellies	1	1	Pretty good!
Average (what the fan art shows)	3/4	81/256 (≈ 1/3)	Eh. Somewhat peaceful.
Excited and eating chocolate	1/4	1/256	Uh-oh. Close to zero.
Trying to stack the blocks on top of the yellow submarine toy, but the blocks keep falling over, and they get upset	2/5	16/256	Not looking good.
The blocks on the submarine fall onto baby John’s head and he wakes up. The first thing he sees is baby Ringo, and he grabs the toy duck and hits baby Ringo in the eye with it. The duck is very precious to baby Paul, and he’s mad at baby John for touching it, so he grabs baby George’s bottle and throws it at baby John. And baby George… we all know what happens when baby Geo’s food is stolen…	0	0	NO CHANCE

S: Beatle babies forever!  Bye!

J: I must show you something before this lesson ends!  Let’s say we had x³ · x⁵.  So, just by the way, that looks like-

S: It’s x⁸.  I knew that already.

J: So, the point is that when you’re multiplying the same variable with different powers, you just add the powers together.  And conversely, for division, if we have x³ ÷ x⁵, we subtract.

S: So, x^-2.  And that was a little bit about exponents!  (And the Beatles as babies based off of a wonderful fan art that my mom says is creepy.)

Quadratics

May 20th, 2017

J: What do you know about quadratics?

S: I know that there is the quadratic formula.  I recently did a lot of quadratics, but I don’t remember exactly what I did.  Let’s just start with a problem using the quadratic formula.  Oh!  I just realized that I memorized it!

(-B ± √-B² – 4AC) ÷ 2A

Now, give me an equation to solve.  And remember: This is Beatles math.

J: George is designing an amplifier horn.  It will be in the shape of a parabola, and the 0,0 origin on the blueprints will be offset for mounting.  George will need to know the two places where the right and left of the amplifier horn are at elevation y = 0.  The equation for the horn is y = x² – 4x + 1.

S: So, I’m trying to get it where y = 0?

J: Yes.

S: Uhhhhhhhhhhhhhh…

J: So, a quadratic equation is in the form ax² + bx + c.

S: Oh!  I remember that!  So, how does that help me solve it?

J: Well, Paul quips that.

S: Quips what?

J: So, the thing I said was in quotes, and after that, the text reads, “Paul quipped”.

S: Oh.  So: ‘”So, a quadratic equation is in the form ax² + bx + c,” Paul quipped.’  Still, how does that help me?

J: Oh, well, Ringo says, “So, for Geo’s equation, ‘a’ would be what?”

S: Oh!  It would be 1.

J: John said.

S: But I said that!

J: “And ‘b’ would be…?” *bad impression of Ringo’s voice*

S: -4.

J: “And ‘c’?”

S: 1.  Having the Beatles teach me math in this story makes it harder to concentrate.  But I want them to keep doing it!  But I feel so weird and fangirlish!

J: And John says, “So, we plug a, b, and c into the quadratic formula.”

S: Oh!  Now I remember how to do everything!  Thanks for jogging my memory, Beatles!

a = 1

b = -4

c = 1

(4 ± √4² – 4 · 1 · 1) ÷ 2 · 1

I’ll solve it with the + first.  *solves it*  That equals 3.7320508075688…

Now for the -.  *solves it*  That equals 0.267949192431…

J: Shall we check those?

S: Okay.

J: So, we could punch into the calculator “(4 + √12) ÷ 2”.

S: And the answer is correct!

J: And now the other one.

S: Also correct.

J: Those two numbers are the roots of the equation, so we can write the equation as y = (x – 3.73) · (x – 0.26).

S: And we still have to make that 0?

J: “We have to make y = 0,” Paul says.  “So, y is 0 when the first subexpression is 0 or when the second subexpression is 0.”

S: Oh!  Because anything multiplied by 0 is 0!

J: “Roight.”

S: This doesn’t make sense because we have no value for x!

J: Ringsy says, “What minus 3.73 is equal to 0?”

S: 3.73.

J: Correct!  And, for the other x, what minus 0.26 is equal to 0?

S: 0.26.

J: Correct!  George finally says, “Thank you, boys!  So, y = 0 when x = 3.73 and 0.26.”

S: Happy endings = gear

J: So, can we graph it?

S: First, what exactly is an amplifier horn?

J: Just an amplifier shaped like a horn.

S: But why is George building one when he could just buy one?

J: Maybe he’s doing it to make an Indian swami happy.  *holding phone up*  Here’s an amplifier horn for an iPhone.

S: Disturbing.  Anyway…graphing.

J: You could do some algebra to figure out when x = 0.  But that’s optional.  Do you want to do it?

S: No, I think we’re out of time.  But why did George need to know the…things for the equation for the parabola-shaped amplifier horn when he could have just drawn a parabola?

J: Because his friend, who’s going to machine it, prefers a parabolic equation.

S: Ah.  Toodles!

Functions

MAY 14TH, 2017

S: First, we should do the table functions, and then the graph functions.  Just an overview for those of you who don’t know: The goal of solving a table function is to find an equation that, when you plug in x, always equals y.  The table gives examples of what y is when x is different values.  A graph function is an equation graphed.

J: John writes a table for Ringo to solve.

x	y
0	0
1	1
2	4
13	169
1,024	1,048,576

What equation does Ringo come up with?

S: x²

J: John makes a harder table for Ringo to solve, as Ringo seems smarter than John thought.

x	y
0	1
1	2
2	7
3	16
4	29
5	46

Ringo gives the problem set to George.

S: George is eating an egg sandwich and doesn’t feel like doing algebra, so he gives it to Paul.  Paul solves it, but he doesn’t know that John made it, so he asks John if John can solve it just to see if he can.  And John concludes from this that Ringo cannot solve the problem.  So, I will solve it.  I will re-write the table and think of the possible answers for each individual pair of numbers, and then see if any of them match.

x	y	GUESSING
0	1	+ 1
1	2	+ 1
2	7	+ 5 · 2 + 3
3	16	· 2 + 10
4	29	· 7 + 1
5	46	· 10 – 4

Gaaaaaah!

J: You might try a column of x² and see how it looks.

S: Okay.

*ends up trying many more columns that Jeff suggests, but only gets to Nowhere Land*

x	y	x2	DIFFERENCE BETWEEN Y & X2	DIFFERENCE BETWEEN X2 AND DIFFERENCE BETWEEN Y & X2
0	1	0	1	1
1	2	1	1	0
2	7	4	3	1
3	16	9	7	2
4	29	16	13	3
5	46	25	21	4

J: Do you want a hint?

S: Yes.

J: Do you see the last column kind of matching some other column?

S: Sort of.  It matches the x column.

J: So, those differences – it’s looking like x – 1 = the difference between x² and the difference between y & x².

S: Except for the first row.

J: Sure.

S: How does that help?  The last column isn’t even y!

J: Sure.  Try a column now where you add that back into the output (y).

S: That makes no sense.  Can I just try by myself?

x² – 1 = k

0² = 0 – 1 = -1, x² + 1

1² = 1 – 1 = 0, x² + 1

2² = 4 – 1 = 3, x² + 3

0² = 9 – 1 = 8, x² + 7

0² = 16 – 1 = 15, x² + 13

0² = 25 – 1 = 24, x² + 21

I cannot do this.  Help!  I need somebody (A.K.A. you)!

J/S:

x	y	x2	DIFFERENCE BETWEEN Y & X2	DIFFERENCE BETWEEN X2 AND DIFFERENCE BETWEEN Y & X2	Y + DIFFERENCE BETWEEN X2 AND DIFFERENCE BETWEEN Y & X2
0	1	0	1	1	2
1	2	1	1	0	2
2	7	4	3	1	8
3	16	9	7	2	18
4	29	16	13	3	32
5	46	25	21	4	50

S: If you compare x to the last column, the equation 2x² works, but only if we pretend that the last column is the output (y), which it isn’t, and only if there wasn’t that stupid zero row that doesn’t match!!!

J: Forget about the zero case.  Zero cases are often the exception.

S: I don’t want to do this anymore!  You should have told me in the beginning that x = 0 didn’t matter!!!

J: Well, shall we go on fast-forward, then?

S: If you put the Beatles in on the next line.

J: Paul points out that y appears to be equal to (y + difference between x² and difference between y & x²) – (difference between x² and difference between y & x²).

S: OHHHHHHHHHHHHHH!!!

50 – 4 = 46

32 – 3 = 29

18 – 2 = 16

8 – 1 = 7

2 – 0 = 2

J: The last column is potentially equal to 2x².  The second-to-last column looks like x – 1.  And so, y appears to be equal to 2x² – (x – 1), or 2x² – x + 1.

S: That’s the equation.  Don’t ask me how it was done.  I can’t do that ever again.  Time to  check to see if it matches, even the zero case.

2x² – x + 1 = y

2 · 0² – 0 + 1 = 1

2 · 1² – 1 + 1 = 2

2 · 2² – 2 + 1 = 7

2 · 3² – 3 + 1 = 16

2 · 4² – 4 + 1 = 29

2 · 5² – 5 + 1 = 46

Everything matches!

Time for part two of the lesson: graphing the function…

*Realizes that graph is terrible because it is not A, even, B, big enough, or C, correct*

J: Where’s your graph paper?

S: I’ll get it…

That is a slightly better graph.  Grand finale time!  Well, what is the grand finale?

J: Paul kisses you, saying, “Thank you!  This is the perfect parabolic hat for my next show!”

S: “Parabolicat”???

J: “Parabolic hat“!

S: Oh.

J: “ParaboliCAT”? HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA HA

S: As you laugh, your daughter is dead.

J: NOOOO!!!  Did the parabolicat eat her?

S: No!  I died of joy because Paul kissed me!

J: Oh.

S: Can’t wait to see you in heaven!  It’s nice up here.

J: So, now you’re up there with John and George, waiting for Paul and Ringo.

S: No!  Because John created the problem for the parabola, and I solved it on the same day!  I went back in time to 1964 or something and died, so they’re all still alive!

J: And you disrupted my conception, so I was never born!

S: And I was never born, so I never died!  Oh, that’s sad.

J: But John never died, and neither did George!  You saved the Beatles!!!

S: No!  It’s still 1964!  Except you’re in the present day, even though you don’t technically exist!  Not even Brian is dead yet!

J: Yes, but you disrupted the timeline enough that they’re all still alive in the present day!

S: Really?

S:

S:

S:

S: YAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAY!!!

I WILL ALWAYS BE REMEMBERED AS THE DEAD FANGIRL WHO SAVED THEM!

But I was never born, and they never died…

Whatever!

At least I know that I saved them, and at least Paul kissed me!🤣

Two Inequalities, Two Unknowns

April 29th, 2017

J: The additive inverse of a number is the negative of that number.  We also have the multiplicative inverse.  The multiplicative inverse of a number is its reciprocal.

S: Okay.

J: So, if we have an equality such as 3x = 4y, we could take the inverses of both sides, and the equality is still true.

S: Let me try that.  But which inverse?

J: Either one or both!

S: 1/(3x) = 1/(4y)

J: That would be the inverse.

S: Dad, just a warning: If the Beatles aren’t in here in two lines, I’m quitting.

J: I’ll see what I can do.  So, that would actually be 1/(3x) = 1/(4y).

S: I wrote what my dad said in his previous line, when he said: “Next line”, but then I had to almost quit because I did go to the next line.  I told him he could get a second chance if he erased “next line” and said something Beatle-y.

J: John erases this.  *erases “next line”*

J: George tickles Starli.

S: AAAAAAAAAA-HA-HA-HA *is being tickled*

J: Okay, so, starting over.  Paul is considering playing Starli’s Yamaha keyboard, but wants to think it over.

S: Because it would immediately become A, very valuable, and B, an heirloom.  And why is he here? *slight fangasm imagining him actually being here and playing a song on my keyboard*

J: Because Starli insisted on Beatles in her math, so it takes ten times as long to learn the same thing.

S: HA.     HA.     HA.     HA.     VERY FUNNY.     NOT.

J: Paul’s piano has 88 keys.  And Starli’s piano has…?

S: 77.

J: Write an inequality involving those two numbers.

S: Okay.  88 > 77.

J: Now take the inverse of both sides of that inequality.  Your choice: multiplicative or additive.  Is it still true?

S: 1/88 > 1/77.  It is not true.

J: How can you fix it?

S: Flip the sign!  1/88 < 1/77.

J: Hmm… So, you flipped the inequality symbol, right?

S: Yes.

J: Good.  Now, write another inequality in terms of those numbers of keys.

S: 77 < 88.  There.

J: Now, multiply both sides by -1, which is the same as taking the additive inverse.

S: 77 · -1 < 88 · -1 ≠ -77 < -88!!!  Flip the sign.  -77 > -88.

J: So, we can conclude from this that when you’re solving an inequality and you’re taking the inverse of both sides, you need to flip the inequality sign.

S: Which I already knew!

J: So, what happens if you have 88 > 77 and you took both the multiplicative inverse and the additive inverse?

S: 88 · -1 > 77 · -1 ≠ -88 > -77 ≠ 88 > 77, but that is equal to its original self.  So, people, if someone asks you to take an even number of inverses, don’t change the equation at all!  Lesson learned; done.

J: Oh.  No.  Don’t change the inequality symbol.

S: But, what I said is also true because… it’s true.

J: No!  If we’re trying to solve an equation by the methods we learned in other lessons, we’ve probably done a multiplicative or additive inverse before.  And when we’re doing this when solving an inequality, we have to be mindful of flipping the greater than/less than symbol.

S: So, we’re both right!!!  Because if you do both, it doesn’t change the number or the inequality sign!

J: If we did both, the numbers are different!

S: Oh!  I see!  Back when I was going to do both inverses, I only did one inverse!!!  So, it would actually be -1/88 > -1/77.

J: Is it true?

S: Well, it’s weird, because there’s a fraction and it’s negative, so the regular wrongness cancels out.  But yes, it’s true.

J: Are we done with that part?

S: Yes.  So, in one sentence, what was the takeaway from that part of the lesson?

J: If you take an inverse, make sure to flip the inequality symbol, but if you take an inverse an even number of times, don’t flip it, but do flip it if you take an inverse an odd number of times.

S: Okay!  Two inequalities, two unknowns, FOUR BEATLES WHO HAVE NOT COME UP IN 28 LINES…  Actually, Ringo hasn’t been mentioned at all!

J: It’s not his time, though, unless you want it to be his time.

S: It’s always his time!  And we need the story problems to be more complicated and interesting so that our Beatlemaniacal audience will be entertained!

J: Okay, well, George and Ringo have been eating mini sandwiches from platters passing by at a party.

S: Oh, that reminds me of that scene in “A Hard Day’s Night”…

J: They’re not sure how many they’ve eaten, but they know a little bit about how many they’ve eaten relative to each other, and they know approximately how many they’ve eaten, because a mathematician friend has been observing and has provided us with two inequalities to solve together.

S: And “we” are Mona and Pete Best.  May I ask where John and Paul are?

J: Having tea with the Queen.

S: Wow.  George and Ringo must feel unhappy and left out.

J: They missed the flight.

S: So George and Ringo aren’t in England?

J: They’re at the fair in Edinburgh.

S: Anyway, John and Paul and Brian and whoever else WOULD CHECK TO MAKE SURE THAT THEY WERE ALL ON THE PLANE.

J: They are all working for MI6, so they’re going undercover and role-playing and sacrificing their usual Beatle-brotherhood for the good of Queen and country.  So, shall we use abbreviations?

S: No.

J: 3george – 2ringo ≥ 5 sandwiches

ringo – 2george ≤ -4 sandwiches

So, use the system of inequalities to find out how many sammiches they ate.

S: “Sammiches”?

J: Yes, sammiches, made by the Earl of Sandwich for their express culinary pleasure.

S: Isn’t there a town called Sandwich in England?

J: *Googles it*  Yes.  It has a population of 4,985.

S: George should live there, shouldn’t he?

J: Of course.

S: So, solving the inequalities…

3george/3 ≥ (5/3 sandwiches + 2ringo)/3

george ≥ 5/3 sandwiches + 2/3ringo

ringo – 2 (5/3 sandwiches + 2/3ringo) ≤ -4 sandwiches

ringo – 2 · 5/3 sandwiches + 2 · 2/3ringo ≤ -4 sandwiches

ringo – 3.33 sandwiches + 1.33ringo ≤ -4 sandwiches

2.33ringo – 3.33 sandwiches ≤ -4 sandwiches

2.33ringo ≤ -4 sandwiches + 3.33 sandwiches

2.33ringo/2.33 ≤ -0.66/2.33 sandwiches

ringo ≤ -0.28 sandwiches

REALLY, DAD?  Ringo ate -0.28 sandwiches?!  Oh!  Ringo ate less than or as much as -0.28 sandwiches.

J: That’s not the result I was looking for.

S: oh no OH NO OH NO

J: Shall I take another look at the inequalities?

S: Okay.

J: *reviews inequalities for ten minutes*  This doesn’t make sense…  Your inequalities are correct, but I know that Ringo ate two sandwiches and George ate three, because that’s what I based the inequalities off of…  *reviews inequalities for another ten minutes*  Oh, I see!  You dropped a sign when you were distributing the -2.  5/3 sandwiches was multiplied by -2, but 2/3ringo was multiplied by 2.

S: So I have to redo it.  Wonderful.  But WAIT!  I know the answer!

J: No, no, no!  It would be un-Beatlesque of you to cheat!

S: You were the one who carelessly told me the answer!

J: But you don’t get to use the answer!

S: Alright, I’ll start with the wrong inequality and write it correctly.

ringo – 2 · 5/3 sandwiches + -2 · 2/3ringo ≤ -4 sandwiches

ringo + -3.33 sandwiches + -1.33ringo ≤ -4 sandwiches

-0.33ringo + -3.33 sandwiches ≤ -4 sandwiches

-0.33ringo ≤ -4 sandwiches + 3.33 sandwiches

-0.33/-0.33ringo ≤ -0.66/-0.33 sandwiches

ringo ≥ 2 sandwiches

Ringo ate two or more sandwiches.

3george – 2ringo ≥ 5 sandwiches

3george – 4 ≥ 5 sandwiches

3/3george ≥ ≥ 1/3 sandwich

George ate 1/3 of a sandwich.

Which is wrong.

J: Shall I have a look?

J/S: *both reviewing*

S: What the heck is going on?  I wrote two inequality signs and dumped the -4!  Okay…

3george ≥ 5 + 4

3/3george ≥ 9/3

George ≥ 3

George ate three or more sandwiches.

Our summary is that Ringo ate two or more sandwiches, and George ate three or more sandwiches.  Good job, Pete!

J: Anytime, Mona!