
024  Video Parody Mashup: 3U

posted Mar 24, 2015, 6:50 PM by Gregory Taylor
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updated Mar 24, 2015, 6:51 PM
]
(yk)^{2}  4p(xh) presents....
"They're Fractals" (*aka "I Love It", Icona Pop)

***
I thought that Mandelbrot and Julia were such a yawn, I saw they won't be on the test. I passed, I'm moving on. I now admit, the truth of it, they're more than denouement. I know they won't be on the test. I don't care! They're fractals! Let's all share!
I thought that Mandelbrot and Julia were such a yawn, I saw they won't be on the test. I passed, I'm moving on. I now admit, the truth of it, they're more than denouement. I know they won't be on the test. I don't care! They're fractals! Let's all share!
{Refrain:} You're on a Hilbert curve, I'm in the Koch snowflake, You want Sierpinski cuts, but Cantor sets I make. Your attractor is strange, Weierstrass can bewitch, You're folding dragons up, but I'm done with this pitch. They're fractals!
They're fractals!
I thought that Mandelbrot and Julia were such a yawn, I saw they won't be on the test. I passed, I'm moving on. I now admit, the truth of it, they're more than denouement. I know they won't be on the test. I don't care! They're fractals! Let's all share! These fractals. Hausdorff it. Be aware! They're fractals. Let's all share!
{Refrain:} You're on a Hilbert curve, I'm in the Koch snowflake, You want Sierpinski cuts, but Cantor sets I make. Your attractor is strange, Weierstrass can bewitch, You're folding dragons up, but I'm done with this pitch. Be aware! They're fractals! Let's all share! These fractals. Hausdorff it. Be aware! They're fractals. Let's all share! These fractals. Hausdorff it. Be aware!
They're fractals!!

posted Mar 14, 2015, 6:26 PM by Gregory Taylor
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updated Mar 16, 2015, 10:49 AM
]
(yk)^{2}  4p(xh) presents....
"It's Probability" (*aka "I'm Getting Over You", Carley Rae Jepsen)

***
I wonder at first glance, Will these events have any chance? So we do some trials, get zero to one for each. I ask now, what's defined? I ask now, can these be combined? And it goes awry, can you see the reason why?
When there's chances, for A and B, Don't just multiply them, check if there's dependency! No denying, Since sets get changed, Just keep glancing for the chances data's rearranged,
When finding Probability! That's right, it's Probability! Sit tight, for Probability! In sight's your Probability! Probably. Probably.
Stuck counting real big sets, And intersections they possess, Know B has this chance but I'm seeing A came first. It's more than a bit strange, That I don't think my chances change, But the more we try, No, this you can't deny...
When there's chances, for A and B, Don't just multiply them, check if there's dependency! No denying (no denying), Since sets get changed (since sets get changed), Just keep glancing for the chances data's rearranged,
When finding Probability! That's right, it's Probability! Sit tight, for Probability! In sight's your Probability! Probably. Probably.
That's right, it's Probability. Sit tight, for Probability. Delight in Probability. Possibly.
When there's chances, for A and B, Don't just multiply them, check if there's dependency! No denying (no denying), Since sets get changed (since sets get changed), Just keep glancing for the chances data's rearranged,
When finding Probability.

posted Dec 18, 2014, 5:44 PM by Gregory Taylor
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updated Dec 18, 2014, 6:52 PM
]
(For Christmas: Dec 18/14)
SINGALONG 04
MathTans present....
"Polar Plot" (*aka "Let it Go")

Official Video: Karaoke: THIS PARODY VIDEO: Christmas Lights Version?
Each grid you try, will relate x and y. Rectangular is that graph. The system is Cartesian, It’s the high school epitaph. No other grids considered, they can be denied. Is there nothing else? What will you decide?
Don’t need to know, don’t need to see, Won’t be tested, but math means more, surely! It’s not for grades, there’s no concern... Well, why not learn...
Coordinate angle view. Polar plot, polar plot! Points the way with distance too! I don’t care, math can be for play, r with theta’s fun. The x kinda bothered me anyway.
If r is now our distance, hypotenuse we find. And the theta’s an inverse tan, our slope we’ve reassigned. To convert back, it’s trig we’ll do, with r, cos theta, and sine too, But there’s no need, as by degree... You’ll see!
Radians can be used as well, Polar plot, polar plot! This circle’s magic spell! Functions curl, on this display, r with theta’s fun!
It’s power can be seen for shapes we’ve not applied, Cardioids, Spirals too as Archimedes once described, And don’t stop searching for the ways these points are new! Square root of minus one? Yeah, we can graph that too!
Polar plot, polar plot! No, you won’t find it on your test. Polar plot, polar plot! Please give your marks a rest. Here’s a grid, Shown in a new waaaay! r with theta’s fuuuun. The x kinda bothered me anyway.  Demi Lovato Version:
Official Video: Karaoke: THIS PARODY VIDEO:
Polar plot, polar plot, Coordinate angle view. Polar plot, polar plot, Points the way with distance too.
Each grid you try, will relate x and y, Rectangular is that graph. The system is Cartesian, It’s the high school epitaph. No other grids considered, they can be denied. Is there nothing else? What will you decide?
Don’t need to know, don’t need to see, Won’t be tested, but math means more, surely! It’s not for grades, there’s no concern. Well, why not learn...
Coordinate angle view. Polar plot, polar plot, Points the way with distance too! Functions curl, On this display! Polar plot, polar plot! The x kinda bothered me anyway.
If r is now our distance, hypotenuse we find. And the theta’s an inverse tan, our slope we’ve reassigned. To convert these points right back, trig will be the key, r cos theta, and sine too, but stay with degrees, you’ll see
Polar plot, polar plot! Radians can be used as well, Polar plot, polar plot, This circle’s magic spell! Functions curl, on this display, Polar plot, polar plot! The x kinda bothered me anyway.
Cardioid. Spiral. Graph them upon arrival. Roots with, a minus. See math that’s oh, so timeless. So much you can show!!
Polar plot, polar plot. Coordinate angle view.
Polar plot, polar plot! Points the way with distance too! Functions curl, on this display, Polar plot, polar plot! The x kinda bothered me anyWAY! Background Vocals: Cardioid. Spiral. Graph them upon arrival. POLAR PLOT! Roots with, a minus. See math that’s oh, so timeless. Cardioid. Spiral. Graph them upon arrival. SPIRAL! POLAR PLOT! Roots with, a minus. See math that’s oh, so timeless.
Polar plot.

posted Dec 25, 2013, 4:56 AM by Gregory Taylor
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updated Dec 25, 2013, 5:32 AM
]
(For Christmas: Dec 25/13)
SINGALONG 03
MathTans present....
"O Factor Tree" (*aka "O Tannenbaum"/"O Christmas Tree", August Zarnack/Ernst Anschutz) O Factor Tree, O Factor Tree, Thy primes are so amazing. O Factor Tree, O Factor Tree, Thy primes are worth some praising.
The rest are odd, but tried and true. O Factor Tree, O Factor Tree, Thy primes are so amazing.
O Factor Tree, O Factor Tree, Least common multiple plea. O Factor Tree, O Factor Tree, Great common factors from thee.
We find those parts, comparing trees; O Factor Tree, O Factor Tree, Those "common" goals you'll bring me.
(Glee version: Primes are not composites...)
O Factor Tree, O Factor Tree, Prime Factory directing, Your structure, it's, whole number bits, Have primes so worth detecting.
Each branching off, each division, O Factor Tree, O Factor Tree, Thy primes are so amazing.
(Option: Repeat first verse)
O Fractal Tree?

posted Nov 27, 2013, 4:19 AM by Gregory Taylor
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updated Nov 27, 2013, 12:14 PM
]
(For Nov 27/13) THIS IS A TRACKBACK LINK FOR COMMENTS
POLYNOMIALS (WITH TRIG) PRESENT...
Para: Think you know when February 29th, 1600 was?
Quinn: Information! Consider that there's different calendars in use even now, on Earth.
CoTangent: And we need to talk about this today because....?
Para: Hanukkah and American Thanksgiving coincide! Kinda. Details below.
Sexi: We'll build to it. Like always.
1: CHINESE CALENDAR
Para: The Gregorian Calendar  from the Leap Year special  is hardly the only one in use. If you DO use it, you're in the year 2013, continually numbering forwards. But let's instead consider a calendar that uses modular arithmetic!
Cubi: Hey, so, wondering what sort of math that is? Hey, it means at some point we'll stop counting forwards, and cycle back to zero  for example, in mod 5, you would count 0, 1, 2, 3, 4, 0, 1, 2... and so on.
Tangent: Actually, a system of that sort exists in computers  think Y2K. As it is, counting forward by seconds since January 1st, 1970, some electronics may experience memory overflow problems in January 2038.
Quinn: More information! The Chinese Calendar has a cycle of 60 years. Every year is named using two stems. The first component is a celestial stem, chosen from 10 available. The second component is a terrestrial branch  or the 12 animals most people know from place mats in Chinese restaurants.
CoTangent: Wait, 10 and 12 stems would imply a cycle of 120.
Sexi: Aha, nice math. Not the way to count it. As an analogy, consider the celestial stem as numbers and the terrestrial as letters. The counting runs: 1a, 2b, 3c, up to 10j. Then the first resets. 1k. 2l. Then the second resets. 3a. This means it's impossible to have a 2a or a YiZi year.
Para: Which is okay, as half the celestial stems are similar. They use woodfireearthmetalwater, but double them up, separating running water from standing water, and the like.
Cubi: Hey, if you're reading, see if you can mathematically link this sexagenary cycle to "least common multiples"!
Quinn: Information! While a chinese year has twelve months, each year only has up to 355 days. This necessitates a "leap year" every few years, whereby an entire MONTH is tacked onto the calendar.
Para: As a Gregorian reference, the 'Chinese New Year' will always occur between January 21st and February 21st. It's connected to the new moon, but keep an eye out in 2015, when two such moons occur within that time frame.
Sexi: As a final aside, there is some argument about when the Chinese calendar first began. Most give 2637 BC as the date, though the actual use of zodiac animals came later.
ArcTan: All of which seems to have no bearing on American Thanksgiving. What's next?
2: ISLAMIC CALENDAR
Cubi: Hey, here's a calendar that's been with us since 622 AD! Hey, after all, that's when prophet Muhammed established the first capital.
Sexi: This means the Islamic Calendar began after 0 AD, but before the Gregorian reforms of 1582 AD. We're currently living in 1435 Anno Hegirae (AH), and it's been this year since November 3rd, Gregorian.
Quinn: Again information! We're talking about the Lunar Hijiri Calendar in this case. Twelve months, in a year of up to 355 days, not unlike the Chinese. But no leap months. To align with the moon, the calendar adds 11 days over the span of a 30 year cycle.
ArcTan: Why do you keep saying 'Information'?
Quinn: Cubi gets an inflection, why not me?
Para: Of course, the Islamic "day leaps" aren't enough to keep them in line with the Gregorian, so we can't peg dates here, like we did with the Chinese calendar. That said, there's two key events that many Western calendars mark: Ramadan, the name for the ninth month, and EidalAdha, which occurs during the last month of the year.
Tangent: Right, and I can see now how those dates would gradually migrate backwards through the Gregorian calendar. Come to think, won't there eventually come a time when the Islamic and Gregorian calendars identify using the SAME number?
Cubi: Hey, yes, try to figure THAT one out at home!
3: HEBREW CALENDAR
Sexi: Now, the Hebrew or Jewish Calendar is lunisolar, like the Chinese. A year is twelve months of up to 355 days, but again, they add an extra month every few years. That said, their year is not aligned with the Chinese.
Para: Also, their calendar days begin at sunset, like the Islamic calendar. We're currently in year 5774, which has been counting since creation. Or, on comparative Gregorian terms, we're looking at a start in 3760 BC.
Quinn: Information! The Hebrew New Year  Rosh Hashana  is observed in their seventh month. This generally occurs in Gregorian September, depending on the new moon.
ArcTan: Stop saying 'Information'! Also, wait, does that mean the Hebrew calendar updates its year... with the start of the seventh month?
Para: Correct. The start of the religious year  Nisan, the first month  occurs (by Gregorian terms) in March or April. But it remains within the same numbered year as the month previous.
CoTangent: Interesting. So is this where we make the connection between American Thanksgiving and Hanukkah  the latter occurring in the ninth month of the Hebrew Calendar? Or, wait, should I have said Chanuka there...
CoTangent: I thought I was articulating, not spelling.
Sexi: Here's the thing. Chanukkah was declared a Jewish national holiday 2178 years ago, and always occurs on the 25th of Kislev. This year, that's in Gregorian November... meanwhile, American Thanksgiving keeps bouncing around from date to date, depending on Thursdays.
Tangent: As Adam Savage might say, 'There's your problem!'
4: ABOUT AMERICAN THANKSGIVING
Para: Ooh. Do you think we've been confusing anyone by constantly saying "American" Thanksgiving?
Cubi: Hey, who cares, people should know there's a Canadian Thanksgiving too. Hey, did you know, Canadians used to celebrate in early November  but in 1957, Parliament shifted the holiday back into October? Hey, they wanted to avoid it ever being the same week as Remembrance Day... that being November 11th.
Quinn: Infor... hm, fun fact?
ArcTan: Tolerable.
Quinn: American Thanksgiving is presently on the FOURTH Thursday of November. That WASN'T always the case, and moreover, the reason for the change is  Christmas Shopping!!!
CoTangent: What now?
Sexi: American Thanksgiving USED to be on the LAST Thursday of November. But in 1939, that meant November 30th. So, to create an extra week of Christmas shopping, in the hopes of stimulating the economy, Franklin D. Roosevelt said Thanksgiving would be November 23rd, 1939.
Tangent: Um. Wouldn't there end up being the same amount of shopping, it merely gets spread out over five weeks instead of four?
Cubi: Hey, don't think like a mathematician, think like a politician.
Para: The upshot was SOME states celebrated the 23rd, and others the 30th. But it gets better. In 1940, the last Thursday and the fourth Thursday were both November 28th. Yet Roosevelt declared Thanksgiving would be the 21st!
ArcTan: Let's go back to talking about global calendars, that made more sense.
Cubi: Hey, it's fine, in 1941 the House put forth a law to say the LAST Thursday would be Thanksgiving.
Sexi: Of course, the Senate amended it to be the FOURTH Thursday before it passed. Despite that, Texas was still celebrating the LAST Thursday as late as 1956.
Quinn: Why is 1956 important, you ask? Information! 1956 is the time previous to today when Hannukah and Thanksgiving coincided! Both were, by Gregorian Calendar, on November 29th, 1956. Sort of.
Tangent: You mean, assuming you lived in Texas. And are counting based on the first candle of Hannukah, not the second  as we are this year, 2013.
Para: For reference, they also coincided on November 28, 1918 and November 29, 1888. Granted, the candle count varies  remember Hannukah starts with sunset tonight, November 27th, whereas American Thanksgiving is tomorrow.
Sexi: Theoretically, the intersection is due to happen again in 2070. Notably, Chanuka is drifting out of November  that's why the dates are becoming less frequent.
CoTangent: Hm. That was interesting. Thank you.
Quinn: Now, with the Thanksgiving holiday dealt with, can we talk Canadian football to counter "American football"? By which I don't mean soccer.
Quinn: My point exactly.
Sexi: Kind of a one dimensional point, don't you think?
 Information from many websites, though primarily:
http://www.webexhibits.org/calendars/calendarchinese.html http://www.timeanddate.com/calendar/info.html http://www.chabad.org/holidays/chanukah/article_cdo/aid/2343364/jewish/ChanukahandThanksgivingABriefHistory.htm

posted Sep 29, 2013, 5:54 AM by Gregory Taylor
ax^{2} + bx + c presents....
"Don't Stop With Riemann" (*aka "Don't Stop Believin'", Journey)
 Official video: Unable to locate (Copyright pull?) Karaoke version: YouTube Here***
Estimate is low, I guess. We want the area underneath this curve.
Just approach from right, Estimate is high, how trite. We want the area underneath this curve.
(Instrumental) (Glee option call out: "Riemann!")
A manner to approximate?A summation to contemplate! Every trial helps us to refine It goes on and on and on and on...
{Refrain:} Segments, added. Up or down along this curve Their width we'll narrow, out of sight. Limit's useful, telling us to stop and observe Gives us answers which are right!
Working with unending sums,Series is what this becomes. Breaking vastness into discrete parts We'll comprehend. Some see now, some years hence, Some may only grasp a sense! But the learning never ends, It goes on and on and on and on...
{Refrain:} Segments, added. Up or down along this curve Their width we'll narrow, out of sight. Limit's useful, telling us to stop and observe Gives us answers which are right!
Don't stop with Riemann, Try some integration! Limits, useful, ohhhhhhh.
Don't stop with Riemann, Try some! Limits, useful, ohhhhhhh!
(repeating option)
Graph pics from this site: Explore with your own sums here: 
posted Sep 25, 2013, 4:19 AM by Gregory Taylor
ax^{2} + bx + c presents....
"When Can We Cancel" (*aka "With Or Without You", U2) Source Lyrics Video: YouTube Here
See the fraction on your page,Feel the brain start to engage. But wait. Think through.
Factor first, and check the whole, To be common that could be your goal. Yet it comes. This next cue.
So can we cancel? When can we cancel?
Through it all, steps must equate, Don't try to rush, you won't be late. Yet I'm worried for you.
When do you cancel? When can we cancel, ahh... I hear screams: Now can we cancel?!
(Instrumental) And it seems to multiply And it seems to multiply And it seems, and it seems, And it seems to multiply...?
You're adding here. Missed factors too, let's stop and check. Sub in an x, and See if it's still true!
And it seems to multiply And it seems to multiply And it seems, and it seems, And it seems to multiply!
When can we cancel? WHY can we cancel  Aha! Check your schemes, For why things cancel! Soooooo... Ohhhhh.... Knooooow....
Why can we cancel? Why can't we cancel, oho! Know the means, For why we cancel, When we will cancel.
Ooooooh!
(Instrumental)

posted Sep 22, 2013, 5:14 AM by Gregory Taylor
[
updated Sep 22, 2013, 5:20 AM
]
ax^{2} + bx + c presents....
(We Can't Find No) "Matching Fraction" (*aka "Satisfaction", The Rolling Stones)

*** [ROOT(2)]: We can't find no... matching fraction. We can't find no... matching fraction. But I try ... and I try ... and I try ... and I try! I can't find one! I can't find one!
When I'm wantin' decimals, And that number has no ratio, (And) I'm soundin' so skeptical! About this exact calculation, Supposed to use new classification? I can't let go! Irrational! Hey hey hey! It's so risque!
[PI]:We can't find no... matching fraction. We can't find no... matching fraction. But I try ... and I try ... and I try ... and I try! I can't find one! I can't find one!
When I'm usin' my circles, And a plan forbids me rounding! First, that is irony! But we, know 2pir, is circumference, That claim is our math decree? I can't let go! Irrational! Hey hey hey! It's so risque!
[PHI]:We can't find no... matching fraction. We can't find no... decimal faction. But I try ... and I try ... and I try ... and I try! I just get ones! I can't find one!
When I'm seein' symmetry, And I'm hearin' this, "golden ratio", And it's found in Fibonacci! This tells me, maybe better attack, the new technique, Soon you see I'm such, a numbers geek, We must let go! Irrational! (But) Hey hey hey! It's so risque!
We can't find one! We can't find one! We can't find one... irrational! No matching fraction... no matching fraction... no decimal faction... We can't find one!

posted Sep 18, 2013, 4:17 AM by Gregory Taylor
ax^{2} + bx + c presents....
"Makin' a Graph" (*aka "Stayin' Alive", Bee Gees) featuring PARAB and EXPONA! (The "Be Exes")

***
Well, you can tell by the way I made my plot, It's Cartesian: four quadrants sought. Make a chart and choose an x, I can solve for y, It's not complex. And now, it's all right, it's okay, 'Cause we will put this on display, So we can try to understand Relationships in this math strand!
Chorus: Whether a relation or in function notation, We're makin' a graph, makin' a graph Points are in the table, and join them if you're able, When we're makin' a graph, makin' a graph X, Y, X, Y, makin' a graph, makin' a graph X, Y, X, Y, makin' a graaaaaaaaph!
Well now curves go low and they go high, And with the equations, you will see why. Got an asymptote that may appear If you have to ask, you may know next year. You know, it's all right, it's okay, We'll plot more points another day, So we can try to understand Relationships in this math strand!
Chorus: Whether a relation or in function notation, We're makin' a graph, makin' a graph Points are in the table, and join them if you're able, When we're makin' a graph, makin' a graph X, Y, X, Y, makin' a graph, makin' a graph X, Y, X, Y, makin' a graaaaaaaaph!
To find solutions: A graph can help you. A graph can help you, yeah. To find solutions... (beat) A graph can help you, yeah. (We're) Makin' a graph!
Well, you can tell by the way I made my plot, It's Cartesian: four quadrants sought. Make a chart and choose an x, I can solve for y, It's not complex. And now, it's all right, it's okay, 'Cause we will put this on display So we can try to understand Relationships in this math strand!
Chorus: Whether a relation or in function notation, We're makin' a graph, makin' a graph Points are in the table, and join them if you're able, When we're makin' a graph, makin' a graph X, Y, X, Y, makin' a graph, makin' a graph X, Y, X, Y, makin' a graaaaaaaaph!
To find solutions: A graph can help you. A graph can help you, yeah. To find solutions... (beat) A graph can help you, yeah. We're makin' a graaaaaaaaph!
To find solutions: A graph can help you. A graph can help you, yeah. To find solutions... (beat) A graph can help you, yeah. We're makin' a graaaaaaaaph!

posted May 19, 2013, 5:54 AM by Gregory Taylor
[
updated May 19, 2013, 6:17 AM
]
Slope: Slope here, to talk to everyone about dividing fractions.
Maud: Oh! I was going to put together a video on that...
Slope: Too late.
Cosine: Um, why bother dressing that way? Everyone knows your other identity now, it's...
Slope: Sis, I love you, but hush.
Slope: Yes, that would make sense.
UNCOMMON DENOMINATORS
Slope: Fractions are a lot like percents. They make NO SENSE unless you have the context of the WHOLE.
Lyn: Give us a "for instance".
Slope: Hey, Circe! What's bigger? Three quarters of a dollar, or half a hundred dollars?
Circe: Obviously three quarters is bigger than a half!
Slope: Ok, thanks, have fun with that then!
Lyn: But hold on. In mathematics, we're basing fractions off of the numerical value 1, in order to be consistent.
Slope: You're suggesting we reject real life examples? I think that's heresy or something.
Lyn: Actually I'm suggesting that for the purposes of this article we assume we're talking about the same "whole".
Slope: Aw, Linear, it's like you can read my mind.
CASE 1: ADDING FRACTIONS
Slope: I know. I said I'd talk dividing. We're building suspense.
Lyn: Okay. Then to add fractions, you need a common denominator.
Slope: No, you don't.
Lyn: Now THAT is heresy.
Slope: What's three quarters plus one half? I draw a picture. It's "two and onehalf" halves, which is one and a half a half, also known as one and a quarter.
Lyn: That's... wait, what? I'm confused.
Slope: THAT'S why we have the common denominator. So that when I give an answer, it's not fractions on top of fractions, or mixed numbers, or some other standard. Lyn: Ah! We want a simplest form.
Slope: More than that. If we have a common denominator, THE WHOLE DOESN'T MATTER ANYMORE. You're no longer comparing dollars to donuts, you're comparing cents to cents.
Lyn: Makes sense.
Slope: Yes. I should know better than to give you straight lines.
Lyn: My lines are already straight.
Slope: So, let's say we add 3/8 to 4/7. Different pieces of one. Messy. But at the same time, that's just 21/56 and 32/56.
Lyn: I see! Now that they're the same parts of one, we ignore the whole. 21 plus 32 is 53.
Slope: In fiftysixths.
Lyn: Yes. Or whatever whole denominator is worked out earlier.
CASE 2: MULTIPLYING FRACTIONS
Lyn: So, multiplying is where we're all "multiply the top and the bottom".
Slope: Actually, we can use a common denominator.
Lyn: We can? I mean, of course we can.
Slope: Just remember that what we're doing is taking a fraction OF a fraction. Here, multiply 1/2 by 1/3.
Lyn: Common denominator means 3/6 by 2/6. Multiply the top, keep the denominator and we get... six sixths? The heck?
Slope: We get six sixths where SIXTHS is the new whole. Thus 1/6.
Lyn: You lost me.
Slope: Then forget the common denominator, we'll come back to it. Given 1/2 times 1/3, even though both are with respect to one, the ANSWER would be with respect to ONE HALF, the first fraction. Lyn: Ah, now I get it.
Slope: No you don't.
Lyn: No I don't.
Slope: 1/2 X 1/3 means HALVES is now my whole, and I carved out a third. Yet that third is a sixth... with respect to 1.
Lyn: Okay... so the answer is 1/3 with respect to halves, or 1/6 with respect to 1.
Slope: Alternatively, consider 1/3 times 1/2. The ANSWER is 1/2 with respect to thirds, or again 1/6 with respect to 1.
Lyn: True, but... sorry. I feel like I should be getting this.
Slope: Let's go again. Do 1/3 times 3/4.
Lyn: Okay. The answer is 3/4  with respect to 1/3. But I want a "whole", so carve up said third into quarters. A third in quarters means I've got twelfths. Oh, but you said we've got three quarters, so I should have multiplied by three... it's 3/12. Or 1/4.
Slope: Exactly. Now consider 3/4 times 1/3.
Lyn: This time I'll start by multiplying numerators. So the answer is 3/3  with respect to 1/4. I carve up my quarter into, uh, a whole. Still 1/4. I guess it makes sense I get the same answer.
Slope: Precisely. Now do 3/8 times 4/7. Lyn: Oh bloody...
Reci: Lyn! Be good. No swearing.
Slope: Hey, Reciprocals! No eavesdropping!
Cotangent: ...Sorry, we're going...
Lyn: Ahem. Well, the answer is 12/7 with respect to EIGHTHS. I carve up my eighths into sevenths, and 12/7th of an eighth is 12/56 of the whole. Or 3/14.
Slope: Now try it with a common denominator.
Lyn: Then as before, we use 21/56 and 32/56. The answer is 672/56ths... but that's not of a WHOLE, it's of a 56th. So 672/3136 of the whole. Or 3/14. Huh.
Slope: Which I grant is the same as if you didn't bother with the common denominator.
Lyn: Or if you simply multiply the top and the bottom. But now it's more clear.
Slope: Which brings us back to...
CASE 3: DIVIDING FRACTIONS
Slope: Here the common denominator is a lot more helpful.
Lyn: I've got this. 3/8 divided by 4/7. Same as 21/56 divided by 32/56. The answer is 21/32 fiftysixths, where... wait, is fiftysixths the whole now? Because the answer IS 21/32.
Slope: That's one way to see it. Alternatively, 1/3 divided by 1/2... you're giving to a "whole", and doing it in "halves". 1/3 + 1/3 = 2/3, which is the answer.
Lyn: Or 1/3 divided by 1/5... you're giving to a "whole", and doing it in "fifths". [1/3 + 1/3 + 1/3 + 1/3 + 1/3] = 5/3
Slope: Now, with 1/3 divided by 2/5, you'd have to cut your answer in half, because you've given to TWO "wholes". 5/3 halved is 5/6.
Lyn: Thus 3/8 divided by 4/7, you're giving to a "whole", and doing it in 7ths, which gives [3/8 + 3/8 + 3/8 + 3/8 + 3/8 + 3/8 + 3/8] = 21/8 but that's four wholes, so [21/8][1/4] = 21/32.
Slope: Four wholes causing a quarter, which explains a bit of why the reciprocal is used in dividing.
Reci: I heard reciprocal. Did you call us back? Circe: Hey, Slope! Did you scam me earlier?!
Lyn: Come to think, we can see fraction division as simply dividing the numerators and dividing the denominators. Which you can even do when dividing two rational functions, with variables... just track restrictions.
Slope: Maybe common denominators were even the norm, until someone with a multiplication fixation noticed that taking the reciprocal of the second helped the common denominator to "cancel out"! Which in truth means there's a multiplication of one.
ArcTan: Did someone call for inverses?
Cotangent: No, reciprocals.
Root: Hey hey, is this a party??
ArcSin: For SCIENCE!
Hyper: Damn it Nisano, stop using my schtick!
Lyn: Aaand we're off track enough to be done here. Except what about the issue of common numerators? Where are they helpful?
Slope: People can try that at home!
 With thanks to @nik_d_maths whose tweet inspired me to finally write this. For more reading:

