## MitPY 11: Asymptotes and Wolfram Alpha

This MitPY comes from frequent commenter, John Friend:

Dear colleagues,

I figured this was as good place as any to ask for help. I’m writing a small test on rational functions. One of my questions asks students to consider the function where and to find the values of for which the function intersects its oblique asymptote.

The oblique asymptote is so they must first solve

… (1)

for . The solution is and there are no restrictions along the way to getting this solution that I can see. So obviously .

It can also be seen that if then equation (1) becomes which has no solution. So obviously .

When I solve equation (1) using Wolfram Alpha the result is also . But here’s where I’m puzzled:

Wolfram Alpha gives the obvious restriction but also the restriction .

emerges naturally (and uniquely) from this second restriction and I really like that this happens as a natural part of the solution process. BUT ….

I cannot see where this second restriction comes from in the process of solving equation (1)! Can anyone see what I cannot?

Thanks.

## Guest Post: The Mystery of Formica’s Triangles

The following story is by frequent commenter and friend, Tom Peachey. Tom requested that we post his story on our site. We’re more than happy to do so, although we think the story is good enough and funny enough to deserve a more respectable home. Here is Tom’s explanation of how he came to write the story:

My GP has kept up an interest in general science. Some years ago he commented that he was surprised to read that the universe is curved. I resolved to write out an explanation and have finally produced what you see below. It was a no-brainer to use Flatland, but my early attempts were pedestrian. Then I added two characters to voice the two sides of mathematics: one free imagination and the other hard proof.

Marty (and my GP) are encouraging me to publish officially but I can’t think where. So I’m now looking for feedback. In particular, what is the target audience? Please do not send out an electronic version further. I can supply a pdf for printing if anyone would like to try it on their students.

## by Tom Peachey

With ideas borrowed from Edwin Abbott, W. W. Sawyer and Frank Dickens.

### 1. Formica the Flant

Once there was a world who’s name is Forgotten. The highest lifeform there were some ants, but not just any ants. These ants were completely flat. And I mean completely – zero thickness. That was okay because their world was completely flat too. These ants could move forward and back, and side to side, but not up and down – there was no up and down. They could not even imagine up or down.

I forgot to tell you, the ants are called flants in their language, Flantspeak. The picture here shows what they look like.

Despite their lack of thickness, flants are quite intelligent and live civilised and interesting lives. The hero of our story is a flant named Formica. He works as a surveyor. But not just any surveyor, Formica is fifth in line for Surveyor General. Currently he is in charge of a complete remapping of the Queendom. This is his big chance to get promotion to fourth in line.

At home Formica lives with his wife Mica. I should explain, when a male flant marries he takes the name of his wife, just adding the prefix For. They have no children yet, although they apply each year for an egg from the Royal Eggbank, but without success so far with the coin flip.

Mica works as a philosopher, trained in this by her maiden aunts, Saken and Skin. Mica’s work involves sitting and thinking deep thoughts. In Formica’s eyes this is not real work. He is a practical flant, scurrying about the Queendom looking useful. This year he was even chosen to measure the foot of Queen Titude CCCXIV. But Formica pretends to be interested in his wife’s work. That work seems to be thinking about impossible questions, such as why time only runs forward, and why is the universe just two-dimensional. Currently she is worrying whether the universe is infinite or bounded. An infinite universe just goes on forever which seems a waste. But with a finite universe, when you come to the end, there must be something beyond?

Long ago the Queendom had kings instead of queens. But they were brought up spoiled and reigned as gluttonous, cruel and dangerous. The “glorious revolution” replaced kings with queens but they turned out to be gluttonous, cruel and dangerous. Then it was decided to appoint queens at birth. Each would reign until the age of 10 when they would be replaced – before they got too dangerous. But the child queens rarely served a full term, dying from a surfeit of sugar. That’s why there is now an “aleatory monarchy”; all decisions are still made by the queen, but only by answering her advisor’s true/false questions by flipping the royal coin.

Despite having no real power, queens are widely adored, especially the cute ones. For example, the basic unit of length, the foot, is still taken as the length of the Queen’s foot. This is measured each year on the Queen’s birthday, and all recorded measurements are then adjusted to the new standard. This is one reason why Formica is so busy; each year he needs to convert all existing records.

† This name means “attached to Gotten” But no-one can remember what Gotten is.

‡ Not actually flipping, it’s more like spinning.

### 2. Mica is not Convinced

As we meet Formica he has just completed multiplying all distances in the database by the factor 0.949, and he has time to prepare for the Great Survey. Today he gets home tired and worried.

“You’re worried” says Mica, “what’s ahead?”

“You know I’ll be starting the Great Survey soon …”

“Yes leaving me alone with no husband and no children. Did you know that Tunate next door has had a new baby every year for 12 years running. What are the chances of that?”

“One in 4096” Formica said automatically. “Anyway, Forgerri and I have been testing the new violet laser prtrctrs. They are meant to be incredibly accurate and are very expensive. But it looks like they’re faulty”.

“So take them back to the shop.”

“Not so easy. They are imported from the Kngdm f Md.”

“Md” exclaimed Mica, “they are so backward”.

Yes but …”

“and dirty!”

“Sometimes muddy …”

“They still have kings!”

“But …”

“And they haven’t invented vowels yet!”

“Yes but they are famous for the accuracy of their instruments. That’s why we paid big money for their prtrctrs.”

Mica was starting to see the problem. “What do these prtrctrs actually do?” she asked.

“They measure angles.”

What’s ahead is Flantspeak for what’s up.

“And you want to measure angles because?…”

Formica was happy to talk about his important work. “This Great Survey will make a grid of triangles across the Queendom. We need to measure the triangles accurately.”

“Why?”

“Well, for example – so people will know exactly which foot of land belongs to which farmer. We measure lengths of our starting triangle and for all the other triangles we just measure angles and work out the sides using trigonometry.”

“So … these prtrctrs are not working?”

“They are working, but not accurately. Forgerri and I measured some large triangles, and the angles did not add up to 180 degrees. Always a bit greater.”

“Do you want them to add to 180 degrees?”

“Of course. According to mathematics they must add to 180.”

“Oh mathematics. I missed school the week they spent on that.”

Formica made a smug smile. One week to learn mathematics! He went to Her Majesty’s Special School for Very Smart Flants where they learned all mathematics in a day.

“But” continued Mica, “Aunty Saken did teach me algebra. Show me why the angles must add to 180. What is an angle anyway?”

Formica took out some chalk and drew a line segment. “This is a straight line.”

“What do you mean ‘straight’?”

Mica was in philosophy mode, but Formica was up for the challenge.

“It’s the shortest path between two points.”

“Um …”

“And it’s the path that light takes.”

“Go on.”

“Let’s rotate this line about one end.” He drew the new line.

“This makes an angle – and the measure of the angle tells you how much it has rotated.”

“Okay.”

“If it rotates all the way back to the start, that is a rotation of 360 degrees.”

“Why 360?”

“Well the ancient Babyloniants used 360.”

Mica was about to challenge, but Formica continued quickly.

“And our Queen has decreed it.”

That always won the argument. Mica just nodded. Formica continued.

“So if we just rotate the line half way, we get an angle of 180 degrees on each side.”

Mica nodded again; she was getting to understand this angle stuff.

Formica was now sounding pompous. “I will now prove that the angles in a triangle add to 180 degrees.”

“What is this prove?”

“A proof shows that something must be true, and shows why it is true.”

Mica nodded.

“Now a triangle is made of three straight lines. Making three angles.”

He drew the triangle and coloured the angles.

“I want to show that these angles add to 180 degrees.”

“Extend one side.”

“And at the red corner draw a line parallel to the opposite side.”

He was colouring the new angles to show that they added to 180,

but Mica interrupted. “What is a parallel line?”

Formica was unsure of this but tried to sound confident.

“Parallel lines never meet.”

“How do you know they never meet?”

“Well, you walk along them and check.”

“So if they are parallel, you could walk forever and never decide.”

Formica stroked his antennas for a while, then stomped to his shed and started hammering some nails.

### 3. A Day Out

The next day was Full Moon Day. No work was done throughout the Queendom. The flants in each village would gather at the local temple to chant prayers. Then each family would walk three times around the temple before adjourning to tend the graves of their ancestors. Mica found this difficult – the other families each had their troop of excited children. She could feel the pity of other flant mothers for her barren family. Or worse. An absence of children was widely believed to be punishment for bad deeds in a previous life. So she was happy when it was time to leave.

Formica was not so happy. The next stop was a feast at his Mother-In-Law’s nest. There, Mica’s mother would quiz him about the chances of grandchildren and discuss at length his shortcomings as a For. But today Formica was not concentrating on his own inadequacies. His mind kept wandering to triangles. Walking home completed a triangle – from home to temple to Mother-In-Law to home. By the time he turned into home he had an idea.

“I have a new proof for the angles of a triangle.” said Formica.

“Does it have parallel lines” replied an amused Mica.

“No. Just walking, and some algebra.”

“Good. I like algebra.”

Formica drew a picture. “This point is H our home.” “Yes.”

“And T is the temple – And M is your mother’s place.”

“Okay.”

“So we have a triangle.” He drew the triangle and coloured the three angles.

“Here T stands for the place of the temple. But I will also use T for the measure of the red angle.”

“And the same for M and H?”

“Yes. I want to show that T + M + H = 180.”

“Now suppose we walk from H to T. Then we turn toward M. We need to turn through an angle of 180 − T degrees.”

Mica inspected the picture. “Yes that looks correct.”

“At M we turn to walk home. This time we turn through an angle of 180 − M degrees.”

“Good again.”

“At home we turn again to face the temple.”

“This time 180 − H degrees.”

“Yes you get it. By this stage we have turned a complete circle, 360 degrees.”

Mica was excited. “Let me solve this.” And she wrote:

“So … they add up to 180.”

Formica reached over and added the standard way to finish a proof.

“Told you so.” he said.

“Let me think” said Mica, “The angles must add to 180. But when you measure them they do not.”

“Correct.”

“It’s as if the straight lines are bent.”

Formica shook his head. “Straight means straight. Bent means bent”.

### 4. A Flash of Inspiration

At work the next day Formica had this brain itch. Is straight really straight?

“Hey Forgerri! I have a new way to test the prtrctrs. No triangles!”

“How?”

“We put the angles together – like this.” And he drew a picture with three angles.

“Easy” said Forgerri, “no walking needed.”

And they set up a prtrctr and measured three angles like in the picture. Forgerri did the adding up.

“Exactly 180” he said – or rather within 1 second of arc, highly accurate. What is going on? When the angles are together they make 180 degrees, but in a triangle they add to more.”

Formica did not reply, he was transfixed, staring into space. Thinking.

“You should turn off the laser”, said Forgerri.

Formica looked across. “It’s not dangerous. If the beam hits a flant then they get a tickle and move out of the way.” He was amused. “As we speak, the laser might be tickling flants in Md – maybe even further.”

Just then Formica felt someone tickling his tail. He turned, but there was no one there. Then he saw the light. It was violet.

That was when Formica discovered how the universe works. Did he kiss Forgerri and do a little dance? Did he shout “Eureka” and run naked in the town? No, Formica was a serious flant, fifth in line for Surveyor General. He did however recite a little poem remembered from his schoolday when they learnt poetry.

“Then felt I like some Trumpian official
Who starts a war o’er something superficial,
Or, like a flea that meets some other fleas,
Silent, upon a Pekinese.”

People have been asking what happened with Mica and Formica. Well, Mica joined the team and went on the Great Survey. While on the road, Mica and Formica developed a new type of trigonometry that works for bent straight lines. It turned out that they were an effective collaboration; Formica would bubble with ideas and Mica would shoot them down, except when she couldn’t. They have become quite famous. The Queen has bestowed a family name – they are now Family Na-Pier. Not that they have time to enjoy their fame. Their life is dominated by baby triplets, three little girl flants: Getful, Give and Eigner. Formica says it must have been a triple-yolker, but Mica disputes this. She claims that someone slipped an extra child into the cot. And she darkly adds “Be careful what you wish for”.

Some readers might be wondering why I have told this story about these flat creatures in this flat world. I’m thinking it might be relevant to a problem in our real 3D universe. As you know, we have at last got messages from our colonies on Alpha Centauri and Lalande 21185. They complete a triangle with our Sun and we now have measurements of the triangle angles. It appears that the angles do not add to 180 degrees, in fact the total is slightly less. Go figure.

Na-Pier is Scots Flantspeak for No Equal.

## Which Republican is More Repulsive?

There’s the Republican who, even after the whipping up of a riot, continues to lie through his teeth, to steadfastly deny that Trump is a narcissistic, sadistic, psychopathic cult leader. Then there’s the Republican who pretends to have finally seen the light, who pretends that they didn’t recognise all along that Trump is a narcissistic, sadistic, psychopathic cult leader.

Which is more repulsive? The answer, of course, is “Yes”. Republicans are loathsome fuckers. All of them.

## Mathematics Books, Free to Good Homes

For a peculiar reason,* we have inherited a very nice collection of mathematics texts. They are mostly calculus/engineering texts and advanced applied mathematics, but there is a range, and there are a number of classics.

We are now looking for homes for these books. So, take a look at the photos below, and comment (or email me), indicating what you might like. Also feel free to pass on the offer to whomever you think might be interested. Here are the rules:

1) There are no rules: this is Calvinball. We’ll make it up as we go along.

2) The books are free, but you could consider making a donation to Tenderfeet. If you’re in Melbourne, we can arrange pick-up or drop-off, and otherwise I can mail the books, and we’ll figure out the postage somehow.

3) You can request as many or as few books/categories as you wish, with whatever levels of enthusiasm and specificity seem appropriate.

4) It is definitely not first come, first served. I’ll do my best to handle overlapping requests in a reasonable manner, with some preference given to starving students.

5) Would you like fries with that? If you express interest in a book, I might suggest similar books from the pile.

6) Most of the books should be identifiable from the photos, but feel free to ask for further details on any book. The books are roughly sorted into topics, so if you cannot identify a book, it’ll probably be similar in content to its neighbours.

7) See Rule 1.

Go for it, and thanks.

*) People are stupid.

UPDATE (06/01/21)

Thanks to everyone for contacting me. There are plenty of books still up for grabs, and people are still welcome to make requests. In a couple days, I’ll look to rationalise the requests thus far, and I’ll contact everyone to arrange the handovers.

UPDATE (09/01/21)

OK, I think I’ve replied to everyone who has requested books, to arrange pick-up/drop-off. If you think I missed you, please give me a nudge with a comment below. (There are still plenty of unclaimed books, particularly on mechanics, and modelling and the like.)

UPDATE (15/01/21)

OK, all the books have now been assigned. The plan is for one big Traveling Salesman tour of Melbourne on Wednesday 20/1. So, unless other arrangements have been discussed, please reply to my email to you, to confirm there’s a safe place to drop the books if need be.

FINAL UPDATE (04/02/21)

All done! All the books have been picked up or delivered or mailed. Along the way, it was great to meet, or re-meet, a bunch of you guys. (Hopefully the mailed books will arrive shortly, if they haven’t done so already. I’m sorry, but it was prohibitively expensive to mail them with tracking.)

Thanks to all of you who offered a home for these excellent but otherwise-God-knows-where books, and I hope they’ll be of interest and use. Thanks also, very much, to those who donated to Tenderfeet.