This MitPY comes from frequent commenter, John Friend:
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
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:
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.
The Mystery of Formica’s Triangles
by Tom Peachey
With ideas borrowed from Edwin Abbott, W. W. Sawyer and Frank Dickens.
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 …”
“Sometimes muddy …”
“They still have kings!”
“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.”
“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.”
“And it’s the path that light takes.”
“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.”
“If it rotates all the way back to the start, that is a rotation of 360 degrees.”
“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.”
“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.”
“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.”