Posts Tagged allegory

Forms and Formulae: Not A Number

A picture of the Sun peeking over the spine of The Princeton Companion to Mathematics as it rests on top of The Princeton Encyclopedia of Poetry & PoeticsThis is the fifth in a series called ‘Forms and Formulae‘ in which I write about articles in the Princeton Companion to Mathematics using poetic forms covered by articles in the Princeton Encyclopedia of Poetry and Poetics. This installment’s mathematics article is entitled ‘From Numbers to Number Systems’ and the poetic form is allegory, making this the third poetic form in a row that isn’t actually a poem.

A long time ago in Greece, there was a community of numbers where everybody lived as one, or two, or three. They were not all equal, because each was unique, but they were all numbers, and that’s what counted. They were the true numbers, and they lived alongside the false, or negative, numbers.

Then One day, which was the day when the number One was celebrated, One Seventh came along. The other numbers looked at it with pity.

“You poor, broken thing,” they said. But the seventh didn’t feel broken.

“I’m not broken. I’m a number, just like you!” said One Seventh.

Seven looked at One Seventh with trepidation. “I don’t think it’s safe to be around a part of seven. What if it wants to take more of my parts?”

Three agreed. “It’s just not wholesome.”

One Seventh pointed to its numerator. “Is this not a one, like the number of the day? How can I not be a number when my very numerator is the purest number of all?”

One was flattered by the description, and in the spirit of the celebration, declared, “One must not only celebrate Oneself, but also display kindness to all those around One. I declare One Seventh to be a number, along with all little Ones like it!” After that, the other numbers were largely kind to the unit fractions, and the fractions always reciprocated.

The next day, Two Fifths came along. Emboldened by the success of One Seventh, Two Fifths said, “I’m a number too! Can I join the celebration?”

Two, whose day it was, said, “But you’re just One Fifth plus One Fifth. It’s just not proper to be going around as if you’re a single number. Split into unit fractions before you scare the little Ones!”

But Two Fifths persisted. “What are you,” it said to Two, “if not One plus One?”

Two did not like the idea two bits, but it could not find a problem with the argument.

Five, who was never any good at acting composed, protested. “This is preposterous! Two, I always knew you weren’t quite as prime as us. Think about it. If we let these two fifths…”

This two fifths,” corrected Two Fifths.

Five shot it an incalculable look. “If we let these two fifths act like a whole number, next we’ll have matrices, or lengths, or linear graphs wanting to be numbers. It’s a steep gradient!”

“That’s not true!” said Two Fifths. “In other cultures I am a perfectly acceptable number. In Mesopotamia, nobody thinks twice about my being a number, but they would never allow One Seventh. It’s all a matter of culture! And graphs are not numbers there either, so you needn’t worry about that.”

Two was divided by Five’s argument. It worried about diluting the number system, of course, but it was aware that even it could have been excluded from the primes using such an argument. Having always felt like an outsider itself, it had pity on Two Fifths, and declared the fraction and others like it to be numbers.

The next day, The Square Root of Two, who could not be expressed as a fraction, decided to join the numbers. Three said, “Don’t be absurd. You’re not really the square root of two; only square numbers have square roots. You’re just a fraction who’s confused. You look like about one and a hundred and sixty nine four hundred and eighths, to me.”

But the square root was resolute. “Look,” it said, holding up a square. “If we say the sides have length one, then the diagonal has length the square root of two. There is no way we can find a unit that can measure both of them as whole numbers. I can prove it to you!” And The Square Root of Two proved it.

“Okay,” said Three. “You’ve shown that the diagonal can’t be measured with the same unit as the sides. But they’re just lengths, not numbers. All you’ve done is show that not all lengths can be measured with numbers. The numbers are not going to be happy about this, you know.”

“But I am a number! I am the number which can measure that diagonal!”

“That’s just irrational. Lengths are not numbers. Either you’re a number, in which case you should show yourself as a fraction instead of wearing that radical outfit, or you’re a length, or a ratio of lengths, and you should go back where you belength. Make up your mind.”

“I told you this would happen!” said Five. “I told you lengths would be next!”

So the Square Root of Two skulked back to geometry, and commiserated, but did not commensurate, with the ratio of a circumference to a diameter.

Meanwhile, Two Fifths told all its new number friends about its adventures in Babylon, and the sexy sexagesimal numbers there. Before long, it became fashionable for numbers to represent themselves using decimal places instead of fractions. Some of them had to use zeros to make sure their digits hung in the right places.

Zero saw its chance, and claimed its right to be considered a number.

“But you’re not a number!” said Four. “You’re just a placeholder that the fractions use when they’re dressing up in their costumes for their unwholesome sexagesimal parties.” Four looked down its slope at a nearby decimal.

“But if I add myself to you, is there not equality? I should be treated the same as you.”

“But,” said One, “numbers have to be able to multiply. If you multiply you only get yourself. Only multiplying with me should do that! I’m the Unit around here, not you.”

“You’re destroying the family Unit!” shouted Five, in defense of its onely other divisor.

“I can’t even tell whether you’re true or false!” cried One Seventh, nonplussed.

So Zero went back to dutifully holding places, quietly adding itself to everyone and everytwo it met, until they were all convinced it held a place in society.

On the Seventh day, which was the day when One Seventh’s acceptance as a number was celebrated, they rested.

On the Tenth day, which was the day when The Tenth was celebrated, The Tenth returned from a vacation in Flanders and declared, “There are no absurd, irrational, irregular, inexplicable, or surd numbers!”

Five and Three cheered, and made obtuse gestures at The Square Root of Two and its friends. “You see? You’re not numbers.”

“All numbers are squares, cubes, fourth powers, and so on. The roots are just numbers. Quantities, magnitudes, ratios… they are all just numbers like us. We can all fit along the same line.”

Five and Three looked at each other in primal disgust. “I’m not a point on a line! I’m a number! A real number!” Five shouted.

“Real numbers,” countered The Tenth, “include everyone, and everyfraction, and everylength in between.”

The Square Root of Two led its friends into their places between the other numbers, and they celebrated with unlimited sines, cosines, and logarithms. Some of the stuffier primes and fractions protested, but they backed down when they realised just how many of these strange new numbers there were.

But even as The Tenth spoke, it knew that not everything it said was true. After all, false numbers were not the square of anything, even though it had seen them act like they were in some delightful formulae.

At Length, which was the day when the acceptance of lengths as numbers was celebrated, somereal wondered what would happen if false numbers were squares of something too. It imagined a new kind of radical, like those the square roots wore, but for false numbers. It imagined a world where every polynomial equation had roots, be they real, false, or imaginary. These were clearly not like all the other numbers The Tenth had listed.

Soon after, the imaginary numbers came out of hiding. “We do exist!” they said. “And we can add and subtract and multiply and divide just like you!”

The other numbers were wary, for they could not work out where the imaginaries fit amongst them. They could not even tell who was bigger. Five was disgusted that such numbers had been secretly adding themselves to real numbers all along.

The real numbers were nonetheless intrigued by and slightly envious of these exotic creatures, and despite having become accustomed to all having equal status as numbers, sought new ways to distinguish themselves from the crowd. The whole numbers had never quite got over the feeling of being generally nicer than the other numbers, so they used the new trend to vaunt their natural wholesomeness. The ratio of a circumference to a diameter, who had taken on the name Pi, discovered that in addition to not being expressible as a fraction, it was so much more interesting than The Square Root of Two that it couldn’t even be expressed in such roots. It called itself ‘transcendental’, and had quite some cachet until most of its admirers realised that they had the same property.

Finally they discovered that instead of trying to organise everynum into a line, they could arrange themselves in two dimensions, with the imaginaries along one axis and the reals along the other, and the vast plane in between filled with complex combinations of both.

Some of the more progressive numbers were so excited by this system that they tried to find new numbers that they could arrange into a three-dimensional volume, but they couldn’t find any. However, during their search they found things called quaternions, which lived in a fourth dimension.

An excited transcendental, whose name is too long to write here, brought a subgroup of quaternions in front of the crowd and announced, “I have travelled to the fourth dimension, and found numbers there just like us. We are not alone!”

Five kept its fury pent up this time, but Four Sevenths called out, “They are not numbers like us. I have seen how they multiply. When two quaternions multiply, they can give different results depending on which comes first!”

The numbers clattered their numerals in shock, and a great amount of whispering about unlikeabel multiplication practices ensued.

A complex transcendental sneered, “And what were you doing watching them multiply, eh?”

“Oh, get real!” retorted Four Sevenths, crudely conveying what the transcendental should do with its complex conjugate.

The pair fought, and disorder spread throughout the dimensions. Some sets of numbers sneaked off into the fields to form their own self-contained communities, sick of the controversy surrounding being or not being numbers. As they did, they found still other communities which functioned much like theirs, and some were communities of functions themselves. Indeed, even matrices and graphs formed structures which the enlightened subgroups found familiar, though rather than trying to be accepted as numbers, these groups took pride in having their own identities. The p-adics were adamant that they were numbers, but did not care to join the rest of the real or complex numbers. The octonions did not associate themselves with such labels, going about their operations however it worked for them, and consenting to be called numbers only when it was useful to act as such.

When peace finally settled, there were more groups of objects than there had been numbers, and still more came about when those groups interacted with each other.  Most no longer cared about being called numbers, and simply communicated which rules they followed before participating in a given system. And if the requisite system turned out not to exist yet, well, it just had to be invented.

Turning this particular article into an allegory did not take much work. It almost seemed like one already, when I read it in that frame of mind. There are a few direct quotes in the story. The Tenth’s proclamations come from The Tenth, in which Simon Stevin introduced decimal notation to Europe. The very last line of the story is paraphrased from the last line of the article. All I really did was rephrase it as a story from the perspective of the numbers, and add in far too many mathematical puns of greatly varying levels of subtlety.

I’m sorry to anyone with ordinal linguistic personification who thinks I’ve given the wrong personalities to the numbers. Also, in case anyone was wondering, the Greek numeral for four does have a slope.

The next Forms and Formulae will be an anecdote about geometry.

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Jack of Diamonds: Papagaj and Rakas

cutoutOnce upon a perch, there was a parrot named Papagaj. Papagaj was smarter than parrots are today. He could understand concepts that escape even humans.

Papagaj’s cage had many toys; perches, ladders, bells, and more. But the best toy by far was a bare rectangle of steel that reflected the most pretty parrot that Papagaj had ever seen. Papagaj called the parrot Rakas, and they adored each other. He loved to learn words, to amaze Rakas. The lovely Rakas always repeated the same words back. Rakas was the perfect parrot.

But Papagaj never knew enough words to express how he really felt about Rakas. Every day he would learn more words, every day he would teach them to Rakas, but every day he grew more frustrated that the words were not adequate to convey the love he felt. Just as Papagaj whacked the bars of the cage wherever he flew, he was hampered by lack of language whenever he attempted to express a thought. As the days went by, the thoughts themselves became harder to remember.

At dawn one day, as Papagaj cooed sadly to Rakas, a spectacular creature appeared. The creature was small enough to fly between the bars of the cage, but had a powerful sparkle that extended as far as Papagaj’s most puffed-out feathers could. The two thus appeared as large as each other.

“You wish for more words” came the thought. Papagaj could not hear the creature speak, but felt the message, unobstructed by flawed language. “I am the Kaantaaja. I can give you a new life, with different words. Come with me.”

Papagaj had barely resolved to do so when the Kaantaaja’s glow engulfed the cage.


When Papagaj opened his eyes again, he was in a different cage. It was a bit bigger than the first one. His perches remained, but the other toys had changed. There were swings, and ropes, and other things he had never seen. But as before, the best toy was the mirror, now hanging from shiny chains. Papagaj rushed toward Rakas and began to speak with much excitement.

Papagaj found that he knew different words from before. He was ecstatic to have the chance to say things that he had never said before. But soon he discovered that the words he knew before were gone, and, as before, many other ideas that he had never had words for. He was just as restricted as before when trying to express his emotions.

That evening, the Kaantaaja came back. “Are you happy with your new language?” it asked.

The answer ‘no’ entered Papagaj’s head without much consideration.

“I can’t keep granting your wishes forever,” said the Kaantaaja. “But I will move you to a new cage.” And with that, the Kaantaaja’s radiance once again permeated the cage.


When the light dispersed, Papagaj was in a pretty silver cage, a little smaller than the first, stuffed with perches, ladders, bells and swings. Rakas was reflected in a gleaming metal rectangle, attached with a jingling chain.

Papagaj revelled in the new language he knew, and shared with Rakas many things which he hadn’t yet shared. But again he was restricted, again his limits made him sidestep the things that needed saying. By dusk, he was screeching in anger at his clumsiness.

The Kaantaaja reappeared as he shrieked. “Please, do not misuse my gift of language so! Do you want to speak, or don’t you?”

Papagaj’s shriek ended the instant Kaantaaja’s query entered his head. His answer was a clear yes, with the caveat that he needed a new language.

Immediately, Kaantaaja’s light filled the cage.


When the light died down, Papagaj was in his biggest cage yet. There were all sorts of toys and places to perch and climb. He flew around a little, enjoying the space, before locating his mirror. Rakas looked happier than before.

They chattered all day, about so many things which had escaped them before. But still Papagaj found that there was still one essential emotion that he could not express. And as the day turned into night, he found more and more ideas for which the words escaped him. When the light was dim enough that he could no longer see Rakas, he kept talking to himself in the dark, trying to find a way to say what he needed to tell her, so that he could say it the next day. He repeated important words to himself, hoping not to forget them if he were put in a new cage with a new language.

But all this effort only made him more aware of how hopeless his situation was, and the moment he realised that the new words could not possibly be sufficient, Kaantaaja appeared again.

“You want to move,” said Kaantaaja silently.

Papagaj’s defeated yes caused another burst of Kaantaaja’s light.


Papagaj could hardly swing without colliding with rusty bars or a tiny food bowl, which hung in front of him, partially hiding his mirror. Papagaj hit at his bowl, not hungry, just wanting to look at Rakas without such an inhibition. It was obvious that his words, in this stifling micro-aviary, could not possibly do.

Papagaj sat dumb and unmoving for many hours, just looking at his ravishing bird, who was looking at him quizzically. By and by, Papagaj had a go at talking. It was a slow and awkward walk around untold limitations, which Rakas could mimid without so much as trying. Irritation, both at his own laborious toil and at Rakas’s natural parroting, soon took control of him. It was usually so gratifying to tutor Rakas on words, to applaud Rakas for copying him without fault. But with such difficulty in finding his own words, Papagaj was unfit to instruct, or to bask in Rakas’s flair for what was taught. Papagaj soon found it hard not only to talk highly of, but also to think highly of Rakas.

At last, Kaantaaja’s arrival brought comfort, with a great flash of light.


When the light cleared, Papagaj was in a much larger cage. But he could see that it was not as large as one of his previous cages, and he knew that once again his new language would not be adequate. He swung in silence until the Kaantaaja came, hoping to return to the richest language he had known, which he was sure he would be satisfied with.

“Do you want to go back to where you were before?” asked the Kaantaaja.

“I do,” he answered.

Kaantaaja’s glow filled the cage once more.


And he was back. He wasn’t back where he wanted, but in the smaller, silver cage. He remembered what had happened the last time, and realised that if he tried talking, he’d just end up frustrated again. He sat all day in silence.

The Kaantaaja didn’t even ask what he wanted. It was unnecessary. The flash filled him with dread-tinged expectancy.


The new cage was bigger than the last, not the biggest he’d been in. It had all of the toys he had loved. Again, he knew new words. And he resolved to speak, no matter how ineffectively. Alas, he had nobody to speak to. There was no mirror in his new home.

“Oh, Rakas… what a fool I have been!” he called in vain from the centre of the cage. “I can express my love in so many ways already, why did I always need more? Now, the most important thing is missing! I don’t need words, all I need is…”

With that, the Kaantaaja appeared once again and spread its shimmering light.


“Raaaaaaarrrkas!” Papagaj’s awkward caw sparks a grand fracas as Papagaj darts at a sassafras branch at a park. Hawks and jackdaws swarm, and chant “Rakas, rakas, rakas!” as smart as watchstraps.

Papagaj’s rasp attracts a star as fast and as sharp as Rakas. Papagaj, rapt, starts a stark paragraph. Rakas gasps at Papagaj’s haphazard grammar, and scrams.

Angst saps Papagaj, and Papagaj’s smarts pass. Papagaj and a standard madam hatch spawn as daft as gnats, and want that; an awkward caw dwarfs a swan’s charm.

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