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.

]]>1. How does geometry relate to the world?

2. How do numbers (especially irrational numbers) relate to the world?

Chapter 1 answers the first and Chapter 4, on the basis of Chapter 2 answers the second.

As presented in Euclid’s Elements, straight lines are infinitely thin, continuous and infinitely straight. Lines on earth, at the microscopic level, are none of these. So how can Euclid’s propositions apply to shapes and relationships in the world and how can his arguments reflect and capture relationships in the world? And if everything that we manufacture is shaped and measured according to our understanding of Euclid’s geometry is that to be expected or is it a happy accident?

Answering these questions is the essential burden of Chapter 1. Some highlights:

• A closed figure with three straight edges is a triangle when, considered as a shape there are three relevant sides with no relevant bending or discontinuities.

• Euclid’s postulates are all primitive measurements, either of distance or direction

• A Euclidean argument is a series of abstract measurements. It is a recipe for establishing the asserted indirect measurement by a series of more direct measurements, reducing ultimately to Euclid’s postulates.

Chapter 1 is the first instance of the broader principle that a) we need mathematics to establish quantitative relationships to support indirect measurement b) we establish quantitative relationships by mathematical arguments that embody a series of abstract measurements. In sum, measurement is both the purpose and the method of mathematics.

]]>**Excursion Day 1: 10/7/2014
Stations:**

Holiday mode (and mood) on! Though we had three main places for tour, there were two extra stopping for photo taking purpose. The first one, itself, was before the penguin tour and took place around 9:30 am-an excellent start for the day! That was probably because we had to get through the winding passage of hill (by bus) before getting into the penguin tour, and a photo session halfway would be breathtaking.

How about penguin tour? Well, since this was not even the formal name of the tour, we should be aware that this tour included some other attractions as well, although the penguin sanctuary was the limelight of the attraction. Moreover, as the saying goes, the best part always come last: it took us some walk before we could enjoy the close-distance contact with the adorable penguins. It should be noted that Justin and I appropriated the pot of luck by witnessing penguins for two times in 3 years (last time being in IMO 2012), a rare opportunity hardly thought by any Malaysian.

We soon departed for the non-tourism station, where we spent our lunch there. Another series of African music show, but instead of Shosholoza, all songs presented were pop songs in English language: something we could understand its contents and appreciate with open heart.

The presentation was eclectic, ranging from the traditional dances:

to the skillful modern dances:

With singing accompanied with dances:

As for lunch, we finally had the chance to enjoy the buffet style dishes after the 2-choose-1 mode for few days. IMO 2014 on Google Plus took some pictures of us, and six of us were jointly captured in two photos:

The final station was the Cape Point excursion, which awarded us the best view of Cape Town. Listening to waves tapping gently, consistently, and rhythmically to the sea, and in front of you was nothing but endless sea. Oh I found myself closer to nature as opposed to the polluted city life!

Of course, here came the second stop near a bay for photo snapping:

And it was a waste not to leave memories with our national flag along:

That was succeeded by the final destination of the day: Cape Point. In the plainest language, “no pain, no gain”. Therefore, even though we had to hike hills with mouth gasping for oxygen and limbs coping for fatigue, we shouldn’t be complaining, since the bird’s eye view in front of us was the symbol of success!

Every contestant was expected to be contented by these excursions by 4pm, and that was when the bus left for UCT (one hour journey). A colourful experience for our little booklet of life, no? Yes, it was.

**Post-excursion
**Tonight we finally managed to watch the replay match of FIFA world semi-finals that we missed after the first exam. There were some extraordinary moments underlying, with a team scoring 4 goals in 6 minutes. In terms of enjoyment, it was futile, however, since we missed the selection of African Dance team to be performing in front of the public in Sport Hall the next day.

**Day 2: Lectures and games
**The morning was another intellectual session with talks to discerning young mathematicians around the world, and there were three, in fact. We had one hour for each lecture, with a short break.

The first one was a former IMO gold medallist discoursing a problem which was still open: is it possible to partition any convex polygon into n polygons of equal area and perimeter? Simple brain activity: meaningless statement for n=1, and for n=2 it was just the matter of using intermediate value theorem. While the topic was elementary idea for any layman to digest about, the insight was beyond what we learnt in school (or even in IMO training program). Using series of obscure mapping theory, the conclusion of validity for any prime power n was justified. How about for other numbers?

The second one, surprisingly, was about physics. Mathematics works hand-in-hand with physics, but sorry, I could understand nothing about it.

The last lecture was a collaboration between number theory and geometry: given a circle of radius 1 and three circles internally tangent to it and mutually tangent to each other, all having radius 1/n. Then all other circles constructed using the same way would have radius 1/n as well.

But for which n would the radius be valid? My brain followed the lecture clumsily and noted the idea of sum of two squares, but the detail was getting me nowhere.

Ok, enough with lectures (whether you can comprehend or not), now take a lunch break.

–

The afternoon program was known as “African Dance and Games”, combining two “sport-like” activities (indeed, if chess is considered as sports, why not board games?) We returned to Sports Hall, with no tables arranged for contest now.

What was the dance about? This time the instructor threw us basic concepts of dancing, with baby steps of expanding and contracting circle, following some rhythm. Having missed the selection test the night before, we paid our debt back by watching the live performance of dance show, with Mojalefa showing is fervor to it by tapping drum.

The board game gave a sense of *deja vu *since the game structure was almost homogeneous to “congkak” in Malaysia (Oops! Name of game forgotten) : marble balls with holes. Though the game rule which differed from original, but who cared? After losing a congkak board during the journey to Argentina for IMO 2012, it was time for compensation to enjoy the game with lost, and learning some modifications from it at the same time.

**The moment of truth
**Where were our leaders and deputy leaders? Unfortunately, they couldn’t be with us since they were involved in marking of our papers: the coordination process. As usual, it went problems by problems. This year, Mr. Suhaimi “appointed” me as agent and sent our scores to us once the coordination was done.

Although coordination of some problems (e.g. P1 and P6 for Malaysia) was unanimous (with total score of 42 and 0 being the most observable difference), leaders had to contend with coordinators on some contentious script. This usually happened in intermediate combinatorial problems, like P2 (and even P5) this year.

The release of score was in mixed feelings: anxious to know what was the outcome, disappointed for having scores docked although one solved it, delighted for scoring more than expected, and eventually, feeling accomplished in the end for splendid overall performance: we scored 129 with everyone scoring at least 2 full credits in total, marking the 5th year in a row in record-breaking.

For me, my heart dipped when I received a 1 for P3 (slightly less than expectation), but a 7 for problems 1,2,4,5 was a blessing to me, clearing all my paranoid of having unexpected cut of score.

Finally, how could the medal boundary be? Looking at the scoreboard of teams, the hope retrieved again: Justin and I had even chance in winning gold medals that we missed for the past two years. While the we had hope for the ultimate accomplishment, that made us nervous and thinking about it from time to time: this ambivalence was worse than that of hopeless.

**10:07 pm, the big fat news.** The jury meeting was held at night, agenda ranging from the formality of approving all scores, coordination and process of IMO, to future consideration and improvement in organisation of IMO. The final agenda, which would extricate contestants from tension would be decision of cut-off points. Mr. Suhaimi then sent his last results-related news to us:

*“Official: Gold 29, silver 22, bronze 16. Good job all.”*Oh what a beautiful night, where Justin and I hit it on the dot! I remembered turning into a complete psycho by exclaiming “Yes! Yes! Yes!” without caring how many people were looking at this monkey recently escaped from the too. As for overall performance, we broke various records in a row, from the total score to overall ranking of 23/101, a noticeable leap from previous 31/97. With ubiquitous WiFi coverage in UCT, Justin, a triumphant soldier, acclaimed our achievement to the rest of the world:

It may be aggrieved for two of us for missing the bronze cut off narrowly by 1 or 2 points, but everyone acknowledged that life was cruel. Zi Song, on the other hand, missed the 3 youngest bronze medalist in IMO history by just a few weeks, but anyhow this proved him as a genuine child prodigy.

**IMO problem as topic of the day, again.
**Still remember the constant c in IMO P6? Yeah, USA leader (Mr. Po-Shen Loh) proved a significantly better result, i.e. c=square root of ln n. A tremendous leap of the boundary, where you can fix it as large as you want. Consequently, there shouldn’t be any surprise that he was asked to present a lecture on it, and it was properly scheduled after the final jury meeting at 10 pm.

Loh is an erudite in probabilistic method, hence the team under his coaching marched victoriously to a brilliant 33/42 in Liar’s Guessing Game, the most difficult problem in IMO 2012. Before showing his big guns, he skillfully proved to us the original IMO problem in 5 minutes, which was already impressive. The improved bound, however, demanded energy both from Loh and from audience, with 90 minutes of verbal explanation and 90 minutes of concentration, respectively. Again, that felt like rocket science to me, and I could only understand basic threshold concept of graph theory employed in it.

–

Finally, for the third night, the “Law” of 10pm-to-bed restriction was abrogated.

]]>Amazingly, though, I *have* encountered people who think this is up for debate.

It isn’t.

]]>In general if interarrival times have (cumulative) distributive function and mean , the following is true.

The probability density of waiting time is

Applying this formula to the above bus example, we have and

Using (1), the probability density function of waiting time is plotted below.

We see that the density function is a rectangle (centred at and having 2/3 of the total probability mass) appended to a triangle (having centroid at and 1/3 of the total probability mass). From this we can say the expected waiting time is minutes. This is more than half the mean interarrival time of 15/2 = 7.5 minutes that one might initially expect.

To prove (1), we shall first look at the length of a particular inter-arrival period that contains a given point , and the waiting time can then be the part of this after time . We make use of the following result from alternating renewal process theory [1]:

Proposition: Consider a system that alternates between on and off states where:

- the on durations are independent and identically distributed according to random variable , and
- the off durations are independent and identically distributed according to random variable .
(However it may be that and are dependent in general.)

Let be the probability that the system is in the on state at time . Then if has finite mean and is non-lattice (i.e. takes on more values than just an integral multiple of some positive number), then

This result is easily understood: the probability the system is on at a given time is the mean on time divided by the mean total cycle time. However it is based on the key renewal theorem, a limit theorem which is not so straightforward to prove.

Now consider a random process with independent and identically distributed interarrival times with common distribution with probability distribution function . Consider the length of the particular interarrival period that contains some point . We shall determine the distribution of . We define an on-off system as follows.

- If is greater than some given , let our system be on for the entire interarrival period (i.e. ).
- If the length of the interarrival period is less than or equal to let our system be off for the interarrival period (i.e. ).

Then by this definition the conditions of the above proposition are met and so

Equivalently, , and if has probability density , differentiating both sides of this equation gives the probability density function of the length of interarrival period containing a point :

Intuitively, longer periods are more likely to contain the point , with probability in direct proportion to the length of the interval. Hence the original density is scaled by the length and then normalized by to make a valid probability density. defined by (3) is known as the **size-biased** **transform** of .

Note that the length of the particular interarrival interval is stochastically larger than any of the identically distributed interarrival times . In other words,

This is the so-called inspection paradox, which can be proved by the Chebyshev correlation inequality (a form of rearrangement inequality) which states than when and are functions of the random variable having the same monotonicity (i.e. both non-increasing or both non-decreasing), then and are non-negatively correlated:

Applying this result with the non-decreasing functions and (which is 1 when and 0 otherwise):

The inspection paradox is a form of sampling bias, where results are modified as a result of a non-intended sampling of a given space. Another example is the friendship paradox, where in a graph of acquaintances with vertices having some differing degrees (number of friends), the average degree of a friend is greater than the average degree of a randomly chosen node. For one thing a friend will never have degree zero while a randomly chosen person might! The degree of a friend is an example of sampling from a size-biased distribution.

How about the distribution of the waiting time for the next arrival? For fixed we consider another on-off system where this time the system is in an off state during the last units of time of an interarrival interval, and in the on state otherwise. That is,

- ,
- .

Alternatively this form can be found by imagining our time to be uniformly distributed within the current interarrival interval, thus partitioning our inspection interarrival interval into and , where is the uniform random variable . Hence the distribution of is found by integrating over the distribution of and using the fact that from (3):

Differentiating both sides of (8) or (9) with respect to leads to (1), the probability density function of waiting time .

Finally, if we wish to find any moments we write where and is independent of the size-biased variable :

In particular

.

Hence we can say that the mean waiting time for the next arrival from any given point in time is more than half the mean interarrival time, unless (i.e. zero variance in , meaning is deterministic).

For example, in the initial bus example where :

- ,
- and
- ,

so and . Hence the waiting time has mean 7.78 and standard deviation of .

[1] S. Ross, *Stochastic Processes*, John Wiley & Sons, 1996.

[2] R. Arratia and L. Goldstein, *Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent?* Available at arxiv.org/pdf/1007.3910.

[3] This answer by Did on math.stackexchange.

]]>Who Created the Rochester Reconstruction initiative and wrote the program and gave the idea to the city

I DID!

Major Mathmatix, Executive, Producer, Manager, wrtiter, Publisher, TV, Film, Director, S.L.A.P TV, Live Wire Records, New York, LiveWire Records, Major Mathmatix, Livewire, LiveWire Empire, LiveWire Athletics, Music, Film, TV, Hip-Hop, Rap, R&B, Major, Keani, Cochelle, Major Mathematics, Mathematics, majormathmatix, major

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Who Created the Rochester Reconstruction initiative and wrote the program and gave the idea to the city

I DID!

Major Mathmatix, Executive, Producer, Manager, wrtiter, Publisher, TV, Film, Director, S.L.A.P TV, Live Wire Records, New York, LiveWire Records, Major Mathmatix, Livewire, LiveWire Empire, LiveWire Athletics, Music, Film, TV, Hip-Hop, Rap, R&B, Major, Keani, Cochelle, Major Mathematics, Mathematics, majormathmatix, major

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