On primes and Pluto

Maths Masters
Nanjing Night Net

Is 1 a prime number? Not according to the Australian Curriculum, which bluntly declares that, whatever else, a number must be greater than 1 to be prime. So, that settles it. We’ll be back next week.

Except, of course, such an answer settles nothing. As many a curious maths student has noted, the apposite question is the follow-up: why is 1 not prime? After all, it is clearly not a product of other numbers. However, the frequent reply is a dismissive “because your teacher (and the Australian Curriculum) said so”. Great help.

There are variations on the don’t-bother-me response. For example, your Maths Masters are often apprised that a prime is a natural number (positive whole number) with exactly two distinct factors, namely 1 and itself; that leaves us with the expected 2, 3, 5, 7, 11 and so on. However this is nothing but legal shenanigans: the contrived qualification “exactly two distinct factors” has been inserted precisely to exclude 1, with still no justification for doing so.

And the justification can’t be all that simple. After all, the number 1 used to be prime. Really.

Exactly 100 years ago mathematician D. N. Lehmer compiled his List of Prime Numbers from 1 to 10,006,721. There, proudly occupying first place, is the number 1:

Was Lehmer a one-off, 1-loving crank? Decidedly no. In the centuries prior to Lehmer many tables of prime numbers were published and some, though not all, began with the number 1. Clearly some explanation is in order, and the explanation begins on Pluto.

Here’s the question: is Pluto a planet?

The answer, as most readers will know, is that Pluto is not a planet but that it used to be. Pluto was discovered in 1930 and was proclaimed to be the ninth planet in our solar system. However, astronomers always recognised that Pluto is very different from the other planets, and in 2006 the International Astronomical Union reclassified Pluto as a lesser, dwarf planet.

The point is that the notion of “planet” is not God-given and it is not set in stone. Rather, astronomers define what they mean by the word. Then, as astronomers learn more about just what kind of things are zooming around out there, they refine their classifications, making “planet” more precise and more useful. And so, as it happens, Pluto is demoted.

The question of which numbers are prime is analogous. We cannot hope to prove that 1 is or is not prime; it is simply a question of how mathematicians have chosen to define “prime”. And, though it is now accepted that 2 should be the first prime number, historically mathematicians have been neither clear nor consistent.

So how did mathematicians come to agree to exclude 1 as a prime, and why did it take them so long to do so? The answers take us into some weird and fascinating history.

The importance of primes is that they are the building blocks of the natural numbers; any composite number (a number greater than 1 that is not itself prime) can be written as a product of primes. For example, 84 is the product 2 x 2 x 3 x 7. Moreover, except for changing the order of the factors, 84 can be written as a product of primes in just that one way. That the same is true for any composite number is the very important (and not so easy to prove) fundamental theorem of arithmetic.

What if we permitted 1 to be prime? In that case, 84 would also have the “prime” factorisation 1 x 1 x 1 x 2 x 2 x 3 x 7. That is, 84 could still be factorised, but it would no longer have a unique prime factorisation.

The upshot is, if 1 is a prime number then describing prime factorisations is more complicated. That would seem sufficient reason to exclude 1 as a prime, and it is the reason your Maths Masters have always accepted. However, both mathematically and historically, that reason turns out to be somewhat wide of the mark.

Consider again the fundamental theorem of arithmetic. Is it such a big deal if we’re forced to replace “product of primes” with “product of primes greater than 1”? Hardly, and no one ever really considered it so. Indeed, until relatively recently the question barely even arose.

The fascinating history of such questions is documented in a wonderful paper by mathematicians Chris Caldwell and Yeng Xiong. They point out (quoting another excellent survey) that prime factorisation was not of any great interest before the 19th century. Yes, division by prime numbers was of practical importance, but not the complete factorisation.

It all really dates from 1801, when the great Carl Friedrich Gauss gave the first explicit statement of the fundamental theorem of arithmetic. Then, mathematicians began thinking seriously about the structure of numbers, and of whole worlds of numbers. And, once they started considering more exotic number worlds, things got very confusing.

In particular, mathematicians discovered very strange worlds in which a number can be factorised into “primes” in fundamentally different ways. (We’ll discuss these bizarre worlds in a future column.) In order to figure out what was going on, mathematicians were forced to be extremely thoughtful and precise with their definitions. A critical component of these definitions was to get the distractions out of the way, to classify do-nothing numbers such as 1 into their own separate group. That was the real impetus for 1 to cease being prime, though it still took another century for the modern definitions to take hold.

That would seem to be pretty much the story, except there is one more, astonishing twist. From the 19th century on there was good reason to exclude 1 from the primes, but what about prior to that? Yes, a number of mathematicians classified 1 as a prime number, but before 1600 it was very uncommon to do so. Why? Because 1 wasn’t a number!

We bet you didn’t see that coming. Certainly your Maths Masters didn’t. In their paper (and in an extensive companion survey coauthored by Angela Reddick), Caldwell and Xiong consider very carefully how mathematicians throughout history have thought of “number”. It turns out that, at least as far back as Euclid, most mathematicians prior to 1600 considered 1 to just be there. By contrast, “numbers” were different, effectively created from 1 by addition. It was only with the emergence of decimals in the late 16th century that excluding 1 as a number began to seem arbitrary and unnecessary.

The amazing fact is, for most of its history 1 has not been a prime number, simply because it hasn’t been a number. If it seems difficult figuring out what mathematics is about now, just imagine what it was like in times past.

Burkard Polster teaches mathematics at Monash and is the university’s resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is helping Barbie smash calculators and iPads with a hammer.


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Chivers puts hand up for Futures League

Promising batsman Owen Chivers will make his ACT and Manuka Oval debuts less than three months after suffering a broken hand while bowling in the nets.
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Chivers hopes to use the three-day match with a NSW second XI starting on Monday ”to make a statement” for Futures League selection next summer.

The 19-year-old from Bowral has made an instant impression in his first season in the Cricket ACT first-grade competition, scoring a double-hundred for ANU in November last year.

But he was dealt a blow in the middle of December when he had to have eight screws and a plate inserted into his right hand after fielding off his own bowling in the nets.

Chivers initially thought he would miss the remainder of the season, but has got full movement back in his hand after playing the past two club games.

”I don’t normally bowl, and I was bowling to one of the lower-order batsmen and copped a ball straight back at me,” Chivers said.

”When I first came back two weeks ago it was 10 per cent weaker, but now it has fully recovered.”

Prime Minister’s XI representative Michael Spaseski will captain an up-and-coming ACT side featuring Chivers, Wests/UC batsman Adam Hewitt and emerging opener Matt Condon.

NSW will use the game to assess Manuka Oval to see if it would be suitable to host the Sheffield Shield final later this month if they win the rights to host it.

It is also an opportunity for fringe fast bowlers Gurinder Sandhu, Josh Lalor and Chris Tremain to push their claims for first-class selection.

”It’s going to be a great experience to see what it’s like to be on the end of a few of those rockets,” Chivers said.

”Sandhu’s going to be coming from a lot of height so I have to figure out how to combat that.

”My goal is to play in the Futures League next season and hopefully I can make a statement tomorrow.”

Play at Manuka Oval starts at 10.30am.

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Discord: NRL’s concussion precautions will protect players from themselves

Players will benefit from the new concussion rules. Photo: Anthony JohnsonUltimate League: Click here to sign up for our Fantasy NRL game 
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Let’s face it, the NRL didn’t do a very good job of proving that players were concussed when their clubs allowed them to play on over the past couple of seasons.

But now, we are being assured, the League is hellbent on proving they are not concussed when – or if – the new concussion guidelines are exploited to get a free interchange.

The bottom line is that it’s a good thing the League is doing something to protect players against themselves, and the sport in this country against bankrupcy which would surely come with an NFL-style class action. An American expert told the club CEOs by video conference recently that the legal action they would likely face would completely ruin them financially.

But the NRL has never taken action against clubs for flaunting the rules as they existed before. Even video of a player being given smelling salts was not considered compelling enough as evidence of an infraction.

Players stumbled around on national television and nothing was done.

So it’s hard to believe that collusion that happens behind closed doors, with just a few people involved, to fake a concussion can be adequately policed by the governing body.

Hopefully everyone will just appreciate what is at stake now, and will do the right thing.Flogging a dead horse

Is deducting points from clubs who go into bankrupcy a bit like executing someone for being dead?

Bradford are on the brink of collapse after their new owners-in-waiting withdrew an offer in response to the Bulls being deducted six competition points for entering administration.

RLF chief operating officer Ralph Rimmer says the would-be owners knew the dangers. Obviously a club going broke is not a good look for the sport and the governing body feels it has the right to respond with some sort of punative measure against those who damaged its brand.

But if there’s a bigger example in professional sport of kicking a dog when it’s down, Discord has not heard it. If the punishment is aimed at clubs who deliberately go into receivership to avoid their debts, why are we punishing the team and the fans, on the field?

Surely we don’t want people who do business in this way involved in our sport OFF the field? Punishing the team by docking points would achieve little but exonerate the RFL of accusations they did nothing.

It’s hard to imagine an NRL club experiencing financial difficulty being docked competitition points. In the past, the administration in Australia has helped clubs in trouble, by either advancing grants or even forwarding loans.

And what of the players still owed money by failed franchises such as the Celtic Crusaders? How does docking competition points help them?

In light of Bradford’s problems, it’s not surprising that the Super League clubs voted against a marquee player system.A word of thanks

Thanks to everyone who commented on Discord last week and Set Of Six on Monday.

Alan said the extended 1997 World Club Challenge was good. Most people would describe it as the most disastrous competition in the history of rugby league! As for his comment that State Of Origin was become irrelvant … Alan we dreamers often overlook the importance of tribalism in our game. Tribalism is why we have eight and a half teams in Sydney and none in Western Australia, South Australia or the Northern Territory.

There is clearly something to it!

Soot says a summer nines tournament may become irrelevant, like rugby union sevens. I’m sure the boffins at Rugby League Central would be happy to achieve that level of irrelevance. It doesn’t matter if the media ignores it, if it keeps the turnstyle clicking over the summer, then the concept will do its job.

DOS called for a PNG team on the NRL. As you may be aware, the PNG Hunters are making their Queensland Cup debut against Redcliffe on Sunday – and I’ll be there. But NRL? Is there a Major League Baseball team in Haiti? Where does the television rights income come from? How do you get players to live there? I have serious doubts it will happen in my lifetime.

Frank from Bexley, I suspect, was taking the mick so I won’t be responding to him.

Taffy said he liked my optimism but I thought last week’s column was largely pessimistic! I disagree that no-one debated union players going to league when union was not openly professional – many column inches were devoted to the subject at the time. And clearly hybrid games are commercial attractive because there are powerful forces pushing for them. You are right, however, to say rugby union in most places would have nothing to gain from rugby league – which makes the prospects I discussed last week even more forboding for league.

I recommend everyone read Friendly_Raptor’s comment at the bottom of last week’s Discord. I agree with Hear The Crow that Eddy Pettybourne should have been sent off on Saturday.

Here’s the forum

Subscribe to the podcast here

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James Courtney wins Adelaide 500 feature as Jamie Whincup makes bad start to title defence

Just when it looked like V8 Supercars champion Jamie Whincup was going to run away with the Adelaide 500, the dream start to his title defence turned into a nightmare after he figured in one of the many dramas of Sunday’s deciding race.
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Whincup failed to complete the final 250-kilometre leg after his comfortable lead at two-thirds distance was erased by a controversial pit lane drive-through penalty that triggered his ultimate demise.

Fighting his way back from the penalty, which dropped him to eighth, his Triple Eight Holden collided with Michael Caruso’s Nissan on the 63rd of 76 laps, snapping the steering. Whincup’s pit stop penalty handed the lead to Holden Racing Team’s James Courtney, who held off Craig Lowndes in the other Triple Eight Commodore to claim the Adelaide 500 trophy.

Lowndes’ second place, on top of his second and first in Saturday’s pair of 125-kilometre races, added to Whincup’s indignity as it left him trailing his veteran teammate in the championship standings.

While the start of Whincup’s bid for an unprecedented sixth V8 title ended in rancour and recriminations, Lowndes left the brutal 3.2-kilometre Adelaide Parklands street circuit with the early lead that most of his previous championship challenges have lacked – and on which they have often fallen short at the end of the season.

He amassed 282 points from the Adelaide 500’s three races, an advantage of 53 points over Whincup (199), who is fourth behind Fabian Coulthard (230) and Shane van Gisbergen, who were consistent scorers over the weekend.

Along with Whincup’s unexpected incidents, Sunday’s slog was punctuated by big accidents that helped define the outcome, causing safety car periods that bunched the field.

Will Davison, in his first event since his risky switch from Ford Performance Racing to Erebus Motorsport, crashed his Mercedes-Benz at the track’s infamous high-speed Turn 8.

After tangling with James Moffat’s Nissan Altima, Davison’s Merc was sent careering into the concrete safety barrier on the exit of the corner, losing its left front wheel and passenger front door.

But that crash was tame compared with the multiple rollover over suffered by Jason Bright, whose Brad Jones Racing Commodore was tipped over in a first-turn scuffle during the restart following Davison’s accident.

Bright was pushed sideways into a kerb, which tripped his car up on its wheels, which dug into the run-off area’s gravel and turned it on its roof.

The Holden slid across the track and slammed into a tyre barrier, sending it spiralling into the air along the safety fence before it slammed to earth upside down.

Bright was uninjured despite just about every bit of the bodywork being crushed or torn.

Whincup avoided the carnage but race officials ordered him to take a pit lane drive-through penalty after they observed that his car controller touched his car, which the rules prohibit.

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Mayor MacKenzie proud of council’s financial health 

PORT Stephens Council’s budget is in better shape than any council in the Hunter.
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The council crawled out of its multimillion-dollar financial black hole of 2009 to record a surplus last financial year.

At a time when other councils are struggling and being forced to make widespread cuts to services and staff, Port Stephens recorded an enviable $1.6 million surplus last financial year.

In 2009 the council had an underlying deficit of more than $10 million. It has spent every financial year since then painstakingly reeling it back in.

Council financial services manager Tim Hazell said the council returned to surplus two years ahead of schedule.

“We’d have to be one of the most envied councils,” Mr Hazell said.

The council also received a positive bill of health from independent auditor PricewaterhouseCoopers. It said the council’s financial position was improving.

However, it had fallen below acceptable industry benchmarks in some areas. These included civic assets, especially sealed and unsealed roads, drainage, kerb and guttering, which were described as being in poor condition.

The council has now turned its focus to asset renewal.

Mayor Bruce MacKenzie said he was very proud of the result, which had been achieved without council having to increase rates by more than the required minimum.

“This is the best council staff-wise and elected people I’ve been involved with for 44 years,” he said.

Port Stephens has the lowest rates in the Lower Hunter at an average of $950, compared with Maitland ($986), Newcastle ($1051), Lake Macquarie ($1141) and Cessnock ($1064).

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