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Posts tagged with arithmetic

Without using a calculator or even pen and paper—or even doing the arithmetic in your head—¿which is the larger product? of:

  • 3.01 × 6.99
  • 2.99 × 7.01

You know 3 × 7  =  21. But what happens if you gently place your finger on the teeter-totter?

 

Although this is “just arithmetic” and so doesn’t require a learnèd vocabulary of higher mathematics, it touches on a few “higher maths” topics:

  • bilinearity being the way I would answer it “conceptually rather than computationally”
    Bilinear maps and dual spaces  Think of a function that takes two inputs and gives one output. The + operator is like that. 9+10=19 or, if you prefer to be computer-y about it, plus(9, 10) returns 19.  So is the relation “the degree to which X loves Y”. Takes as inputs two people and returns the degree to which the first loves the second. Not necessarily symmetrical! I.e. love(A→B) ≠ love(B→A). * It can get quite dramatic.    An operator could also take three or four inputs.  The vanilla Black-Scholes price of a call option asks for {the current price, desired exercise price, [European | American | Asian], date of expiry, volatility}.  That’s five inputs: three ℝ⁺ numbers, one option from a set isomorphic to {1,2,3} = ℕ₃, and one date.    A bilinear map takes two inputs, and it’s linear in both terms.  Meaning if you adjust one of the inputs, the final change to the output is only a linear difference.  Multiplication is a bilinear operation (think 3×17 versus 3×18). Vectorial dot multiplication is a bilinear operation. Vectorial cross multiplication is a bilinear operation but it returns a vector instead of a scalar. Matrix multiplication is a bilinear operation which returns another matrix. And tensor multiplication ⊗, too, is bilinear.  Above, Juan Marquez shows the different bilinear operators and their duals. The point is that it’s just symbol chasing.    * The distinct usage “I love sandwiches” would be considered a separate mathematical operator since it takes a different kind of input.
  • wiggly numbers
    image
  • ± ε
  • statistical robustness
  • tensor product of modules
    Tensor product of modules.png
  • isoperimetric inequality

I’ll leave the question/reasoning unanswered with a space for you to answer which is larger and why.

3±ε  ×   7∓ε    =    21±¿?

(Source: Wikipedia)




The Speenhamland allowance scale enacted in 1795 effectively set a floor on the income of labourers according to the price of bread.

When the gallon loaf cost 1s, the laborer was to have a weekly income of 3s for himself. … Weekly wages of 3s are equal to …3.72 pounds of bread per day for a single labourer. This is an important figure to remember as the Speenhamland allowance.

As a pound of bread provides about 1100 calories, the allowance gave the labourer a total of 4100 calories per day. An agricultural labourer doing 8-10 hours of vigorous work can easily require 3000 calories/day. It is evident that the Speenhamland allowance provided just above the bare means of subsistence.




Commutativity is an easy property to render into English: it means order doesn’t matter.

http://2.bp.blogspot.com/-kqL0IAUkfQE/Tb8ik3KuGHI/AAAAAAAAAGc/1iHHkGauUfQ/s1600/rubiks+layout+001.jpg

Watchmen

For example “three groups of five rocks” totals the same as “five groups of three rocks”. In fact a general proposition is true: “L groups of R rocks” totals the same as “R groups of L rocks” for any L,R. Which is surprising if you think about arraying the stones in piles or spirals or circles


recursive "Square" function

but less so if you think about arraying them in grids

image

  

But what about associativity? It’s a basic assumption of category theory and every monoid or semigroup is associative. Since functional programming (F#, Haskell, etc) is based in these composition-friendly mathematics of braids, string diagrams, monads, and monoids, the associative assumption will come up in functional programming as well.

Parametric Monad

If the property holds then we can do things like this:

http://25.media.tumblr.com/8796d491890057c0ab488ccf4589fc34/tumblr_mpck6hmZC11qc38e9o1_1280.jpg

but what does it mean for associativity to hold? Just like I didn’t understand why the classical poetry I read in school was considered good until I read some truly bad poetry, I need some examples of non-associative things before I can understand what it means to assume some process is associative.

The sedenions aren’t associative

File:HypernumbersPowerOrbitW.gif

and neither are the octonions.

http://upload.wikimedia.org/wikipedia/commons/thumb/2/2d/FanoPlane.svg/1000px-FanoPlane.svg.png

But these two algebras are unusual so in order to explain why they don’t translate evaluation parentheses, you have to first figure out how they work. Same with Okubo algebras, Jordan algebras, Poisson algebras, and vector cross-products. How about a more commonly understood subject matter?

Here is one: arithmetic of exponents.

The Berkeley calculator evaluates 2^2^2^2^2 right-to-left.

To access the Berkeley Calculator: Type bc -l at the terminal in Linux or Mac. In Windows get PuTTY and ssh new@sdf.org.

If the evaluation order doesn’t matter (associativity), then the square root of 2^2^2^2^2 should be the same as the base-two log of 2^2^2^2^2, since one cancels “from the bottom” and one of them cancels “from the top”.

image

image

But guess what?

The square root of 65536 is 256, but the log_2 of 65536 is ≈16. Since they’re not the same, exponentiation is non-associative.

Playing around with exponents of a few two’s and different evaluation orders can be done either in bc -l or on paper.

4^4^2
(2^2)^(2^2)^(2)

(2^2^2^2)^2
(2^2^2^2)*(2^2^2^2)

65536^2 = 4294967296
2^65536 = ridiculous

(2^3)^2 = 8^2 = 64
2^(3^2) = 2^9 = 512

And now that I’ve played around I have a plain-English description of what associativity means: "Order of evaluation doesn’t matter"




I had to tell someone what £20−£12.25 was. At first I thought she was stupid. You really don’t know? But then I realised that many people can’t do mental arithmetic of the variety £20−£7.75; they just don’t admit it or ask for help. Then I thought she was smart.

image

I read somewhere that dyslexia is overrepresented among CEO’s. The person who pointed it out speculated that it’s because dyslexics are used to asking others for help. No matter how brilliant you are, you can’t be good at everything a large organisation needs to do. How are you going to be the person at the top if you’re more focussed on being brilliant yourself rather than seeking help from someone who’s smarter than you?




Readers of isomorphismes, you might enjoy powers of two tumblr.

powersoftwo:

2100 = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.




@portereduardo and I were discussing redistributive taxation on twitter the other day.

Mr Porter wrote a piece in the NY Times about hyper-taxing the tippy top margin of American income to reduce the US’ yearly government deficit. Economists Pikkety & Saez (famous for assembling a widely-quoted data set about top incomes) estimate that $400 billion per year could be raised by putting the US’ top marginal rate back to 80%—about ten times as much as I thought could be raised, that’s actually a significant chunk of the yearly (spending − revenue) deficit.

 

Mr Porter used the diminishing-marginal-utility argument for redistribution: poor people value money more than rich people. (It’s been said that exponential increases in money only beget linear improvements in happiness.)

If you believe that rich Americans should give money to poor Americans, surely it follows that Americans in general should give most of their money (at least their earnings above a certain level) to the poorest in the world—people like the Aboubakars. 
The Aboubakar family of Breidjing Camp in Chad.Food expenditure for one week: 685 CFA Francs or $1.23
Nothing speaks to me like this photo series. The families and their food.
This famous photo of the Aboubakars (taken in 2006 I believe) inspired me to eat more legumes and beans over the past few years. I figured—if 6 of them can get by on ~ $1/week, I can definitely lower my expenses by working what’s in those bags into my diet — crowding out the rich, expensive food (meats, pâté, cheese, hummus, butter, pre-made stuff).
Always fighting the hedonic treadmill. Thanks, Aboubakars (and Peter Menzel).
RELATED: Global Rich List, Angus Maddison’s History of the World Economy, Hans Rosling’s 2010 TED talk

According to the World Bank, the world’s yearly output is worth $63 trillion, and there are 6.8 billion people on Earth. That means an income per person of about $9250. Let’s pretend that it were possible to just “take that money and put it over there” — let’s imagine that people worked every day just as they do now, got their paychecks, and when they did so the various world governments had shuffled all of everyone’s monies around so that the weekly amount was $177.88. What would the world look like then?

 

What follows is my own speculation about what the world would be like at $9000 parity. A lot of this logic actually requires not-quite-parity because no-one would be swayed by a salary offer of $150,000 over $100,000 if they’re going to keep exactly $9,000 of it either way. So in order for market forces to allocate labour to the places where it’s currently going and thus keep production stable, you would need to do something more akin to:

  • taxing every dollar earned over $5,000/year at a 90% rate
  • (After all, if you live in a tent and all you eat is oatmeal, vegetables, legumes, and vitamins/salt, $96/week is far beyond what’s necessary to stay alive. And who are you that you should get foie gras and Mario Kart, when others eat rice gruel?)
  • taxing every dollar earned over $50,000 at a 99% rate,
  • taxing every dollar earned over $500,000 at a 99.9% rate,
  • and so on,
  • with everyone guaranteed at least $1,500/year and those making less than $9,000/yr getting their wages supplemented by a negative earned-income tax.
  • Phew.

Thus a law professor making $250,000 would keep $200,000×1% + $50,000×10%+$5,000 = $11,000, a hefty 22% above the average and a supernal 633% above the lowest earning minimum for non-workers. More than enough to represent status symbols that would motivate him/her to keep teaching the law to young people, right?

Plus, if the world output were fungible, we would have $9000 per person, not per working person. So the usual dynamics of unemployment, childhood, and retirement would take place. The income of working people might be more like $18,000 or more and then split back down depending how many dependents they had — respecting the bar that the same number of people need to keep working the same number of hours at the current jobs in order for world production to stay constant.

 

But, it’s easier to talk about if we said everyone just gets $9,000/year even. Here are my speculations about what that world would be like:

Things that would not exist if everybody’s incomes were equal:

  • High Fashion — The world would be a drabber place.
  • Sotheby’s — No Rothko pieces auctioning for millions.
  • Expensive houses — Obviously, this is where most of rich people’s money goes right now. One could certainly do time-shares on an expensive place, though — or pack a lot of people into one expensive house. The prices of houses would have to drop as well due to falling demand.
  • Disney World — The capital costs can only be sustained by a steady flow of traffic spending between $2,000 and $20,000 per week at Disney World. However you could still go to Knoebels and ride the Red Baron once for a buck (4% of your daily nut). You would have to pay $40 entrance fee ($45 now at Holiday World!) which would mean forgoing two days’ worth of income (though still paying for rent and food on those days). I think we <link> would still have amusement parks and rides, since Americans weren’t so rich when Coney Island was built.
  • Dubai — Obviously.
  • A variety of restaurants — unless prices dropped dramatically due to the redistribution, people couldn’t eat out like contemporary New Yorkers do. A sandwich from Jimmy John’s costs $5.50, $8.83 if you get soda and chips. That’s 34% of your daily nut, not counting rent/utilities. And the San Francisco “yuppie food stamp” ($20 bill gets you lunch) would amount to 79% of your daily costs — surely putting you over the limit for the day once necessities are paid.

    I predict (or should I say, counterfactualdict) that some people would still choose to make a living by serving food to others—they would probably serve the same two or three foods to everyone all day (perhaps varying the amount per person but less choice)—the economy of scale coming through not having to cook several times or manage different ingredients. I used to go to a place in Dublin run by Hari Krishnas, I’m thinking it would be something like that — or like the comedores of Latin America—you can have eggs, beans, and tortilla made any of 5 different ways—con café; pero crema, no tenemos.
  • Universities. Currently, US universities mostly get money from two places: tuition, and research grants. Adults believe so strongly in the value of education that they will postpone consumption for 10-20 years and lump all of that savings into payments to professors and administrators over a 4-year period. But there’s no way somebody earning $177.88/week is going to pay $80,000/year to send anyone to NYU. However professors would ostensibly be willing to share their knowledge for the same money everyone else is making—so what would the total result be?
  • Cars. How are you going to afford insurance and a car on $770/month? Perhaps it could be done but I think people would just bike / walk / take the bus. Horses might even make a comeback as transportation for normal people. (But rich countries already have the pavement infrastructure so I think people would bike there.) Then again, people might still use motors to get around but much cheaper vehicles, maybe smaller motors or old old cars (think Cuba). After all we still have the human and factory capital to make automotive vehicles, and we still have a lot of cars about. Although our V8 technology is much better than the 2-stroke technology.
  • Banks. Well, they would change, wouldn’t they? If all of their customers made $9e3 / year, it seems their business would have to be very different.
  • Orthodontia. Too high of capital costs. Smile, everyone’s teeth are fugly now.
  • Insurance. Once you start accumulating a lot of stuff while continuing to make buckets of income, what does it make sense to do? Protect your stuff. If the top incomes were hacked off, there would be much less demand for property insurance—after all, we wouldn’t be driving expensive cars or living in expensive homes anymore.
  • Pharma research. At least the business model would have to change. I don’t know exactly what medicines / procedures / medical devices are purchased exclusively by the wealthy, but if the customer base was wider and shallower, that has to make a difference.
  • Medical care. Again I don’t know what exactly would change, but medical care would be quite different for the same reasons.
  • Whole Foods. If everyone were living on $9000/year, then here are some of the ways food consumption might change. Out-of-season fruit might be a luxury; pre-packaged food (not frozen peas but like sun-dried tomato hummus) … Right now OECD grocers waste produce that doesn’t look perfect; I predict consumers would buy tomatoes with bruises and such.
  • Jetskis. As fun as these things are, they have to go. They’re rich boy toys. I guess you could trade used jetskis since they’ve already been produced—but I can’t imagine the continuance of demand for such a luxury item. Then again —maybe you could 
  • Overtime. I believe the business world would slow down if the opportunities became less lumpy—if consumers were more homogeneous, business might be more of a “steady flow” — like a factory. Also, the opportunity cost of leisure would go down so people would be less inclined to work overtime.
  • Classical music. Symphonies have been the provenance of the wealthy since Mozart’s time; just look at who’s sponsoring your local lyric opera today, chances are an élite financial firm’s name is among the biggest donors. Not only does the demand depend on wealth, but the supply does as well. Tanglewood and Julliard don’t come cheap, so we don’t get Nina Simone anymore.
  • "I’m so broke; let’s go to a cheap bar." Maybe I’m being a little wishful. 

Things that could still exist if everybody’s incomes were equal:

  • Billion-dollar fighter jets. Don’t forget that after you earn $100,000 and give $91,000 to the government for redistribution to the poorest, the government still gets to take another $3000 for the purposes of running the government! Maybe the size-of-government would be relatively smaller in a world at parity, but wouldn’t the likelihood of war be greater? We needs dem fighter jets to keep de peace.
  • Research grants. Again, the government still gets to take yet more of your earnings after the redistribution. So they could fund cancer research and so on. The cost of labour in the U.S. might go down quite a lot given the post-tax redistribution, meaning the research would be more labour-intensive and less capital-intensive — so maybe more grants for theorists and fewer experimental machines built?
  • Doctors. Cuban doctors already give up most of their earnings because they’re from a communist country. That didn’t stop them from wanting to be doctores. A Cuban doctor saved my life one time.
  • Two-income households. Sure, all the jobs at Subway and Starbucks will have disappeared, but with companies trying to adjust to 3 billion new customers there will be 
  • Plumbing, showers — You can currently get this for fairly cheap.
  • Apartment buildings — Shared housing is the easiest way to make rent go down. I expect extended families would move in together, or there would be a lot more Craigslist ads and living in even more cramped quarters in a big city where mostly yuppies live. Landlords would essentially be f*$ked over by parity, but isn’t it their fault for trying to siphon money out of the wealthiest instead of serving the poorer customers for a business?
  • Colourful clothing. People figured out dyes hundreds and thousands of years ago. It stands to reason that there was enough demand then, when we <link> were so much poorer, for colour, so there would be demand for it now.
  • Computers. Do you know how cheap an old computer is? Very. Forget One Laptop Per Child, you can find enough parts at a charity shop to get yourself running Puppy Linux on $177/week.
  • Trains? I’m not sure how goods would be moved from point to point in a world at parity. Both trains and lorries can be pretty efficient ways of moving goods from point A to point B; in this Uniform Factory Society I’m imagining I’d guess the returns to providing plain goods at lower cost would exceed the returns to imagining more stylish <link substance of style> or imaginative goods. Trains have been around for quite a long time so that’s an argument that they would still be affordable 
  • Sports. Sure, you wouldn’t have athletes making $20 million per contract, but people played sports long before that happened.
  • Beer. People invented beer long ago, when we were much poorer. Thank heavens we don’t have to give that up. 
  • Lumpy pools of wealth. Just because you equalise individual incomes, doesn’t mean that people still can’t choose to combine their savings and earnings together. People could still aggregate their savings in mutual funds and pick an investment manager to decide which of the new factories churning out staples (or the service companies delivering the staples, or the research companies figuring out how to configure the production resources) was going to grow the fastest and generate the most profits. If the capital gains tax were also 90%, people could still increase their wealth by investing in such a mutual fund. From the investment manager’s perspective, the AUM game would no longer be about convincing a few rich-o’s to invest (although I only said I would tax income, not existing wealth…so I’m contradicting myself here) but more like a high street bank’s problem: get tens of millions of people to open a savings account with you.
  • Envy. This is another question I posed on twitter (@leighblue responded). Does the quantity of envy adjust to relative circumstances? If not, then people would be demonstrably happier if incomes were equalised. If envy quantities do adjust, though, then I would feel as much envy about that law professor’s $11,000 after redistribution as I currently do about her $250,000.
  • Careers. What would you do if money didn’t matter? We would find out the answer to this question very quickly. What effect would it have on the macroeconomy, other than fewer corporate litigators?
  • Cabbage. Of course money, but I also mean literal cabbage. I think cabbage, beans, legumes, barley, potatoes, and other cheap-yet-nutritive foods would make a comeback — whether people cooked at home or ate at a comedor. I also predict vegetarianism would rise as the steak-eaters noticed vegetarian families having more of everything else. (Or maybe Americans would return to “meat one day a week”.)

Some more caveats. It’s arguable that economic growth could be faster in a world of parity. After all, there would be another 3 billion customers to serve—and those people would no longer be wasting their time walking firewood from the hilltop down to the stove.

It’s also arguable that economic output would be cut into 1/10th or worse. Perhaps the great economic efficiency we experience in the world today is directly dependent on the prices of labour being able to vary from $1/day to $100,000/day. You can see in a few of my examples above (e.g., the university) that very radical changes would have to take place — perhaps so radical that people wouldn’t continue doing their current jobs. And where would we be without corporate litigation firms that charge $800/hour?

Firms would all be competing to serve a more uniform group of customers, so production that otherwise went into expensive goods (real estate development in Aspen) would just go into continuing to improve the lives of everyone—like engineering a super-cost-efficient factory and distribution system that has a thousand times the scale advantages that the biggest past monopoly ever had.

Anything else you think I’m missing? A gross, lumpy price deflation as the world adjusts to parity? Massive unemployment or entrepreneurship or single-firm dominance as firms that produce staples suddenly have a lot more orders and firms that produce finery are shuttered? Would typical corporate profits go up or down? Operating costs?

The idea of redistribution is not a new one, if you know of some excellent scholarly discussions of the issue please link to them in the comments and excerpt a quote or two.

One last caveat: I’ve always lived in a capitalist country, so I don’t any personal experience with government-equalised incomes. Anybody who grew up or lived and worked in a communist country, I’d love to get your 2p.




When I was in kindergarten, we would argue about whose dad made the most money. I can’t fathom the reason. I guess it’s like arguing about who’s taller? Or who’s older? Or who has a later bedtime. I don’t know why we did it.

 

  • Josh Lenaigne: My Dad makes one million dollars a year.
  • Me: Oh yeah? Well, my Dad makes two million dollars a year.
  • Josh Lenaigne: Oh yeah?! Well My Dad makes five, hundred, BILLION dollars a year!! He makes a jillion dollars a year.
    (um, nevermind that we were obviously lying by this point, having already claimed a much lower figure … the rhetoric continued …)
  • Me: Nut-uh! Well, my Dad makes, um, Infinity Dollars per year!
    (I seriously thought I had won the argument by this tactic. You know what they say: Go Ugly Early.)
  • Josh Lenaigne: Well, my Dad makes Infinity Plus One dollars a year.




I felt so out-gunned. It was like I had pulled out a bazooka during a kickball game and then my opponent said “Oh, I got one-a those too”.

image

Sigh.

Now many years later, I find out that transfinite arithmetic actually justifies Josh Lenaigne’s cheap shot. Josh, if you’re reading this, I was always a bit afraid of you because you wore a camouflage T-shirt and talked about wrestling moves.

Georg Cantor took the idea of ∞ + 1 and developed a logically sound way of actually doing that infinitary arithmetic.




 

¿¿¿¿¿ INFINITY PLUS ?????

You might object that if you add a finite amount to infinity, you are still left with infinity.

  • 3 + ∞   =   ∞
  • 555 + ∞   =   ∞
  • 3^3^3^3^3 + ∞   =   ∞

and Georg Cantor would agree with you. But he was so clever — he came up with a way to preserve that intuition (finite + infinite = infinite) while at the same time giving force to 5-year-old Josh Lenaigne’s idea of infinity, plus one.

to infinity, and beyond

Nearly a century before C++, Cantor overloaded the plus operator. Plus on the left means something different than plus on the right.

1 + \infty \ \ = \ \ \infty\ \ < \ \ \infty + 1

  • ∞ + 1
  • ∞ + 2
  • ∞ + 3
  • ∞ + 936

That’s his way of counting "to infinity, then one more." If you define the + symbol noncommutatively, the maths logically work out just fine. So transfinite arithmetic works like this:

All those big numbers on the left don’t matter a tad. But ∞+3 on the right still holds … because we ”went to infinity, then counted three more”.

image

image

By the way, Josh Lenaigne, if you’re still reading: you’ve got something on your shirt. No, over there. Yeah, look down. Now, flick yourself in the nose. That’s from me. Special delivery.




 

#### ORDINAL NUMBERS ####

W******ia's articles on ordinal arithmetic, ordinal numbers, and cardinality flesh out Cantor's transfinite arithmetic in more detail (at least at the time of this writing, they did). If you know what a “well-ordering” is, then you’ll be able to understand even the technical parts. They answer questions like:

  • What about ∞ × 2 ?
  • What about ∞ +  ? (They should be the same, right? And they are.)
  • Does the entire second infinity come after the first one? (Yes, it does. In a < sense.)
  • What’s the deal with parentheses, since we’re using that differently defined plus sign? Transfinite arithmetic is associative, but as stated above, not commutative. So (∞ + 19) + ∞   =   ∞ + (19 + ∞)
  • What about ∞ × ∞ × ∞ × ∞ × ∞ × ∞ × ? Cantor made sense of that, too.
  • What about ∞ ^ ? Yep. Also that.
  • OK, what about ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^ ∞ ^  ? Push a little further.

I cease to comprehend the infinitary arithmetic when the ordinals reach up to the  limit of the above expression, i.e.  taken to the exponent of  times:

\lim_{i \to \infty} \ \underbrace{{{{{{{{ \infty ^ \infty  } ^ { ^ \infty} } ^ {^ \infty}} ^ {^ \infty}} ^ { ^ \infty  }} ^ {^ \infty }} ^ {^ \infty}   } ^ {^ \ldots  }   }_i

It’s called ε, short for “epsilon nought gonna understand what you are talking about anymore”. More comes after ε but Peano arithmetic ceases to function at that point. Or should I say, 1-arithmetic ceases to function and you have to move up to 2-arithmetic.





 

===== SO … WHAT COMES AFTER INFINITY? =====

You remember the tens place, the hundreds place, the thousands place from third grade. Well after infinity there’s a ∞ place, a ∞2 place, a ∞3 place, and so on. To keep counting after infinity you go:

  • 1, 2, 3, … 100, …, 10^99, … , 3→3→64→2  , … , ∞ + 1, ∞ + 2, …, ∞ 43252003274489856000   , ∞×2∞×2 + 1, ∞×2 + 2, … , ∞×84, ∞×84 + 1,  … , ∞^∞∞^∞ + 1, …, ∞^∞^∞^∞^∞^… , ε0,  ε+ 1, …




Man, infinity just got a lot bigger.

PS Hey Josh: Cobra Kai sucks. Can’t catch me!