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Aug1217
  The Wanderer: What if there were three kinds of candles: green, vanilla, and turquoise. The green candles take five seconds to light and three of them to light another candle. The vanilla candles are wet and need sixteen seconds to light. And the turquoise candles are trick candles that take five seconds to light, but once lit there's only a 1/3 chance they'll remain lit long enough to light another candle. If one vanilla candle is lit in the center of a large crowd of people who haven't seen how the various candles work yet, what will happen? 

Aug1217
  al wazir: <If one vanilla candle is lit in the center of a large crowd of people who haven't seen how the various candles work yet, what will happen?> I never use vanilla candles. Make it chocolate instead. 

Aug1417
  Sneaky: Sneaky: <The speed of light can neither be calculated nor measured. Its value is a *defined* quantity> Pi and the golden ratio are defined quantitiesthe speed of light is and has always been measured empirically. All you've done is pointed out that the meter is (currently) defined using the speed of light. (If I recall, at first it was defined in relation to the earth's circumference.) Anyhow, c certainly can be measured; famously measured first in the 19th century, by studying the motions of the moons of Jupiter with telescopes and stopwatches. So you're not wrong saying it's exactly 299,792,458 m/s  but you're forced then to confess you don't know the exact length of a meter! 

Aug1417
  zanzibar: Glad to see the correction about the speed of light being measurable. All you have to do is count to one:
c = ħ = 1.
(Now, if only we could just set π = 1, we could cut down mistakes by at least ½, if not 1) 

Aug1417
  al wazir: <Sneaky: So you're not wrong saying it's exactly 299,792,458 m/s  but you're forced then to confess you don't know the exact length of a meter!> Oh, but I do: <1/299,792,458 of the distance light travels in a vacuum in one second.> http://www.dictionary.com/browse/me... 

Aug1417
  john barleycorn: yes, and 1 second is
299,792,458 m divided by 299,792,458 m/s 

Aug1417
  beatgiant: <Sneaky>
<Pi and the golden ratio are defined quantities>
Most <units of measurement> are arbitrarily defined, but the quantities associated with Pi and the golden ratio are not. They represent universal mathematical relationships that can be estimated or represented exactly by infinite series or continued fractions etc., and would be the same for intelligent beings on Proxima Centauri b, if they exist and care about math. 

Aug1417
  john barleycorn: Pi and the golden ratio are real numbers not per se units of measurement. they act for example as multipliers or divisors in measurements. like 299,792,458. 

Aug1417
  beatgiant: I understand the state of Indiana actually did almost pass a law redefining pi as 3.2. If they were really able to do that, it would make life a lot easier for schoolchildren all over the galaxy... 

Aug1417
  john barleycorn: <beatgiant> guess why pi is called an irrational number. Kudos to the pragmatic approach of the state of Indiana. I think I read it in one of Martin Gardner's books but with 4 (instead of 3.2). 

Aug1417
  beatgiant: <john barleycorn>
You can't make this stuff up https://en.wikipedia.org/wiki/India... Say, I wonder if <Sneaky> is from Indiana. The ability to square the circle would explain why he's doing better than both of us in the ChessBookie game. 

Aug1417
  john barleycorn: <beatgiant: <john barleycorn> You can't make this stuff up ...> only in the US of A :) 

Aug1417
  Sneaky: <beatgiant>
<Most <units of measurement> are arbitrarily defined> Like the inch, the ounce, the hour? Of course. And that's why they aren't perfectly defined. If I told you that a "Sneaky foot" was the length of my right foot, you'd never be able to know its length in centimeters until you meet me and do the measurement, and even then, it's going to have an error margin. <but the quantities associated with Pi and the golden ratio are not.> Of courseyou can't compare an exact value like pi to the "Sneaky foot", they are cut from a different cloth. I fully agree they are not *arbitrarily* defined, but they are most certainly *defined* and that's all my small point was that I directed to <al wazir>. Pi can be defined in at least three ways that I know of: you can define it conceptually, such as "the ratio of a circle's circumference to its diameter" (and even though that's the one we learn in school you could define it in terms of things that seemingly have nothing to do with circles!) Or you can define it as an infinite series and Euler, Ramanujan, and countless others did. Or you can define it as a recursive fraction like this graphic shows on Wikipedia: https://wikimedia.org/api/rest_v1/m... There probably are dozens of other ways to define it using geometry, probability, topology, and branches of math far over my head. On a quasiphilosophical note, I don't believe any of these definitions are more "primal" than the others. We've all been taught to think of it as the circumference/diameter but if you want to think of pi as (4/1)*(4/3)*(8/3)*(8/5)*(12/5)*(12/7)*(16/7)*(16/
9)*... that's equally as valid, IMO. It's not fair to say that one is a consequence of the other, and not vice versa. In our minds it may work that way, but math itself is disinterested in our emotions. Perhaps on Proxima Centauri b the Ramanujan series is their initial introduction to the topic and only later do they learn that it also happens to be the ratio of circumferences to diameters. 

Aug1417
  john barleycorn: <Sneaky: <beatgiant> <Most <units of measurement> are arbitrarily defined> Like the inch, the ounce, the hour? ...> I hink <beatgiant> has inch vs. centimeter etc. in mind. The units are indeed arbitrary. what is important that they can be converted from one measurement system to the other. 

Aug1417
  beatgiant: <Sneaky>
Well, that's a bit like arguing that the square root of two could be defined as an infinite series or continued fraction first, and only then derive the fact that it happens to yield two when multiplied by itself.Suffice it to say, if you have defined things like circles, gaussian probability distributions etc., pi will come out of those without further invention. But I don't want to go on and on about this and take the forum off topic. 

Aug1517
  john barleycorn: <<sneaky> On a quasiphilosophical note, I don't believe any of these definitions are more "primal" than the others.> Being more "primal" is not a criterion for a definition. In fact, the "primal" ones may be clumsy and being replaced but it is a matter of taste where you put the difficulties. In the definition or in the theorems. 

Aug1517
  beatgiant: <where you put the difficulties. In the definition or in the theorems.> Or both. If, as <Sneaky> suggests, we define a circle as a figure whose distance around is a certain Ramanujan sum times its distance across, and then prove a theorem that the points on a circle are equidistant from a center, I would have difficulty with both the definition and the theorem. 

Aug1517   ughaibu: The probability of two randomly selected nonzero natural numbers being coprime, is six divided by pi squared. So, the number of ways to define pi is, presumably, infinite and the ways to define it, arbitrary. 

Aug1517
  beatgiant: <ughaibu>
The original controversy was <pi is a defined quantity> versus <pi represents universal relationships>. Showing that there are lots of things that are related by pi tends more toward the latter. 

Aug1517   ughaibu: Is there really a dilemma? 

Aug1517
  beatgiant: <ughaibu>
The original contrast was speed of light (measured) versus pi and the golden ratio (defined). But it all depends what <Sneaky> means by those terms. 

Aug1517   ughaibu: Okay, I guess I should read through the earlier posts. 

Aug1617
  al wazir: <ughaibu: The probability of two randomly selected nonzero natural numbers being coprime, is six divided by pi squared.> Prime numbers become increasing sparse with increasing size (the number of primes .lt. n scales asymptotically as n/ln n as n → ∞). Accordingly, composite numbers become increasingly dense as n → ∞. But as n → ∞ the number of primes that *might* divide n grows without bound, so it is not intuitively obvious that the probability that two randomly chosen natural numbers are coprime is nonzero. I looked up the proof of this remarkable theorem (http://www.cuttheknot.org/m/Proba...). There are two elementary demonstrations there. Both have statements like "the probability that k divides a is 1/k." But one thing troubles me. If k > a, the probability that k  a is *zero*. The simpler of the two proofs goes on to say
<Now, k was just one possibility for the greatest common divisor of two random numbers. Any number could be the gcd(a,b). Furthermore, since the events gcd(a,b) are mutually exclusive (the gcd of two numbers is unique) and the total probability of having a gcd at all is 1 leads to> 1 = ∑q/k^2,
where the summation runs from 1 to ∞.
But it seems to be that for any *particular* pair (a, b) the sum should be truncated at min(a,b). What am I missing? 

Aug1617
  beatgiant: <al wazir>
I don't think you are missing anything. To get the result over all numbers, one takes a limit as min(a,b) goes to infinity. 

Aug1717   ughaibu: Here's some fun in the phi, pi and (almost) squaring the circle story: https://www.goldennumber.net/wpcon... 



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