Number 6 Fair View,
The Garth Chambers,
Dexter University Chapel,
Dexter.
NORREX
DX1 1AY
My Dearest niece Josephina,
thank you for your recent letter, which was thought provoking. Geometry on a pair of chopsticks is indeed novel. May I suggest you consider just one chopstick to start with and consider it to be a truncated cone. No, it is not a silly thing to think about. Mathematics is what you make of it, and if you choose to consider something to be true and then see where that leads you, you are following in the footsteps of the very greatest of mathematicians. Even if what you choose to consider to be true clearly isn’t, it is a legitimate academic exercise to ask, but what if it were true? As long as you consistently keep to the rules of your mathematics your conclusions will be valid. So do pray continue with your explorations.
To answer some of your questions. The value of π does not remain constant if one defines it to be the ration of a circles circumference to it’s diameter. Many assume it cannot change from the anticipated 3.1415926..., however that is only true if one assumes an essentially Euclidian surface on which the circle resides. I say essentially Euclidian, most would say flat, but then one gets into boggy ground as to what that means.
Rather than spoil your fun with your chopstick geometry, I’ll demonstrate what I mean by considering the following gedankenexperiment. If you have a conventional piece of paper four inches by four and a half inches (the extra half inch is purely to provide a gluing tab). On it you draw a circle of two inches in radius that touches three of the sides with a conventional compass and you mark the centre. You can then roll up your paper and glue it into a cylinder using the tab. The circle now touches the top and the bottom of the cylinder and meets itself at the glue join. Clearly if the value of the ratio of the circumference to the diameter, known as π, were 3.1415926... for the ‘circle on the cylinder surface i.e. the paper’ before it still is, so we conclude the cylinder is essentially a Euclidian surface, though it is no longer flat.
This is true provided we stay on the surface of the cylinder. However things change if one considers the projections of the circle on the cylinder onto a surface or allows lines within the cylinder. Let us consider just the latter.
Ignoring practical limitations like the thickness of the paper and the width of the circle line, not to mention inaccuracies in construction, the circumference of the cylinder is the diameter of the circle, 4 inches, so its diameter is 4/π ≈ 1.273 inches. The centre of the circle you marked is thus 1.273 inches from the point at which the circle touches itself if one measures through the space inside the cylinder rather than the 2 inches going round the cylinder surface.
If one says the distance from the centre of a circle to a point on the circumference is the radius, r, then going from our marked centre parallel to the axis of the cylinder r is 2 inches.
However. if one goes across the inside of the cylinder to where the circle touches it self r is now 4/π ≈ 1.273 inches.
The circumference of the drawn circle is given by C = 2πr = 2*π*2 = 4π ≈ 12.566 inches.
If one accepts that the Circumference, C, is twice the radius times π, (C = 2πr)
Then π = circumference divided by twice the radius, (π = C/2/r)
When r = 2 then π = 4π /2/2 = π or the usual 3.142... as expected.
But when r = 1.273 then the value of π, let us use φ for our new value of ‘π’, is given by,
φ = (2 * π * 2)/ 2/ (4/ π) which tidying up a bit gives
φ =2 * π / (4/ π) = 2 * π / 4 * π = 0.5 * π ^ 2 ≈ 4.9348
In fact if you started with r = 2 and move round the circle r would decrease to a minimum of 1.273, where the circle touches itself and the increase back to 2 again at the start. The calculated value of ‘π’ would similarly start at π and increase to φ and then decrease back to π again.
You may wish to do the exercise above and see what it makes you think.
After all that brain work,
Now I need a drink alcoholic of course. Counting the number of letters in the words, gives you 31415926. A useful mnemonic.
But what of other surfaces or other universes? If one draws a circle on a sphere, which of course cannot be made flat without distortion, a problem that has vexed the minds of cartographers time out of mind, the value of π decreases.
Similarly on the curve of a saddle, only there the value of π increases.
Here’s a couple of puzzles for you.
Does the value of π remain constant over the entire surface of a sphere, or does it change according to how big the circle is or where one draws it? You already know the answer to that one. What you already know and the contents of this letter are all you need to arrive at the solution
A trip that takes in a quarter of the equator goes up to the North Pole and then back to the starting point could be considered to be a triangle, let us call it a spherical triangle, which on a Euclidian surface has an angle sum of 180⁰. What is it the angle sum of your trip which has three ‘straight lines’ enclosing the three angles now?
Here’s something for you enjoy. In 1 Kings 7:23 it says a basin was 10 cubits in diameter and 30 in circumference. That would on first sight imply the value of π were 3. However, if the circumference inside were indeed 30 cubits and the diameter from outside to outside were ten. How thick was the basin?
I look forward to your visit on the 13th to the 20th. I have borrowed my friend Algy’s theodolite so we can do some 3D geometry on our trips out. The company I ordered your geometry instruments from have included a navigator’s parallel rulers. They should arrive in time for your birthday. I enclose a copy of ‘Flatland: A Romance of Many Dimensions’ by Edwin Abbott Abbott for your entertainment.
My very best to my dear sister, your father and yourself,
Your loving Uncle Peter
P Halthorp
Professor P. Halthorp
P.S. And here’s another old joke,
There are three kinds of people in the world. Those who can count and those who can’t.
.
Notes.
Flatland: A Romance of Many Dimensions by Edwin Abbott Abbott is an interesting read and available from ebay for £3 including packing and postage.
For any interested in the non Euclidean values of ‘π’ the following is interesting.
https://www.researchgate.net/publication/259702188_Calculati...
Comments
Heh
Math jokes, I love it. :)
Consider all positive odd integers to be of the construction 2n +an-1*2n-1 +an-2*2n-2...+a1*2 +1 where ax can have only the values 0 or 1 and n can be as large as necessary to create a number. Call this the obverse formula.
Consider 2n+ a1*2n-1+ a2*2n-2+...an-1*2+1. Call this the reverse formula.
Consider the obverse but turn all values of a that were 1 to 0 and all values of a that were 0 to 1. Call this the inverse formula. Do the same for the reverse and call it the converse formula.
Consider that if one of these four formulas produces a prime number, is it true that one of the others must also produce a prime number (that may be identical to the first one)?
(I hope I got the HTML right.)
This is a conjecture I came up with in my third year of taking a math degree. :) So far as I know, no one has ever proven or disproven Melton's Conjecture though it is true up into the high 6-digit numbers as far as I can tell.
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
4 formulae
The HTML is clear and I understand what the nomenclture is conveying. I just wish I could manipulate HTML like that! It must I imagine take serious patience. As for the conjecture, I have no idea and the material is unfamiliar to me. A brief skirmish in the literature and in my own resources has turned up nothing as yet. I'll do a deeper search some time. Interesting conjecture though. Number theorist always have asked interesting question, often simple questions, many with deep answers, many with no answers as yet which of course are called conjectures. Long live Riemann whose hypothesis is not a hypothesis, but is a conjecture of the first magnitude.
Regards,
Eolwaen
P.S. The title box will not accept π as valid input and gives the same error message I referred to regarding the interrobang.
Eolwaen
Figured out
I figured out what the problem with the title boxes is. They are used as indices in the database and it's the database code that is rejecting them. The funny thing is that they go through a hashing algorithm to turn them purely alphanumeric before actually being used as indices but the code tests them for suitability before running the hash.
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Unicodes?
So presumably there are rather a lot of unicode symbols that are 'forbidden'.
Regards,
Eolwaen
Eolwaen
If
If you consider the great majority to be rather a lot, then yes. :) The acceptable codes are a tiny subset, a mere penninsular exuberance of the continent of unicodes. :)
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Unicodes
As I suspected. Thank you.
Regards,
Eolwaen
Eolwaen
Melton's Conjecture
10011 = 19 -- the original
11001 = 25 -- the reverse
11101 = 29 -- the inverse
10111 = 23 -- the converse
The first and last number are always one? Only the inner numbers are flipped or reversed?
Okay, i misread the question. "Is it true that one of the others must also produce a prime number?" So not necessarily all of the others.
101111011011110000110101110101111 = 6366456751 is prime
110000100100001111001010001010001 = 6518445137 = 73 × 383 × 233143
111101011101011000011110110111101 = 8248901053 = 19 × 577 × 752431
100010100010100111100001001000011 = 4636000835 = 5 × 19213 × 48259
I hope that I transformed these numbers properly. I started with a long string of randomly punched ones and zeros, and converted the number as prescribed. Of course, none were prime. But one happened to factor into 73 and a huge prime, so I used that huge prime. Of course, it's possible that the prime factor site got it wrong, and 6366456751 might not actually be prime. I'm currently checking its factorization on a different site -- one that's going a lot slower than the first site that I used.
Very sorry, Joyce/Erin.
-- Daphne Xu
Empirical Approach
I love it. Threw some numbers in and see what happens. Ramanujan was slated for it, but not for long!
Regards,
Eolwaen
Eolwaen
Bigger Prime
That's a bigger prime than I had ever tested it with. :)
Of course, I came up with this idea 50 years ago and factoring had to be done by hand. :)
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Conjecture failures
I happened to have a Java sieve prime generator lying around.
In this age of computers, it is simple to check the conjecture. About 25 lines later I had a program to test it.
The first failure was: 139 209 245 175 10001011 11010001 11110101 10101111
Running a test of the first million (2^20) numbers there are 82025 primes with 37038 failures.
What's the size of the first failure?
What's the value of the first failure? This still might be interesting and informative.
Thanks for doing the nasty work.
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
π
I prefer to keep the Euclidean definition of π as the ratio of the Euclidean circumference to the diameter, and treat non-Euclidean surfaces based on the information available.
As the initial part of the Professor's thought experiment points out, role up a plane into a cylinder, and the geometry on the plane remains unchanged. Intrinsically, without looking or going off the plane, the surface is flat. One notices nothing until one has gone far enough in a direction to see the repetition -- the easiest way, one has gone around the circle. Extrinsically, looking or going off the plane, the cylinder is curved.
On the other hand, consider an ideal spherical Earth. A circle at the latitude of φ (for example, 30 degrees = π/6 radians) will have its circumference = 2πR cos(φ) = πR sqrt(3). Go north -- perpendicular to the circle -- through the north pole and back down to the other side of the earth. The radius of the circle is R(π/2 - φ) = Rπ/3.
So circumference/diameter = πR sqrt(3)/(2Rπ/3) = 3 sqrt(3)/2.
The general expression in terms of φ is 2πR cos(φ)/[2R(π/2 - φ)] = π sin(π/2 - φ)/(π/2 - φ). Let φ approach π/2, and the (sin(x)/x) term becomes 1, and the ratio becomes π. So the surface of the sphere is locally Euclidean -- pick a small enough region, and it's approximately flat.
I hope that if I were given only two-dimensional coordinates (such as longitude and latitude) and the appropriate metric as a function of the coordinates, that I would be able to reproduce that result, using only the internal geometry. But I would still like to understand where the rolling-up might occur, and how it would affect things.
-- Daphne Xu
Surfaces, Manifolds?
It is the nature of any surface (Is this a conjecture? Or merely part of the definition of a surface?) that locally it appears Euclidean. That is the property that cartographers rely on to produce flat maps on a piece of paper that portray portions of the oblate spheroid that is Earth. The key word in the preceeding sentences is of course 'locally'. The larger the propotion of the Earth's suface they map the greater the unavoidable distortions they introduce. A street map of a small area is probably near enough Euclidean for such distortions not to be noticeable. On the other hand Greenland, for this and other reasons too it must be said, is nowhere near as large as most maps shew it because for most purposes it is considered to be better to have the distortions where they matter to the least extent. Maps of Greenland do exist that are centred on the centre of Greenland and the point is clearly seen looking at them. Many (all?) spaces cannot be fully described from within them. A simple example is a Möbius strip which can be created by taking the flat strip and after twisting it gluing it together, but it could have been twisted clock- or anti-clockwise which to a creature living on the strip is a meaingless concept. It is only meaningful from without the Möbius universe.
Regards,
Eolwaen
Eolwaen
The Melton Ring
The Melton Ring
Everyone is familiar with a Moebius Strip, a length of paper glued to its own end with a half-twist so that a line drawn down the middle of the paper goes twice around before meeting itself.
But a strip of paper is a real three-dimensional object. Imagine instead of being flat and thin that this paper has a thickness equal to its width. Now the Moebius strip has two edges, one with a line drawn down the middle and one without.
Now imagine that the cross-section of this object were five-sided instead of four-sided before the line was drawn down the middle. Now the line drawn down the middle goes five times around before meeting itself.
What if the object had seven sides? Nineteen? Six hundred and forty-one?
What if it had an infinite number of sides? In other words, a circular cross-section.
And you gave the ends a twist that was irrational to the circumference of the cross-section before gluing the ends together?
Now if you start anywhere on the surface of this object and draw a line perpendicular to the cross-section, that line will be a closed loop infinite in length before it comes back and meets itself. Will that line touch every point on the surface? If not, why not?
Do you have a three-dimensional object with a one-dimensional surface? If not, why not?
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Interesting
The polygon you describe is what I referred to in my lecturing days as an infinitagon. Since presumably as the number of sides approach infinity you are suggesting (or I think you are) that the width of each side approaches zero. At that point we have to consider the ultimate geometric concepts, points with zero dimensions, lines with only one, surfaces with only two and of course solids with three. That would indeed give an apparently three dimensional object with a surface created from a one dimensional line, though one is playing with definitions here. A similar situation exists in the famous 'butterfly wings' image of chaos theory fame where the apparently two dimensional surfaces of the wings are generated by a line that alternates around each wing. When you say a twist that was irrational to the circumferance, I am assuming you are saying that which of the sides are glued to which is random? I can't see that it would alter the situation, what would alter is what order the line segments (each one once around the strip) were assembled in to create the total length of line. Lovely ideas.
Regards,
Eolwaen
Eolwaen
Arbitrarily Close
"Will that line touch every point on the surface? If not, why not?"
I think that the line will be dense in the surface, in that it will approach every point arbitrarily closely. If the twist is a rational fraction of the circumference (m/n), then after n passes, the line will touch where it began. It will have made m cycles around the circumference. The spacing between adjacent paths will be the circumference/n. As n approaches infinity, the spacing approaches zero.
Perhaps that does mean that for an actual irrational number, the entire path will touch every point after an infinite number of passes.
-- Daphne Xu
Irrational
It's not clear to me what you mean by 'irrational number'. If you mean, as I think you do, one approching infinity, then in the limit since the line (a one dimensional concept being the locus traced by a point of zero dimensions) has to go around an infinite number of times, the surface must be filled by definition before the point can arrive back at where it started from. No matter how fast the point tracing the line travels all that would take an infinite amount of time, which can't happen because all this is based on the idea of infinitely divisible spce - there being no possible smallest size since it could always be halved leaving futher space for the point to have to fill - which scientists say is not the structure of the universe. But what do they know? This is mathematics we're playing with and reality has nothing to do with us. Let the scientists get their hands dirty with facts. Mathematicians can roll around in the stuff of their lives all day without even needing a shower, though there may be other perfectly valid reasons to take one.
Regards,
Eolwaen
Eolwaen
Donuts
Okay. You have your cylinder with a straight line drawn straight down the cylinder on one side. Say that the cylinder is 20 meters long, and its circumference is 12 cm. You connect the two ends of the cylinder to make a donut or torus. No twist, the two ends of the line meet. Now a 1/3 twist. One end is 4 cm away from the other. Continue the line parallel to the first. It will go around 4 cm away from the first end, or 8 cm away from the start. Continue again, the line will go another third of the way around the circumference -- 12 cm total -- and will meet the start.
Now for a 2/3 twist: the line will go 8 cm from the start on the first cycle, 16 cm or 4 cm from the start on the second, and 24 cm, 12 cm, or 0 cm from the start on the third. It reached the starting point in three cycles, and went around the 12-cm circumference twice.
Replace 2/3 with m/n: the line reaches the starting point in n cycles (maybe earlier as well, if m/n isn't the simplest form). During those n cycles, it makes m circles around the circumference.
m/n is a rational number, of course. If you twist it an irrational number of circumferences, the line will never meet its starting point. Not only that, an irrational number may be approximated by m/n. A better approximation requires a greater n, meaning that the lines are progressively closer together, and the line will reach its starting point in more cycles.
-- Daphne Xu
Irrationality
Got you. Perfect sense. The same result is obtained when the number of sides of Erin's polygon tends to infinity as a limit and the width of the sides approach zero as a limit.
Regards,
Eolwaen
Eolwaen
Exactly :)
Now as for the irrationality of the twist, all I meant was that the twist was not a ratio, not a fraction that can be expressed in integers, of the circumference. But will the line not completely fill the surface unless the twist is infinitesimal, only one side of the infinitagon? If not, why not?
Consider a multi-sided polygon as the cross section of our ring. If the number of sides is an odd number, then it is possible to choose a twist such that a drawn line will appear on every outer surface before reconnecting. If the number of sides is a prime number, then it is impossible to choose a twist that does not result in a line appearing on every surface before reconnecting.
Is infinity a prime number? If not, why not?
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Infinity
Infinity is not a prime number because it is not a number. It is a concept that finds application in endless branches of mathematics, not least the theory of limits. Given that, the line can get as close as one likes, (arbitrarily close) to filling the space but is still infinitely far away from so doing, because any two adjacent lines have room for an infinite number of lines between them as was demonstrated with the image of the butterfly's wings in a way that was beautifully explained in 'Chaos: Making a New Science' by James Gleick which was published in 1987.
Regards,
Eolwaen
Eolwaen
If you need
If you need to spin your head around, consider https://en.wikipedia.org/wiki/Aleph_number :)
Now consider, what is the cardinality of the set of all cardinal transfinities? It's a trick question. :)
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Cardinality
Which set would that be? It's a trick answer.
Regards,
Eolwaen
Eolwaen
Amazing
I reckon there're c. five times as many words in the comments as in the piece!
W^5 Which Was What We Wanted. English version of QED which means Quantum ElectroDynamics? I googled it to check the spelling, and that's what I got! Which just goes to shew you can't trust anything. I mean who'd have thought you could get dodgy information off google. Quod Erat Demonstrandum.
Regards,
Eolwaen
Eolwaen
Joyce discussed the Melton
Joyce discussed the Melton Ring on her blog.
I see this "infinitagon" ring as the dual of a spring ring, but one which wraps around infinitely many times before meeting itself. This gives us a space-filling curve on a torus without the "twistyness" of a Serpinski curve.
Andrew
Space filling
One needs to be cautious with a term like 'space' (3D) if the line (1D) lies on the surface (2D) of a torus, and even more so with the term filling when the concept is one that moves towards a limit, because if it takes infinitely many terms it can't meet itself. Like division by ever decreasingly small numbers one still can not divide by zero. Division by zero is undefined as is the line meeting itself.
Regards,
Eolwaen
Eolwaen
I was using the phrase "space
I was using the phrase "space-filling curve" as it is used in https://en.wikipedia.org/wiki/Space-filling_curve - but for the surface of a torus rather than a unit-square.
I don't think this curve is a limit; we don't take a sequence of curves with ever smaller twists, but pick an "irrational" twist in the sense that the ratio of the step moved for each loop to the total circumference is irrational. For example, take a cylinder of radius 1unit and join the ends to make a torus, but rotate the ends by one unit; the ratio in this case is 1/pi.
Limit
In the senses that have been used in these comments, by starting from a rational twist, 1/3 was the initial twist, and worrking up say 1/5. 1/7...1/143 &c. an irrational twist is the limit of that process. The rational twists, be the denominator however large, do return to the start. Any irrational one can not do so, in that sense it is the limit of the process, never returning to the start.
If one could assess the Housdorff dimension for a particular case and it were 2, then it is 'area' filling, but again that is based on the idea that in the limit it fills the 2D surface. Clearly if it were <2 it does not fill the 2D surface. I can't see how it could be >2, but that would need more thought, and it is fascinating concept, an over filled suface, shades of super parallel lines!
Regards,
Eolwaen
Eolwaen
Sub-atomic
There are some interactions in sub-atomic particles involving quantum dynamics that require calculations of spaces that are larger inside than outside. These are usually solved by the insertion of elements expressed as fractional extra dimensions. If you count all the extra dimensions needed for all the variations of these equations then you need either 11 or 23 dimensions to describe the real world!
It actually significantly simplifies things to assume that the "real world" is a three dimensional simulated projection of an 11-dimensional space, like a hologram of a globe.
Math. If it doesn't make your head hurt, you're not doing it right. :)
Hugs,
Erin
= Give everyone the benefit of the doubt because certainty is a fragile thing that can be shattered by one overlooked fact.
Head Pain
Ain't that the truth.
Regards,
Eolwaen
Eolwaen
calculations of spaces that
calculations of spaces that are larger inside than outside
Ah. That must be how Dr Who's TARDIS works.
Differential Manifold
I think that the surface or manifold has to be differentiable at least -- continuously differentiable. Piecewise continuously differentiable, perhaps, being locally Euclidean only where the surface is continuously differentiable.
There's also the locally Minkowskian or locally Lorentz manifold, used in General Relativity. General Relativity can only give the intrinsic curvature of space-time. Nothing prescribes the extrinsic curvature, the equivalent of the plane being rolled up, or possibly recombining in a Möbius strip. In my view, that's a defect in GR, accurate though GR is.
-- Daphne Xu
Discontinuity
Regarding your first paragraph. It is hard to envisage, or at least I can't, how a local Euclidean approximation could be used around or touching a discontinuity, so I agree with your remarks concerning differentiability and local Euclidean approximations.
As to General Relativity, it's now much more sophisticated than in its original 'simple' form, which is to say most of the time it's adequate but under 'rare', read bizarre, conditions tweaks have been applied to cover for some of its shortfalls in its applicability, though some yet remain. Those tweaks are usually too much trouble to bother with.
At a lower level, if two cars doing 30mph each have a head on collision, we know the impact speed is not 60mph but a bit less. It's just the headache involved in calculating that bit isn't worth the difference, unless of course you enjoy the fun you're having doing it! Then it's worth paying for.
Regards,
Eolwaen
Eolwaen
Velocity Combination
A little bit less? (30 + 30)/(1 + 30^2/(3600*186000)^2) = 60/1.0000000000000020072969257846122 = 59.999999999999879562184452923508 mph -- about 1.2e-13 mph away from 60 mph. The difference is a bit over ten times the diameter of an atomic nucleus every second. Typically, 30 mph is a course approximation -- or maybe a fine approximation if accurate to the nearest 0.1 mph -- 30.0 mph in other words -- the correction doesn't mean anything.
Suppose v = 0.1c. How important would one think relativity is? Or rather, how far from straight Newtonian (Galilean, traditional) calculations would we be? How far from one is gamma? First approximation: gamma - 1 = v^2/2c^2, or about half a percent from one. For kinetic energy, we need that term. So now the question is, how far from gamma - 1 is v^2/2c^2? We can have v/c somewhat greater than 0.1, and still have the following inequality (if v isn't 0):
3/8 (v/c)^4 < gamma - 1 - 1/2 (v/c)^2 < 1/2 (v/c)^4
We're talking really small numbers here. (Of course, for something such as the global positioning system, one needs to account for both speed time dilation and gravitational time dilation to get the right answers. Electronics and atomic clocks have no problem picking out a loss of a microsecond -- even a nanosecond.)
-- Daphne Xu
A bit less
Like I said. It's a little bit less.To say 'the correction doesn't mean anything' is not a statement a mathematician could make. To me it means everything, simply because it's there. That 30mph + 30mph ≠ 60mph is unarguably true. One either is a mathematician which by definition implies a pedant demanding 100%, not 100% - δ (be δ however small), or one is a scientist or engineer with high level mathematics skills. Sense, reason, and convenience do not enter in to it.
Regards,
Eolwaen
PS [added later] it would be better to say that 30mph + 30mph < 60mph is unarguably true.
Regards,
Eolwaen
Eolwaen
Gotchas
I'm concerned that demand for absolute (and phony) precision is used for "gotchas" and obfuscation rather than enlightenment and understanding.
But let's begin with precision: 30 mph + 30 mph = 60 mph, period. Instead, the relativistic combination isn't that sum.
Then consider that "30 mph" means within an interval containing 30 mph. [29,31] mph, [29.5, 30.0] mph, [29.9,30.1] mph -- it doesn't matter. The additional precision from the relativistic calculation is patently false, because it presupposes a number that is patently false (such as exactly 30 mph). That is to say, 59.99999999999988 mph is not more accurate, it is patently false.
Finally, do we teach kids to add the velocities? Or do we teach the "more accurate" formula? The former can involve understanding, visualization, etc. The latter is strict mindless memorization.
-- Daphne Xu
Ya'll are way smarter than many of us
I read Flatland earlier this year and while it made some sense and was entertaining, today's math lesson is driving me to drink. Now I need a drink alcoholic of course. 31415926. Clever. I memorized pi decades ago because my first calculator did only the 4 basic functions and didn't have a button just for pi. Shows my age. Sadly.
>>> Kay
Age
Amongst other things I collect sliderules, sad I know. I personally used them, when they were expensive and ownership of a quality one was envied. But a top of the range one had not just π on them, but 1/π, π^2, Ln(π) and Log(π) too, and cost half a month's salary for a working person. Interestingly British Thornton, the largest manufacturer in the UK, had laid down all the tooling for their latest and most sophisticated model when the electronic calculator hit the marketplace. Not a single new model slide rule was ever produced. That was how fast calculators took over. I only know of one other product where that happened so rapidly. The introduction of tights [US panty hose?] wiped out stockings in a couple of months. A friend whose father was a market trader had just bought a bulk consignment which took him the best part of 20 years to sell.
Regards,
Eolwaen
Eolwaen
Math
I agree with Barbie. Math is hard.
Mathematics
That's the whole point, Melanie! That's why we do it. It's an intellectual version of eating a super hot sambal oelek chile dish, it turns the act of living into an act of courage.
Eolwaen