[HN Gopher] A trick to eliminate 2p (sometimes)
___________________________________________________________________
A trick to eliminate 2p (sometimes)
Author : mgunyho
Score : 170 points
Date : 2023-06-28 16:05 UTC (6 hours ago)
(HTM) web link (marci.gunyho.com)
(TXT) w3m dump (marci.gunyho.com)
| gunnihinn wrote:
| This notation maybe makes some things in trigonometry or Fourier
| analysis easier to do. Then a wild polynomial appears and all of
| the sudden we have to write
|
| (d bar) x = 1 / 2 pi.
| icapybara wrote:
| Much ado about nothing. Hiding 2pi behind an abstraction layer
| just makes things harder to debug later.
| broses wrote:
| People have proposed introducing a symbol for 2p before, most
| often t. I like to go a step further and introduce a symbol for
| 2pi. I use pi with a dot above it, pronounced "pi dot". Pi dot
| can be defined as the period of the exponential function (which
| can be defined in terms of its Taylor series). Then 2p is pi dot
| / i, and p is pi dot / 2i. Of p, 2p, and 2pi, 2pi is probably the
| most natural, even though it's imaginary. I suppose that depends
| on the type of math you're doing though.
|
| On a similar note, when doing quantum physics, I like to
| introduce h dot, which is i x h bar. There are tons of formulas
| where you either get i x h bar or -i / h bar, but these are just
| h dot and 1 / h dot, so this removes a little sign confusion and
| saves a little handwriting.
|
| People will argue that real constants are more natural, but maybe
| they're not. Maybe radians are naturally imaginary, so if h bar
| is meant to have dimensions of energy time per radian, then it's
| better to use the imaginary h dot.
| dan_fornika wrote:
| Would it be more appropriate to call 2pi "tau dot"?
| linuxdude314 wrote:
| This is the way!
|
| Introduce all the constants you need.
|
| Don't redefine functions and operators for syntactic sugar.
| DavidSJ wrote:
| To address the problem they discuss at the end with defining Th =
| e^2pi, they could instead define Th(x) = e^2pix, the circular
| analog to the exponential function exp (which is really more
| fundamental than exp(1) = e anyways).
| abecedarius wrote:
| Note there's an existing notation which I've mostly seen in
| lower-class settings like high-school textbooks: r <angle-sign>
| theta, for the complex number r e^(i theta). Optionally leave
| out the r.
|
| So you have that "most beautiful formula in all of
| mathematics":
|
| <angle-sign> tau = 1
| raldi wrote:
| I was once lucky enough to take a physics class taught by the
| head of the department, and I remember one of his policies on
| tests or homework was that if you got an answer that was off by
| 1/2 or 2*pi or anything like that, he'd nonetheless issue full
| credit because "you got all the physics right."
| thumbuddy wrote:
| Clever!
| NotYourLawyer wrote:
| Interesting, but I wonder if you'd get all the same benefits by
| just measuring angles in units of turns instead of radians. That
| seems cleaner than the weird dbar differential stuff.
| gorkish wrote:
| This is more or less "the point" for those of us who argue for
| tau instead of pi.
|
| I should note that using this "trick" of prescaling rotations
| by 2pi so that they are in the 0..1 range is de rigueur in
| computer graphics programming.
| JdeBP wrote:
| For goodness' sake, don't tell Matt Parker that one can write
| computer programs where 2p is rescaled to 1!
|
| We'll never hear the end of it. There'll be some Python
| program that takes four months to calculate
| 1.00000000000000000000000000000001 . (-:
| kccqzy wrote:
| That's actually exactly the question I asked my math teacher
| when I first learned about radians. I mean, I learnt degrees
| when I was very little, at an age when one tended not to
| question why, but I learned radians at an age old enough to
| question why. The answer I received was about making
| trigonometric identities cleaner: the derivative of sine
| becomes "just" cosine rather than a hypothetical turn-based
| sine (called usin by the article) having a derivative of a
| turn-based cosine multiplied by 2pi.
|
| But this article seems to do a good job explaining that a lot
| of those 2pi factors appear when you deal with differentiation.
| So it seems useful to have _both_ turn-based trigonometric
| functions and this new differentiation operator.
| paulddraper wrote:
| That's a really good answer.
|
| And justification in general for "why radians" vs degrees,
| gradians, turns, whenever
| linuxdude314 wrote:
| Due to the fact differential operators are linear and the
| nature of the accumulation of constants of integration it's
| pretty easy to prove that the differential operator proposed
| is in fact equivalent to the standard derivative operator.
|
| Source: used to tutor calculus and differential equations in
| college.
|
| Generally speaking this is not a useful trick.
|
| There are _plenty_ of amazing ways to leverage Euler's
| identity but I fail to see how this is one of them.
| scythe wrote:
| You can always define angles in turns. But the problem is that
| it conflicts with the definition cos(x) = Re{e^(ix)}. Trig is
| not so easily separated from the rest of mathematics.
| NotYourLawyer wrote:
| Yeah, you're right. And radians are what make the trig
| identities involving derivatives work out nicely.
| adrian_b wrote:
| The simpler derivation formula was important when such
| symbolic computation was done by hand.
|
| Now, except perhaps for school exercises, anything
| complicated is done with a computer and this advantage is
| much less important.
|
| The increased accuracy and simpler formulas in other places
| when measuring angles in cycles a.k.a. turns vastly
| outweigh the advantage of radians for differentiation.
|
| In real applications you almost always compute the
| derivative of sin(a*x), not of sin(x), so you have to carry
| a multiplicative constant through derivations anyway and
| the single advantage of the radian vanishes.
| adrian_b wrote:
| There is no need for that definition.
|
| It is possible to completely remove the e^x function from
| mathematics without losing anything.
|
| It is possible to express everything using a pair of
| functions, the real function 2^x and the complex function
| 1^x.
|
| Then the cosinus and the sinus are the real and imaginary
| parts of 1^x (where x is measured in cycles a.k.a. turns).
|
| The only disadvantage of this approach is that symbolic
| differentiation and integration are more complicated, by
| multiplication with a constant.
|
| In my opinion the simplifications that are introduced
| everywhere else are more important than this disadvantage.
|
| The real and complex function e^x was preferable in the 19th
| century, when numeric computations were avoided as too
| difficult and simple problems were solved by symbolic
| computation done with pen and paper.
|
| Now, when anything difficult is done with a computer, both
| the function e^x and associated units like the radian and the
| neper are obsolete.
|
| When programming computations on a computer, using 2^x and
| 1^x results in simpler and more accurate computations.
|
| Moreover, when doing real physical measurements it is
| possible to obtain highly accurate values when the units are
| cycle and octave, but not when they are radian and neper.
| kandel wrote:
| Mind explaining how to express some simple functions? Im
| interested
| adrian_b wrote:
| All mathematical formulas can be inter-converted based on
| the identity:
|
| e^(x + i*y) = 2^(x/ln2) * 1^(y/2Pi)
|
| Most formulae from textbooks are written in such a way to
| be simpler with e^x and its inverse, but it is almost
| always possible to move the constants ln2 and 2Pi between
| various equations so that in the end they will disappear
| from most relations, with the exception of the derivation
| or integration formulae.
|
| In most applications, more equations are simplified than
| those which become more complicated.
|
| A very important advantage of 2^x and 1^x versus e^x is
| that for the former the reductions of the argument to the
| principal range where the function is approximated by a
| polynomial can be done with perfect accuracy and very
| quickly, unlike for the latter. Moreover, for the former
| it is easy to verify the accuracy of any approximation,
| because for any argument that is represented as a binary
| number the functions 2^x and 1^x can be computed with a
| finite number of sqrt invocations (based on the formulae
| for half angle) and sqrt can be computed with any number
| of desired digits. Computing e^x with an arbitrary
| precision is trickier, because it requires criteria for
| truncation of an infinite series.
|
| It should be noted that 1^1.0 = 1, 1^0.5 = -1, 1^0.25 =
| i, 1^0.75 = -i
| kandel wrote:
| so you express x (the simple polynomial) as ln(2^(x/ln2)?
| or as an infinite series expansion?
| movpasd wrote:
| Thing is, angles don't really have units (the technical term is
| they are dimensionless). They are a length (the subtended arc
| of a circle) divided by a length (the radius of the circle).
| When you want to do something like get a sine wave of period T,
| you inevitably have to include a 2p somewhere.
|
| Speaking as someone who had to write down many 2p's in
| university (especially as I find angular quantities like
| angular frequencies ugly and unintuitive to work with), I think
| this notational trick would've been very useful!
| Misdicorl wrote:
| This is not true. Angles very much have units and it's why
| you can express the same concept with different numbers. Pi
| equals 180 degrees equals 0.5 turns.
|
| 1 radian has different units than 1 steradian and if they
| didn't there wouldn't be a need for two different words to
| denote them.
|
| The quantity is a ratio of two lengths, and the length
| measure does "drop out". But it's not just any ratio, it's a
| very particular ratio, and the unit defines the
| particularness of that ratio.
| jacobolus wrote:
| The reason it is confusing is because an angle measure is a
| kind of logarithm of a rotation, and logarithms (sort of)
| have a unit: the base.
|
| The appropriate canonical representation of a rotation is a
| unit-magnitude complex number _z_ = exp _ith_ = cos _th_ +
| _i_ sin _th_ , which has a planar orientation (whatever
| plane _i_ is taken to represent; if you want to represent a
| 3D rotation you can replace _i_ with an arbitrary unit
| bivector) but is unitless.
|
| Such a rotation _z_ can be thought of as the ratio of two
| vectors of the same magnitude: _z_ = _u_ / _v_ satisfies
| _zv_ = _u_ , i.e. is the object by which you can multiply
| _v_ on the left to obtain _u_. Whatever original units your
| vectors _u_ and _v_ had gets divided away.
|
| This is similar to the way the "ten" in "scale by ten" is
| unitless, but if you take the logarithm you get "scale by
| 10 decibels" or "go up by 3 octaves and 3.9 semitones",
| which have the base of the logarithm as a kind of unit.
| Misdicorl wrote:
| I think I fundamentally disagree with you. Angles do not
| have a "base" any more than meters do (being embedded
| inside some metric space could be considered a base I
| suppose).
|
| But you seem to be drawing a distinction between meters
| and angles in your analogy where I assert none exists.
| The base of a number system only affects representations.
|
| This is not true for divisions of lengths. 1 meter
| divided by 2 meters is 0.5 as a number. But it is only
| 0.5 radians under (1ish) specific arrangements of those
| lengths in a particular metric space
| jrockway wrote:
| Yeah, units that algebraically reduce to 1 are always very
| interesting to me. Consider a chart showing how many CPU
| seconds you're consuming per second. The unit is
| seconds/second, which is equal to 1, but it is still a
| distinct concept from radians.
| 6gvONxR4sf7o wrote:
| Angles aren't dimensionless any more than lengths are
| dimensionless (feet per second makes just as much sense as
| rpm). It's just that angles have symmetries that lengths
| don't, which is where 2 pi comes in. Do you want units where
| your symmetries are expressed in multiples of 1, 2, or 2 pi
| (for turns, half-turns, and radians, respectively)?
| linuxdude314 wrote:
| If you mean units instead of dimensions I agree.
|
| There is a natural reason for pi occurring in physics that
| makes little sense to ignore.
|
| Treating it as something to be dealt with misses the forest
| for the trees.
|
| Radians are the naturally occurring Euclidean unit of
| angular measurement.
| eigenket wrote:
| Angles are absolutely more dimensionless than lengths are.
| For an easy check you can't add quantities where the
| dimension differs, which means it doesn't make sense to add
| a length to its cube. On the other hand it _does_ make
| sense to add an angle to its cube - this is a necessary
| component of computing sin(angle) by the power series
| sin(angle) = angle - (angle^3) /6 + ...
| 6gvONxR4sf7o wrote:
| How can we compute angle - (angle^3)/6?
| 360 - (360^3)/6 = -7M degrees
|
| or is it this? 2*pi - (2 * pi)^3 / 6 =
| -35 radians = -2k degrees
|
| Or maybe this? 1 - (1^3)/6 = 0.8 turns
| = 300 degrees
|
| They're wildly inconsistent because I'm not taking the
| units into account and we have to take the units into
| account.
| eigenket wrote:
| Units are not the same as dimensions, something can have
| a dimension of 1 (which is what we usually mean by
| "dimensionless") and still have different units, just as
| something can have a dimension of length but still be
| measured in meters or feet.
|
| As far as you three examples go, which is "correct"
| depends on what you are trying to calculate - if you want
| this to approximate the power series for sin close to 0
| you should use radians. Otherwise you use something else.
| adrian_b wrote:
| Actually the dimensions are by definition the same as the
| fundamental units, i.e. the units that are chosen freely,
| independently of all other units.
|
| A dimensional formula of a quantity just writes its unit
| as a function of the fundamental units.
|
| In any equality of physical quantities, in the two sides
| not only the dimensionless numeric values must be equal,
| but also the units must be equal, which is usually
| expressed by saying that the dimensions must be the same,
| and it is verified by writing in both sides the
| dimensional formulae, i.e. the units of both sides as
| functions of the fundamental units.
|
| A dimensionless quantity is a ratio of two quantities
| that are measured by the same unit, so that the units
| simplify during the division.
|
| There may be different but related dimensionless
| quantities, which are differentiated by different
| definitions of those quantities, but a dimensionless
| quantity cannot have different units.
|
| This is just meaningless mumbo-jumbo that has been sadly
| introduced in the documents of the International System
| of Units, in 1995, after a shameful vote of the
| delegates, who have voted automatically, without thinking
| or discussing, a vote equivalent with establishing by
| vote that 2 + 2 = 5.
| 6gvONxR4sf7o wrote:
| Sure, but that's orthogonal to the "angles don't really
| have units" assertion and the "it does make sense to add
| an angle to its cube" assertion, which are the ones I'm
| responding to.
|
| As another example for the second assertion, you can
| compute e(-t) via power series too, adding seconds to
| seconds squared and seconds cubed, etc, which comes up
| all the time. But that doesn't mean `dimensionless +
| seconds + seconds^2` implies seconds are dimensionless
| any more than sin's series with `angle + angle^3` implies
| that angles are dimensionless.
| orangecat wrote:
| _you can compute e(-t) via power series too, adding
| seconds to seconds squared and seconds cubed_
|
| The argument of exp does have to be dimensionless,
| exactly because adding seconds to seconds squared doesn't
| work. If t has units of time, there has to be another
| factor with units of inverse time, for example continuous
| compound interest is exp(rate*time).
| eigenket wrote:
| If you say "angles have units" I agree with you -
| obviously you can measure them in degrees or radians or
| whatever you want. I was responding to the claim
|
| > Angles aren't dimensionless any more than lengths are
| dimensionless
|
| They are dimensionless, but they still have units. The
| concepts are orthogonal.
|
| As for the question about adding an angle to its cube, I
| would say the enormous usefulness of computing trig
| functions by power series suggests strongly that this is
| meaningful.
| adrian_b wrote:
| > They are dimensionless, but they still have units. The
| concepts are orthogonal.
|
| The concepts are not orthogonal, they are incompatible.
|
| A dimensionless quantity (which angles are not) is by
| definition the ratio of two quantities that are measured
| with the same unit.
|
| When you compute the ratio by division, the two identical
| units disappear from the result, therefore the result is
| indeed dimensionless.
|
| There is no way to choose a unit for a dimensionless
| quantity in the usual sense.
|
| At most you could define a new different dimensionless
| quantity, as the ratio of two dimensionless quantities,
| i.e. as a ratio of ratios, but because it needs a
| different definition this should better be viewed as a
| different quantity, not as the same quantity with a
| different unit.
| eigenket wrote:
| Ok, you've replied to quite a lot of my comments, with
| some fairly confusingly worded responses. Some of your
| claims appear to be incompatible as far as I read them.
|
| For example you claim that angles are not dimensionless,
| but that dimensionless quantities are formed by the ratio
| of two quantities with the same unit. Since the angle
| subtended by an arc in a circle is the ratio of the arc
| length and radius, it would seem that these two claims
| contradict each other.
|
| I do agree that without some aditional work the power
| series argument I wrote above does seem to be wrong.
| adrian_b wrote:
| No, even if you have reproduced a definition of the plane
| angle that is encountered in many textbooks "the angle
| subtended by an arc in a circle is the ratio of the arc
| length and radius", this definition is very incorrect.
|
| This very wrong definition forced upon many students is
| the root of all misconceptions about plane angles.
|
| The reason why this definition is wrong is because some
| words are missing from it and after they are added it
| becomes obvious that its meaning is different from what
| many teachers claim.
|
| First there is no relationship whatsoever between the
| magnitude of a radius and the magnitude of an angle. What
| that definition intended to say was:
|
| "the angle subtended by an arc in a circle is the ratio
| of the arc length and of the length of an arc whose
| length is equal to the radius".
|
| By definition, a radian is defined as the angle subtended
| by an arc whose length is equal to the radius.
|
| To explain how plane angles are really defined would take
| more space, but the only thing that matters is that the
| characteristic property of plane angles is that the ratio
| between two plane angles subtended by two arcs of a
| circle is equal to the ratio of the lengths of the two
| arcs.
|
| Introducing this characteristic property of the angles in
| the so-called definition from above reduces it into the
| sentence "the angle is measured in radians". This is
| either a trivially true sentence when the angle is indeed
| measured in radians, or it is a trivially false sentence
| when the angle is measured e.g. in degrees. It certainly
| is not a definition.
|
| If that had been the definition of plane angle, that
| would have meant that the plane angle was discovered only
| in the second half of the 19th century, together with the
| radian, while in reality plane angles have been used and
| measured with various units for millennia.
|
| To measure plane angles, it is necessary to first choose
| an arbitrary angle as the unit angle, for instance an
| angle of one degree.
|
| Then you can measure any other angle by measuring both
| the length of the corresponding arc and the length of the
| arc corresponding to the chosen unit angle. Then the two
| lengths are divided, giving the numeric value of the
| measure of the angle.
| 6gvONxR4sf7o wrote:
| > I would say the enormous usefulness of computing trig
| functions by power series suggests strongly that this is
| meaningful.
|
| That argument holds just as true for dimensional
| quantities frequently computed by power series though,
| which means it can't be valid.
| eigenket wrote:
| I would argue that dimensionful quantities are never used
| as arguments to power series, however you are correct
| that this does not imply that angles are dimensionless,
| since (like we do with other quantities, we can divide by
| whatever unit we like to get something dimensionless). I
| withdraw that argument.
|
| A better argument that angles are dimensionless is that
| dimensionless quantities are formed by the ratio of two
| quantities with the same dimension, and the angle
| subtended by an arc in a circle is given by the ratio of
| the arc length and the radius.
| Someone wrote:
| I'm not sure it will work everywhere, but for _sin(x)_ ,
| one can write sin(x) = x/(1
| radian)! - x3/(3 radians)! + ... = x/(1 radian) -
| x3/(1 radian x 2 radians x 3 radians) + ...
|
| That makes the 'radians' units cancel out.
| adrian_b wrote:
| Your argument is wrong.
|
| Any physical quantity, for instance length, can appear as
| an argument of a nonlinear function that can be developed
| in a Taylor series. So your example would be identical
| for any other quantity not only for angle. I can make an
| analog computing element where a voltage is equal to the
| sinus of another voltage, so after your theory, voltage
| is dimensionless.
|
| The reason why this is possible is that the arguments of
| such nonlinear functions are either explicitly or
| implicitly not the physical quantities, but their numeric
| values, i.e. the ratios between those quantities and
| their units, which are dimensionless.
|
| In the case of the nonlinear sinus function, what is
| usually written as sin(x) is just one member of a family
| of functions where the arguments are angles implicitly
| divided by units of plane angle:
|
| sin(x) is the sinus function with the angle implicitly
| divided by 1 radian
|
| sin(x * Pi/2) is the sinus function with the angle
| implicitly divided by 1 right angle
|
| sin(x * Pi*2) is the sinus function with the angle
| implicitly divided by 1 cycle a.k.a. turn
|
| sin(x * Pi/180) is the sinus function with the angle
| implicitly divided by 1 sexagesimal degree
|
| It is very sad that the logical thinking about angles of
| most people has been perverted by what they have been
| taught in school, which is just a bunch of nonsense
| copied again and again from one textbook to another.
| mgunyho wrote:
| > sin(x) is the sinus function with the angle implicitly
| divided by 1 radian
|
| This to me sounds like the most natural explanation. For
| example, in a sibling comment someone mentioned that "you
| can calculate e^(-t)", but I disagree: in physics it's
| always e^(-t / T), where T is some time constant, so that
| the argument of the exponential is dimensionless. Same
| applies to sin(x): usually we write something like
| sin(2pi f t), where the units of f and t cancel out, and
| the 2pi is there to cancel out the invisible implicit 1
| radian. sin(ft) would be wrong, at t = 1 / f you wouldn't
| have advanced by a full cycle.
| adrian_b wrote:
| The claim that plane angle, solid angle and logarithms are
| dimensionless quantities is a horrendous mistake and many
| generations of physicists have been brainwashed by being
| taught this aberration without ever stopping to think whether
| this claim can be proved.
|
| I will discuss only the plane angle, because it is the most
| important, but the situation is the same for solid angle and
| logarithms.
|
| The justification commonly given is that the plane angle is
| dimensionless because it is the ratio of two lengths, the
| length of the corresponding arc and the length of the radius.
| This justification is stupid, because that is not the
| definition of the plane angle, but it already includes the
| choice of a particular unit.
|
| As formulated. this justification only states the trivial
| truth that the numeric value of any physical quantity is the
| ratio between that quantity and its unit. By the same wrong
| justification, length is dimensionless, because it is the
| ratio between the measured length and the length of a ruler
| that is one meter long.
|
| Correct is to say that the plane angle is a physical quantity
| that has the property that the ratio between two plane angles
| is equal to the ratio between the lengths of the
| corresponding arcs.
|
| This is a property of the same nature like the property of
| voltage that the ratio of two voltages across a linear
| resistor is equal to the ratio of the electric currents
| passing through the resistor. This kind of properties are
| frequently used in the measurement of physical quantities,
| because few of them are measured directly but in most cases
| ratios of the quantities of interest are converted in ratios
| of quantities that are easier to measure.
|
| This property of the plane angle allows the measurement of
| plane angles, but only _after_ an arbitrary unit is chosen
| for the plane angle. Because the choice of the unit is
| completely free, i.e. completely independent of the units
| chosen for the other physical quantities, the unit of plane
| angle is by definition a fundamental unit, not a derived
| unit.
|
| The freedom of choice for the unit of plane angle is amply
| demonstrated by the large number of units that have been used
| or are still used for plane angle, e.g. right angle (the unit
| used by Euclid), sexagesimal degree, centesimal degree, cycle
| a.k.a. turn, radian.
|
| The fundamental units of plane angle, solid angle and
| logarithms must never be omitted from the dimensional
| formulae of the quantities, otherwise serious mistakes are
| frequent & such mistakes have delayed the progress of physics
| with many years (e.g. due to confusions between angular
| momentum & action; the Planck constant is an angular
| momentum, not an action, as frequently but wrongly claimed).
| This is a problem especially for the unit of plane angle,
| which enters in the correct dimensional formulae of a great
| number of quantities, including some where this is not at all
| obvious (e.g. magnetic flux).
| cycomanic wrote:
| The argument about why not to include the i in 2 pi i is
| incorrect. The problem is he says (e^x)^i2pi = e^i2pix does not
| work because ln(e^i2pi)=0. But he needs to use the complex
| logarithm. And for the complex logarithm ln(e^z) =z for z in C.
|
| If that wasn't the case calculation rules of logarithms and
| exponentials would depend on if arguments are complex or real, a
| lot of physics would become much more complicated suddenly.
| enriquto wrote:
| You can also eliminate the constant in some of the integral
| formulas by using dx instead of dx. I'm surprised the author does
| not propose this.
|
| However, some constants will still remain. Most conspicuously,
| the 2p constant in the very definiton of the Fourier transform. I
| once took a personal crusade to eliminate _all_ such constants in
| the elementary Fourier formulas (plancherel-parseval, convolution
| theorems, commutation with derivatives), and it turns out to be
| possible by using the Lebesgue measure divided by sqrt(2p) in all
| the integrals. Thus it may seem that defining dx=dx /sqrt(2p) can
| be a better choice.
| BlueTemplar wrote:
| A bit similar to how pi is only half of a turn, so you need to
| apply the transformation twice (so _square_ root) to get back
| where you were(ish) ?
| mgunyho wrote:
| > You can also eliminate the constant in some of the integral
| formulas by using dx instead of dx. I'm surprised the author
| does not propose this.
|
| In the post I propose doing that for Gauss' theorem and
| Cauchy's formula, because there it's convenient, heh. But to me
| it feels better to use Th^ix than a scale factor in front,
| since the 2pi is always present in the exponential, while the
| prefactor can be avoided in Fourier transforms if you keep the
| 2pi in the exponential (or hide it inside Th). Does this not
| apply also to the elementary formulas you mention?
| enriquto wrote:
| > Does this not apply also to the elementary formulas you
| mention?
|
| Oh, you are right! I disliked the 2pi factor in the
| exponential because it messes with derivatives. But if you
| define your scaled derivative then the factor disappears
| again. So cool!
| scythe wrote:
| If you're really slick about it, you even "fix" the Gaussian
| integral this way.
|
| Let e = e^sqrt(2pi), dex = dx/sqrt(2pi), and we have
|
| int_{R}(e^(int_0^x(t det)) dex)
|
| = int_{R}(e^(sqrt(2pi) x^2/(2 sqrt(2pi))) dx/sqrt(2pi))
|
| = 1/sqrt(2pi) int_{R}(e^(x^2/2) dx)
|
| = sqrt(2pi) / sqrt(2pi)
|
| = 1
| enriquto wrote:
| > exp(sqrt(2pi))
|
| Now, this is a number that I don't recall having seen before.
| The letter e seems strangely fitting for it
|
| e = 12.2635111...
| scythe wrote:
| I was with him until this point:
|
| >This notation could be abused even further by denoting _dx_ = 1
| /(2p) _dx_ , which can then simplify some integral formulae,
|
| But now you're screwing up all of your previous integral
| formulae!
| version_five wrote:
| I once read a book that proposed a "new" trigonometry that iirc
| worked with the hypotenuse squared and maybe the sin^2 of an
| angle as it's base quantities, and the author showed how easy it
| was to do stuff. This feels about the same. Not wrong, but not
| really useful once you've learned the usual way to do it, not
| easier to learn, and you'll be forever confused if it's all you
| learn.
| kevin_thibedeau wrote:
| Trigonometry is really about circles and rotations. The
| triangles are just an artifact of static diagramming. Zeroing
| in on triangle properties suggests a lack of fundamental
| understanding.
| alecst wrote:
| Responding to this part in the article:
|
| > I'm not entirely sure about the intuitive meaning of "taking
| the derivative and dividing by 2p". Is there some sort of
| fundamental connection to periodic functions?
|
| If you have a function f(x) where x is measured in radians, and
| there are 2pi radians per turn, then you can change variables.
|
| Let t represent turns. One turn is 2*pi rad, and you want t = 1
| when you've gone all the way around in x, so t = x/2pi.
|
| By the chain rule,
|
| df(x)/dx = df(t)/dt dt/dx = 1/2pi * df(t)/dt
|
| So I think this might be the meaning you're looking for when you
| do the rescaling of the derivative.
|
| You're using turns as units instead of radians.
| cos(x=2pi)=cos(t=1)=1, and so on.
| killthebuddha wrote:
| FWIW they mention this in the article:
|
| > I like this, because it kind of eliminates the need for
| radians: the x in usin(x) has the unit of "turns". I think this
| is conceptually much simpler.
| zackmorris wrote:
| I wonder if this would help for elliptic integrals. They are
| notoriously hard to solve, and I keep hitting them in my hobbyist
| calculations around magnetic fields. This is maybe the best video
| I've found to make them approachable:
|
| https://www.youtube.com/watch?v=SJtbeg_PZ30
|
| If anyone has any abstractions that might help, maybe around the
| nome the author mentioned, I'd love to hear them!
|
| https://en.wikipedia.org/wiki/Nome_(mathematics)
| evanb wrote:
| dbar is pretty common in lattice field theory notes (though I've
| never seen it in a book), where it is used because fourier
| integrals naturally come with a 1/2p.
| [deleted]
| [deleted]
| phonebucket wrote:
| Easy trick to eliminate difficulties arising from pi: fix them
| via legislation [0].
|
| [0] https://en.wikipedia.org/wiki/Indiana_Pi_Bill
| linuxdude314 wrote:
| Is this really not an elaborate troll?
|
| Arguing that a literal mathematical equivalence (e^i _x = e^2pi_
| ix) that you then use to "redefine" trig functions so you can
| reformulate physics to mitigate making errors dividing my a
| constant is completely absurd.
|
| In situations when there are strings involving 2pi where this
| makes any kind of sense typically a new constant is introduced to
| incorporate it.
| orangecat wrote:
| Interesting. It's probably not worth defining a new constant for
| exp(2p), but this is a further demonstration of the Tau
| Manifesto's argument that 2p is much more of a fundamental value
| than p.
| garbagecoder wrote:
| Fundamental sounds like a value judgment. Pi is transcendental.
| 2 isn't. That's really the distinction. Unless there were other
| finite factors in pi, that is the number that's always going to
| have to be approximated in computation.
| ohwellhere wrote:
| "Tau is transcendental. 0.5 isn't."
|
| It's definitely a value judgment. But the value judgment is:
| Is the radius or the diameter more fundamental to a circle?
| BlueTemplar wrote:
| As the Tau manifesto points out, tau/4 = pi/2 = lambda is
| pretty fundamental too ! (The "orthogonality constant" ?)
| garbagecoder wrote:
| Geometry isn't the only application of pi. Nor is the
| geometric interpretation the only way to determine what is
| "fundamental."
|
| Why not use Darians? http://proper-pi-manifesto.com
| bmacho wrote:
| Fundamental sounds _meaningless_ to me.
|
| Talking about statements and words with _meaning_ I am sure,
| that Tau _is_ the more useful circle constant out of these
| two: less symbols, conceptually clearer equations.
|
| Also turn is generally a more useful measurement unit for
| angle than radian or degree, but radian has its own merits,
| and not disposable, unlike pi.
| sfpotter wrote:
| "[...], and it's dimensionless, so you can't easily check if you
| forgot to divide or multiply by it."
|
| For what it's worth, it's often the case that a factor of 2pi is
| the difference between something being in terms of cycles/sec or
| rad/sec. In an experimental context, it usually isn't too
| difficult to judge which of these "units" a quantity you're
| looking at is in...
| failuser wrote:
| I'm surprised by the number of positive replies. Obviously the
| current notation is made up like all notations, but this is just
| a waste of time, 2p naturally arises in so many places. The h-bar
| for the Plank constant is just a product of not agreeing what
| constant to denote. sin(x)=x+o(x) is just too nice to give up.
| Switching units needs a way greater benefit than this.
| ctoth wrote:
| Unrelated to the content but complaining about the website is a
| popular thing to do here so I'd like to share my experience, as a
| blind user.
|
| Here is what my screen reader sees for this page:
|
| Recently, I came up with a trick that can get rid of
|
| in many cases. It's pretty simple, but it has some interesting
| implications. This is the trick: I just define a new derivative
| operator, like so:
|
| That's all. You just take the derivative and then divide by
|
| . I call this derivative operator with the bar on the upper
|
| the reduced derivative.Now, why is this interesting? To start
| off, we'll note that the unique function
|
| ... The unfortunate truth is a ton of math content on the web
| reads like this. It has crippled me as a blind user who would
| like to appreciate math for over a decade. In university I was
| forced to pursue a degree other than CS because the math program
| used software which produced output like this and refused to
| change.
|
| There has been technical progress, and many sites are starting to
| work better--Wikimedia most fantastically, but this old bugbear
| made me want to speak up and beg people to try and review their
| math content with a screen reader before publishing (I think
| MathJax has some built-in accessibility now?).
| mgunyho wrote:
| Author here, I'm sorry to hear that it doesn't work well with a
| screen reader. I tested it with the reader mode of Firefox,
| which renders MathML perfectly, although I don't know how that
| would translate to a screen reader. Safari reader mode renders
| the math inline, like this: I just define a
| new derivative operator, like so: dxd f(x)[?]2p1 [?]dxd f(x).
| That's all.
|
| while Chrome's reader mode just fails to recognize the content
| entirely, even though it's the most basic <body><div
| id="content"><p> ... structure possible. I have basically zero
| web dev experience so I don't know how to fix this, maybe I
| need to tweak the KaTeX settings.
|
| I think it's quite sad that math is so difficult on the web.
| While setting up the blog, I looked around and it seemed like
| FF is the only browser with proper MathML support, but I think
| that was also being phased out because it's apparently buggy
| and hard to maintain. IMO, the screen reader version should
| just basically be the LaTeX source, which is probably kind of
| awful when read out loud, but at least it would be unambiguous.
| zamadatix wrote:
| Chrom* browsers just got decent MathML support finally, so I
| don't think it's on the way out quite yet. Some still like
| other solutions more though.
| hoten wrote:
| This does seem like a bug/missing feature in Chrome's a11y
| tree, which you can see in the DevTools with an
| experimental feature enabled:
| https://i.imgur.com/ckWnrlL.png
| hoten wrote:
| You used the Katex library for rendering math symbols, which
| currently emits code that supports MathML and falls back to
| an HTML rendering (this is my understanding after reading the
| HTML for a bit). The MathML content is marked up correctly
| such that browser should be able to construct the a11y tree
| correctly. I didn't check, but apparently other browsers like
| FF and Safari construct the a11y tree from the MathML markup
| correctly.
|
| Chrome _just_ added support for MathML, so the lack of a11y
| support here is not surprising. I found this bug report which
| I believe covers this: https://bugs.chromium.org/p/chromium/i
| ssues/detail?id=103889...
|
| In short - you didn't do anything incorrect, and AFAIK any
| temporary fix to improve a11y for Chrome users would involve
| some heavy lifting in the Katex library.
|
| I suggest interested parties to star the above bug report.
|
| ___
|
| Other commenters mentioned the aria-hidden property being
| set. This is intentional, as without it Safari/FF/compliant
| screen readers with MathML support would double-read the
| content. I'm honestly not sure what the ideal markup would be
| to support both types of browsers - should a solution exist,
| it may involve using JavaScript to change the markup based on
| the user agent detected.
| codazoda wrote:
| Yes, I'm pretty sure it's that aria label. The first line of
| this code. <span class="katex-html" aria-
| hidden="true"> <span class="base">
| <span class="strut" style="height:0.6444em;"></span>
| <span class="mord">2</span> <span class="mord
| mathnormal" style="margin-right:0.03588em;">p</span>
| </span> </span>
|
| My first inclination was to simply run `say` on my Mac
| comamdn line and paste the line in. That read it correctly.
| The other commenter called out the `aria-hidden` attribute
| and I'd guess that's it. It's explicitly hidden.
|
| Apparently this is all intentional as outlined in this bug
| report.
|
| https://github.com/KaTeX/KaTeX/issues/38
| mgunyho wrote:
| But note how right above this span there is the <span
| class="katex-mathml">, which is not aria-hidden, and which
| contains the MathML representation of the equations. I
| would have assumed that a screen reader would look at that,
| since it contains the semantical information.
| zerocrates wrote:
| I bet the "reading" modes of the browsers are just using the
| HTML that's visible on the page, but that's marked explicitly
| as hidden from screenreaders, using the aria-hidden
| attribute.
| mgunyho wrote:
| From what I can tell it's the opposite (on Firefox): the
| math is in the source HTML twice, once in <span
| class="katex-mathml">, which is hidden using CSS in the
| non-reader mode but is rendered in the reader mode, and
| <span class="katex-html" aria-hidden="true">, which is what
| FF displays normally, and it's missing from the source when
| I inspect it in reader mode. Having two spans like this
| comes directly from KaTeX, whose authors I'm sure have
| thought about accessibility. I imagined that MathML would
| be somewhat standard and screen readers would understand
| it.
| zerocrates wrote:
| Huh, you're right about what the Firefox reader mode
| does. I'm somewhat surprised it uses the MathML. I'd just
| assumed what was happening from the 2pi in the header
| becoming plain text but that header is probably a special
| case and just from the page title rather than the h1.
| mgunyho wrote:
| Yes, I manually put 2p in the html <title>, while the
| <h1> has the KaTeX rendered math in it :)
| noman-land wrote:
| Thank you for sharing your experience. Web accessibility has a
| long way to go and reminders like these help.
| jovial_cavalier wrote:
| I'm curious how blind people normally engage with math. For me,
| engaging with math almost always means conjuring up a visual
| representation in my mind. Failing that, an equation.
|
| Since visualization is so fundamental to doing math, and since
| mathematical symbols and equations are a written language for
| which there is no spoken analog, I really can't imagine
| engaging with math without my eyes. Even reading equations
| aloud verbatim is not reliable. "X plus B squared" can mean (x
| + b)^2 or x + b^2
| kybernetikos wrote:
| The way I've heard those distinguished in spoken math is x +
| b^2 is said "x plus b squared" and (x + b)^2 is said "x plus
| b all squared. There's a similar approach for divide "x plus
| b over 8" vs "x plus b all over 8". That was often enough but
| if it wasn't you'd be reduced to pronouncing brackets.
| Sharlin wrote:
| Using postfix operations in the Way of Forth would be
| unambiguous and of course otherwise superior as well as is
| well known _[citation needed]_. "x b plus squared" vs "x b
| squared plus". Well, at least as long as it's agreed on
| whether "x b" means two variables or one with a two-letter
| name. But the latter don't really exist in math. You just
| expand to new alphabets when you run out of letters.
| nilstycho wrote:
| I had a math graduate student teaching my linear algebra
| class. He taught dot and cross products entirely
| algebraically, never drawing vectors as arrows, but as arrays
| of numbers. When I suggested after class that teaching the
| visual representation might help some students, he pushed
| back. Visual understanding, he explained, was a crutch best
| avoided, because visual intuition could break down in higher
| dimensions. I thought that was a surprising perspective from
| a math graduate student, of all people.
| jameshart wrote:
| Seems like an odd choice when talking about the cross
| product, since the cross product is only a thing in 3D. You
| can define analogous things in other dimensions but it
| becomes clearer and clearer that it's not meaningfully a
| 'product'.
|
| So it doesn't matter if your visual intuition for a cross
| product breaks down in higher dimensions - a cross product
| is only a thing in three.
| nilstycho wrote:
| Good point; you're right. I might be misremembering the
| cross product. I do remember that he didn't even teach
| the geometric interpretation of vector addition.
| jacobolus wrote:
| This is very off topic, but the wedge product absolutely
| is "meaningfully a product", generalizes fine to
| arbitrary dimension, and has a perfectly reasonable
| visual/spatial/geometric interpretation.
|
| (Indeed, we should entirely scrap the cross product in
| undergraduate level technical instruction and replace it
| with the wedge product; one happy effect will be
| replacing students' misleading spatial intuitions with
| better ones.)
| jameshart wrote:
| Or bivectors, or k-vectors, or blades...
| kandel wrote:
| I can see where he's coming from. Geometric intuition is an
| useful tool but in this semester's linear 2 class I
| developed more because I stopped using it. It's too strong
| of a tool and blots out "dryer" intuition and methods, and
| also as you progress you find more and more places where
| it's not useful. What's the geometrical intuition for
| whether two circles intersect in Q^2? who knows?
| anvuong wrote:
| This is quite true, especially true when talking about
| direction (gradient) in high dimensional space. I don't
| think this can be avoided, since after all we are creatures
| living in 3D space where left right up down are quite well-
| defined, just need to make a mental note every time you
| have to deal with more than 3 dimensions.
| ctoth wrote:
| In terms of reading the notation out loud, if there is verbal
| ambiguity, remove it?
|
| For instance, for your example: "X plus B quantity squared"
| or "x + b squared"
|
| You can also do things like change the pitch of speech as
| symbols are nested, play specific tones to represent symbols,
| pan things across the stereo field to represent groupings,
| and otherwise make the symbolic equation into a multimodal
| experience.
|
| But of course, you can't do any of this if the semantic
| representation of the equation is lost and it is rendered as
| strictly graphics or whatever.
|
| I'm a bit confused to your original point about visualization
| because how ever in the world could I program if I couldn't
| abstractly manipulate symbols? I suppose not visually, but
| there's something non-word-oriented happening in my head.
| jovial_cavalier wrote:
| >I'm a bit confused to your original point about
| visualization
|
| I guess I find that programming is somewhat more word-
| oriented than pure math. For instance, how do you think
| about complex exponentiation, or a rotation matrix? Do you
| bring to mind the sensation of spinning around? For myself,
| I bring to mind the image of the entire complex plane
| rotating and stretching along a spiral, but I'm led to
| believe that those who are blind from birth aren't really
| capable of doing that.
|
| Harder examples might include fourier series, convolution,
| gradient descent, etc.
|
| I think you could almost consider the visualization of
| these things like a crutch. I wonder if not being able to
| visualize them might remove preconceived notions about how
| they behave, and give you different insights.
| sebzim4500 wrote:
| The equations are in mathml, I feel like the issue is with the
| screen reader.
| [deleted]
| RogerL wrote:
| But what is the answer? What should I do differently? I get
| don't use images, but then what? I can't imagine all screen
| readers have the same capabilities, or that there is a base
| common ability, so what should we do?
|
| Googling says MathML is the answer (e.g.
| https://www.washington.edu/doit/how-do-i-create-online-math-...
| this site uses MathML and your reader isn't handling it. So now
| what? (alt-tags? something else?)
| 1-more wrote:
| When I view source (not the DOM) I can see that all of your
| math has aria-hidden="true". That seems to carry over into
| the parsed/generated math markup: run
| `document.body.innerHTML += '<style>[aria-hidden="true"]{
| outline: 3px solid red; }</style>'` to see all the hidden-
| from-screenreader things highlighted. That may be enough.
|
| For content that really cannot be written in a way that
| screen readers can handle, there is always the idea of Screen
| Reader Only content. It's a hassle, but let's jump in and
| give it a shot
|
| For instance for the first math thing you can have
| <style> .sr-only {
| position:absolute; left:-10000px;
| top:auto; width:1px;
| height:1px; overflow:hidden; }
| </style> <p class="sr-only" id="definition-of-
| reduced-derivative"> The reduced derivative with
| respect to x is denoted crossed-d over dx of the function
| f(x). It is equivalent to the the derivative with respect to
| x (denoted d over dx) of the function f(x) all over 2 pi.
| </p>
|
| Then you make sure that the element that wraps up your first
| equation has aria-describedby="definition-of-reduced-
| derivative" so that the SR reads out that content. I think
| you may need to not have "aria-hidden" on that math wrapper,
| but I'm not sure.
|
| This is not an authoritative answer; I'm just some asshole
| who writes front-end code a lot. More of a Cunningham's Law
| situation that anything really. You don't want to end up
| creating one experience for sighted users and completely
| different one for screen-reader and refreshable-braille-
| display users. But this can maybe get the wheels turning for
| how to address it? Also again maybe TOTALLY unnecessary once
| you un-hide the math markup.
| bjornasm wrote:
| One option would be to do testing with screen-readers and
| then find out if the content works, and if not find out why.
| patrec wrote:
| I think most people are fine with the idea that some of the
| costs (monetary and otherwise) arising from disabilities
| should be transferred from the people suffering from them
| to society at large. But I don't think it's reasonable for
| producers and users of broken screen readers to expect
| everyone (including authors of private blogs posts) bending
| over backwards to accommodate them, nor is it a remotely
| efficient use of societal resources. It also creates
| completely perverse incentives.
| User23 wrote:
| If there is a bug in the screen reader then that's on the
| vendor.
|
| However the overwhelming number of cases where a website
| is unusable with a screen reader are due to lack of
| proper semantic tagging, like alt text and so on. The
| last thing a screenreader user wants to hear on a site is
| "button," "button," "button..." There's certainly room
| for tooling to help though. I definitely think it sucks
| that many accessibility linters are nonfree and thus will
| never be used by site devs who aren't worth suing.
|
| It's worth remembering that accessibility helps everyone,
| not just the disabled. In fact one of the more popular
| arguments against accessibility is that it allows non-
| disabled persons to do more than intended.
|
| So while there's a good argument to be made for being
| charitable to those with a frankly really lousy
| condition, if you just want to be self-centered you still
| benefit from properly tagged data.
| patrec wrote:
| Did you even bother looking at the website source before
| writing this?
| psychoslave wrote:
| It depends on what public you care about. If readability did
| general public is a concern, just get rid of all these
| symbols and go with plain prose text, possibly using images
| as preferred illustrations over any ideographic way to encode
| ideas.
| marginalia_nu wrote:
| I don't think it's really viable to just stop posting
| equations altogether. Yeah it's more accessible, but it
| would also completely cripple any sort of discussion
| between mathematicians, physicists, etc.
|
| Equations and mathematical notation aren't simple
| illustrations. They're in many ways more important than the
| stuff around them.
| marcosdumay wrote:
| You can't discuss math while getting rid of the math
| symbols. That's not a reasonable proposal.
|
| Math on the web is broken, the the affected people should
| be up complaining about that. This site did the most
| accessible thing possible; the fact that every tool broke
| here, just like they do for every other method is not
| really the author's fault.
| psychoslave wrote:
| > You can't discuss math while getting rid of the math
| symbols. That's not a reasonable proposal.
|
| If we assert that some topic can't be discussed with
| plain prose, then the only logical conclusion is that you
| can't discuss the topic at all.
|
| > Math on the web is broken, the the affected people
| should be up complaining about that.
|
| There is really two different topic there.
|
| One is, how can screen readers deal appropriately with
| symbolic notations -- be it astrological esoteric formula
| or mathematical abstruse formula.
|
| An other one is how you chose to express ideas. Using
| symbols only is always possible, whatever the topic.
| Using prose only is always possible. Using multiple
| representations is also always possible, including audio
| record, alphabetical text, ideograms, pictures, video.
|
| Note that I didn't blame the author for any fault here. I
| just pointed out that, if one take as a goal to be the
| most accessible as possible to general public, using
| academic symbols is not the way to go. It doesn't mean
| that people with some matching academic curricula might
| not prefer to have a document full of esoteric symbols,
| be it for real actual communication advantages or mere
| bigotry and vulgar elitism.
|
| We all know here, I guess, that anything written with
| this kind of symbols can be just as well and without any
| ambiguity transcribed into a programming language which
| use exclusively mundane words or turned into a series of
| two signs that no one can grasp instantly.
|
| That conversation make me think about comments in
| https://news.ycombinator.com/item?id=36433212
| [deleted]
| zamadatix wrote:
| Why is "plain prose" so logically the litmus test for
| what can be discussed? It certainly wasn't designed
| around the idea it should cover every possible concept,
| or do so remotely efficiently either, so the repeated
| assertion it logically is the only way to know if a
| concept can be discussed is a bit hard to follow. It does
| cover most concepts though, and conveniently.
| HappMacDonald wrote:
| > If we assert that some topic can't be discussed with
| plain prose, then the only logical conclusion is that you
| can't discuss the topic at all.
|
| Please bear in mind that mathematical notation _is a
| language_ , and that mathematical formulas _are_
| perfectly valid plain prose _in that language_.
|
| I imagine that some screen readers will fail gracelessly
| when faced with Chinese script or Hindu as well.
| Especially if they're not unicode compliant.
|
| But once you hit that threshold you are simply in a
| battle of dueling accessibility concerns. Not everyone is
| sighted, but neither does everyone rely on English as a
| primary language.
|
| Nor should they as the English language was not designed
| to convey all concepts accurately. It excels mostly in
| conveying concepts germane to anglophone cultures. And
| three guesses what concepts are _not_ popularly relevant
| to your standard anglophone? That 's right.. calculus and
| theoretical physics.
|
| That's not to bash on English as a language. It really is
| a flexible beast with a far reaching vocabulary. But
| there literally exists no language that is ideal for
| encapsulating every single idea under the sun. One must
| have a way to support many of the most diverse ones at
| the same time. And modern mathematical notation belongs
| on that short list along _with_ English.
| gbear605 wrote:
| It's not entirely impossible; mathematicians did it for
| thousands of years (just read Newton's Principia). It
| does use very specific language to do so, and some of the
| language might not exist for some advanced mathematical
| concepts, but I think that this whole article could be
| written that way.
| jacobolus wrote:
| > _mathematicians did it for thousands of years (just
| read Newton's Principia)_
|
| What mathematicians did for thousands of years arose in a
| culture of oral proofs supplemented by prepared diagrams:
| Euclid's proofs were meant to be recited aloud in front
| of an audience while pointing at an image labeled only
| with single letters/numerals - _not_ read in a book. The
| society was substantially illiterate, there was no access
| to paper or good pens, algebra had not yet been invented,
| and all arithmetic was done mentally or using fingers or
| physical tokens.
|
| Compared to mathematical notation, natural language
| expressions are often incredibly large and cumbersome,
| can make following the argument extremely difficult, and
| make many kinds of symbolic manipulations all but
| impossible.
|
| Providing a visual way to interpret and manipulate
| mathematical expressions was a revolution in mathematics
| without which most modern mathematics would never have
| appeared. Eliminating that is comparable to writing
| computer programs via punched cards because "that's how
| they used to do it".
| marcosdumay wrote:
| Yeah, let's talk about the difference in accessibility
| between the text on this site and Newton's Principia...
| mgunyho wrote:
| I think parent is referring here to the fact that math
| used to be written without symbols (other than numbers)
| up to the 1300s according to Wikipedia: https://en.wikipe
| dia.org/wiki/History_of_mathematical_notati... (very
| interesting article!) However, I would say that there is
| a reason why notation tends towards terse symbols: it's
| much more efficient and unambiguous.
| SAI_Peregrinus wrote:
| Even Newton's Principia has symbols, they're just
| different. His notation used dots and lines & boxes in
| various positions.[1]
|
| [1] https://en.wikipedia.org/wiki/Notation_for_differenti
| ation#N...
| psychoslave wrote:
| Maybe ctoth or some other user of screen reader might
| have a look at https://en.wikisource.org/wiki/The_Mathema
| tical_Principles_o... and tell us how it goes.
| curtis3389 wrote:
| > review their math content with a screen reader before
| publishing
|
| I second this for anything you put on the web. I opened the
| webapp I work on with a screen reader, and it was an incredibly
| valuable experience. You get to see your site from a different
| perspective, and various issues stand out like a sore thumb,
| and I honestly found fixing the accessibility issues extremely
| satisfying.
| jacobolus wrote:
| What screen reader do you use? As a sighted person with no
| screen reader experience I tried Apple's VoiceOver on some math
| heavy Wikipedia pages and found that it completely mangled the
| formulas there, e.g. not distinguishing between numerator and
| denominator of fractions, not giving any indication of the
| difference between a coefficient versus an exponent,
| pronouncing invisible formatting commands, and so on.
|
| Are there any websites with extensive, complicated mathematical
| formulas which are accessible to screen reader users? What's
| the current state of the art?
|
| In general, would you prefer a typeset formula navigable in the
| usual way by a screen reader, or an explicit English language
| fallback explicitly pronouncing the formula the way a lecturer
| would read it to a class?
|
| I ask because from what I can tell there are few if any screen
| reader users among authors of Wikipedia technical articles. I
| at least have quite a poor understanding of how to make those
| articles accessible.
|
| Do you mind if I email you to ask questions about screen
| readers and mathematical formulas?
| hoten wrote:
| Do you have any additional examples of webpages that fail to be
| readable for you (math-related or otherwise)? I work on an
| accessibility tool for web developers, and would like to see if
| we are detecting those bad cases correctly.
| jraph wrote:
| Wow, I'd be interested in knowing how to fix this. I don't
| currently write math on the web but if I had to, I would be
| tempted to do it exactly like this: bare HTML with MathML, no
| JavaScript. And would do it thinking about blind people relying
| on accessible content to read it.
| xingped wrote:
| Hi! I'm trying to work on understanding how to make websites
| more accessible right now and probably the first thing people
| generally think of are blind users. However, I'm having a hell
| of a time figuring out how to use screen readers. Is there any
| screen reader or documentation/tutorial you might recommend for
| sighted users to learn how to use screen readers?
| IIAOPSW wrote:
| You should call it d/dx bar.
|
| Anyway, I love the choice of theta because a while ago I came up
| with a nice notation for sin and cos and this fits it really
| well. When I first learned trig, it was by way of skipping into
| physics early. I only understood cos as the magic button for
| getting x components from angles, and y as the button for y
| components. So my notation is based on this very literal brute
| understanding. All the symbols are circles with lines on the
| appropriate sides.
|
| sin = -O- (should be overbar)
|
| cos = O|
|
| -sin = _O_
|
| -cos = |O
|
| Why did I make symbols for the negative versions of the same
| functions? Is minus sign too good for me? No. I did it because
| you can differentiate by just rotating the symbols clockwise and
| integrate by rotating counter clockwise. d/dx O| = _O_.
|
| The way you defined theta, and the graphical depiction of theta,
| fits nicely.
| JBorrow wrote:
| It says that in TFA :)
| smaddox wrote:
| > You should call it d/dx bar.
|
| Or put the modifier on the denominator so that the product and
| chain rules are obvious (the modifier only persists on the dx,
| not on the dy)
| H8crilA wrote:
| Also, p is the wrong constant. The very definition is awkward:
| the ratio of two radiuses to the circumference. How about one
| radius?
|
| It is much more natural to work with 2p. Some people use the
| letter t (Tau) to denote 2p, and it simplifies almost all
| naturally occurring expressions. For example, what is more
| elegant?
|
| e^(p*i) = -1
|
| e^(t*i) = 1
| hughes wrote:
| Coincidentally, happy Tau Day! (6.28)
| mgunyho wrote:
| Not a coincidence :)
| justincredible wrote:
| [dead]
| tgv wrote:
| t = 0, got it.
| H8crilA wrote:
| Well, I can say p := 3p, and -1 still works. The point is
| exp() is that 2p*i periodic.
| tgv wrote:
| The original post missed the _i_...
| bobbylarrybobby wrote:
| e^pi + 1 = 0, of course
| bauble wrote:
| e^i*tau = 1 is pretty elegant, too.
| JdeBP wrote:
| Lots of focus on the Greek letters, at the expense of an
| important Latin one, there. (-:
| floatrock wrote:
| > t simplifies almost all naturally occurring expressions
|
| basically the point of the Tau Manifesto:
| https://tauday.com/tau-manifesto
|
| And would ya look at the date today... 6/28...
| hinkley wrote:
| dang, the article says 2, not 2n
| mabbo wrote:
| `Th^i = 1` does make this really slick, imho.
|
| I often wonder if someday when we meet alien intelligences,
| they'll have a completely different set of constants, derivable
| from our own but different. Th=535.491... may be such an example.
| munchler wrote:
| I don't think Th makes for a good fundamental constant, though,
| because it's composed of pi and e, which must persist as
| distinct concepts in the end.
| crdrost wrote:
| I've occasionally played with the notations that maybe @ = e^i
| or even 1 = e^{2pi} to simplify these sorts of expressions
| before. So for example a forward moving wave can be written,
|
| 1^{x/l - nt}
|
| with no particular ambiguity or even parentheses. The choice of
| constants should give you some pause though--we don't have a
| great way to talk about "true wavenumber" k so we have to talk
| about wavelength, and we use "f" for a lot of other things
| while Greek nu looks like an English V so that can sometimes be
| confusing... it's not _bad_ but it's weird enough that it's not
| obviously better.
| ajkjk wrote:
| I've thought of this as well, and sorta agree. But I think the
| real source of the 2pi-s is a bit more subtle? Which is basically
| that anywhere 2pi shows up is a quantity that is supposed to have
| different 'units' than the rest of the equation it's in, but for
| whatever reason we have erased all the units so we keep finding
| quantities multiplied together with a conversion factor of 2pi.
| Now one way to handle that is to write Tau or something in its
| place... but another is to, somehow, erase all the 2pis entirely
| but keep track of the 'units' on everything.
|
| In fact it is very hard to find places in math where 2pi shows up
| 'on its own', added to other quantities that are not also in
| angular units of some sort. That is, most of the pis show up in
| calculations that involved pi, or circles, in some way (often
| quite sneakily). Of course it shows up on its own in the
| circumference/area of a circle, but you can express the area in
| terms of the circumference, so really it's just about the
| circumference that has a special value. And I wonder about
| whether it's possible to just... pick a different value for the
| circumference, an arbitrary symbol with no value, and then
| expressing every other use of pi in terms of that one without
| ever being forced to pick its value.
|
| (Of course when you tie a string around a circle and measure it
| and it comes out to 2 pi r, yeah, you're forced to pick the
| correct value. Oh well.)
___________________________________________________________________
(page generated 2023-06-28 23:01 UTC)