And by 'binary pulsars' you really mean 'the only binary pulsar system that we've every seen'. While there are plenty (read: dozens) of examples of pulsars in binary systems that have a main sequence or giant star as a companion, there's just the one known example of a binary pulsar system (i.e. two pulsars orbiting each other). And yes, it's one of the best tests for the relatavistic corrections gravity that's every been measured (and the reason that Gravity Probe B got pushed back, again).
Oh, and as to the gravity problem...even exotic objects like a one solar mass black whole with feel like a one solar mass star (i.e. the sun) from the outside of the object (go do a Wikipedia for "Guass' Law"). The composition, shape, density, color, sexuality, of the object has nothing to do with the gravitational potential.
I think you're absolutely right in a lot of this. One point that I'd like to push further on is this: The author writes things like, "So, if I measure the quadrance", or "When I measure the spread" without seeming to have a good grasp of what it means to 'measure' a quantity.
What he MEANS is that he can calculate values for his toy model of geometry based on things that he can measure out in the real world. The unfortunate thing for this particular toy model is that in the real world we don't have any means of accessing the quantities that he wants to view as 'fundamental'; for example, there's no really good way to directly measure the square of a distance or the spread of a height of wall. These are quantities that are just not physically realizable. Just to get into the 'right' set of coordinates requires us to make a measurement and then make a calculation. Then to get back out to something we can measure in the real world requires us to undo the transformation into his set of coordinates back out into our physics (locally linear) world.
Too much effort for not enough pay off. In addition, once you refuse to teach/learn trigonometric functions the first time around you also have the problem that you're gonna have to learn them eventually if you ever decide to take calculus or (at the ABSOLUTE latest) a course in complex analysis (as so many/. readers seem to have done). So all you're really doing is shooting yourself or your students in the foot if you decide to go haring off on some scheme like this to make 'math seem easier'.
I'd rather see people spending this much effort writing a GOOD middleschool math text.
What I think the guy meant by reducing things to simple formulas is this: Why teach students a whole set of werid formulas that seem to be handed down from on high (i.e. angle addition formulas, addition of squares of sines and cosines, etc ) when you could teach them just ONE formula that's handed down from on high (exp(ix) = cos(x) + i sin(x)) from which they can easily derive ALL of the other formulas. There's no real need for students to understand functions of a complex variable to do it this way and it'll make integrals of trig functions exponentially (sorry) easier later on in their careers. The concept of 'i' should really not be any harder for students to swallow than the concept of a non-repeating fraction or even something like 'infinity'.
I think that's mostly because of the vocabulary of the topics: 'Rational Numbers' just FEELS more intuitive than 'Complex Analysis' before you even crack the textbook just based on the titles.
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And by 'binary pulsars' you really mean 'the only binary pulsar system that we've every seen'. While there are plenty (read: dozens) of examples of pulsars in binary systems that have a main sequence or giant star as a companion, there's just the one known example of a binary pulsar system (i.e. two pulsars orbiting each other). And yes, it's one of the best tests for the relatavistic corrections gravity that's every been measured (and the reason that Gravity Probe B got pushed back, again). Oh, and as to the gravity problem...even exotic objects like a one solar mass black whole with feel like a one solar mass star (i.e. the sun) from the outside of the object (go do a Wikipedia for "Guass' Law"). The composition, shape, density, color, sexuality, of the object has nothing to do with the gravitational potential.
I think you're absolutely right in a lot of this. One point that I'd like to push further on is this: The author writes things like, "So, if I measure the quadrance", or "When I measure the spread" without seeming to have a good grasp of what it means to 'measure' a quantity.
/. readers seem to have done). So all you're really doing is shooting yourself or your students in the foot if you decide to go haring off on some scheme like this to make 'math seem easier'.
What he MEANS is that he can calculate values for his toy model of geometry based on things that he can measure out in the real world. The unfortunate thing for this particular toy model is that in the real world we don't have any means of accessing the quantities that he wants to view as 'fundamental'; for example, there's no really good way to directly measure the square of a distance or the spread of a height of wall. These are quantities that are just not physically realizable. Just to get into the 'right' set of coordinates requires us to make a measurement and then make a calculation. Then to get back out to something we can measure in the real world requires us to undo the transformation into his set of coordinates back out into our physics (locally linear) world.
Too much effort for not enough pay off. In addition, once you refuse to teach/learn trigonometric functions the first time around you also have the problem that you're gonna have to learn them eventually if you ever decide to take calculus or (at the ABSOLUTE latest) a course in complex analysis (as so many
I'd rather see people spending this much effort writing a GOOD middleschool math text.
What I think the guy meant by reducing things to simple formulas is this: Why teach students a whole set of werid formulas that seem to be handed down from on high (i.e. angle addition formulas, addition of squares of sines and cosines, etc ) when you could teach them just ONE formula that's handed down from on high (exp(ix) = cos(x) + i sin(x)) from which they can easily derive ALL of the other formulas. There's no real need for students to understand functions of a complex variable to do it this way and it'll make integrals of trig functions exponentially (sorry) easier later on in their careers. The concept of 'i' should really not be any harder for students to swallow than the concept of a non-repeating fraction or even something like 'infinity'.
I think that's mostly because of the vocabulary of the topics: 'Rational Numbers' just FEELS more intuitive than 'Complex Analysis' before you even crack the textbook just based on the titles.