COSMOS: A Spacetime Odyssey, Starts March 9
It is a fact that we repeatedly observe masses attracting each other, or more properly, we observe that the distance between them shrinks. It is a fact that we observe the strength of this attraction decreasing as the square of the distance between the masses. It is a fact that we repeatedly observe masses revolving around other masses. It is not a fact that this is caused by a force mediated by gravitons. Despite the fact that would explain everything we have observed and meaured, it's still only a theory. We have another theory called general relativity that says gravity is caused by a curvature of space and time. That also explains everything that we have observed and measured. If Einstein hadn't proposed that extremely counterintuitive ridiculous sounding notion, Tyson would probably be up there on TV saying that the quantum theory of gravity is a fact.
He could have informed without misleading. He could have said something like, "While nothing is absolutely certain in this world, virtually all scientists have become as certain that we share a common ancestor with monkeys as they are certain that the sun will rise tomorrow." Of course that may not be true because if I gave all scientists a choice between having me bet their life savings on the monkey theory, or them betting their life savings on the sun rising tomorrow, but they had to drive an hour to bet on the sun, I'll bet almost all of them would be making that drive.
There are some teachers who believe they need to tell lies to students even in elementary math classes. I don't think much of this teaching philosophy. I think it's unnecessary, counterproductive, and due to laziness and teaching ineptness. They will tell their students that you can't take the square root of a negative number, or that you can take an odd root of a negative number and get a real result. They could easily add a sentence or two indicating that there are other areas of math in which what they are saying would be different, but for now it is true in a limited context. For mediocre students, that kind of stuff goes in one ear and out the other anyway, but good students will perk up at that and become interested in what these other areas are, and that might encourage them to take more math classes. Most high school teachers probably don't even know that (-1)^(1/3) = -1 is technically not true, and for good reason. If they did it on their calculator and got an error, they'd probably would have no idea why. But they tell their students lies either because they don't know any better themselves, or because they don't want to open a can of worms. They figure the students have enough on their plate, so it's best to lie to them and get them to learn what they need to learn for now, and then if they go on in math they can learn something different. I think that the teaching of math and science should be the telling of a cohesive story, not one filled with half-truths, contradictions, and recantations. Whether a student goes on in math and science or not, he should come away with truths, not lies. When he learns that he was lied to, it makes the whole subject look suspect.
He could have informed without misleading. He could have said something like, "While nothing is absolutely certain in this world, virtually all scientists have become as certain that we share a common ancestor with monkeys as they are certain that the sun will rise tomorrow." Of course that may not be true because if I gave all scientists a choice between having me bet their life savings on the monkey theory, or them betting their life savings on the sun rising tomorrow, but they had to drive an hour to bet on the sun, I'll bet almost all of them would be making that drive.
There are some teachers who believe they need to tell lies to students even in elementary math classes. I don't think much of this teaching philosophy. I think it's unnecessary, counterproductive, and due to laziness and teaching ineptness. They will tell their students that you can't take the square root of a negative number, or that you can take an odd root of a negative number and get a real result. They could easily add a sentence or two indicating that there are other areas of math in which what they are saying would be different, but for now it is true in a limited context. For mediocre students, that kind of stuff goes in one ear and out the other anyway, but good students will perk up at that and become interested in what these other areas are, and that might encourage them to take more math classes. Most high school teachers probably don't even know that (-1)^(1/3) = -1 is technically not true, and for good reason. If they did it on their calculator and got an error, they'd probably would have no idea why. But they tell their students lies either because they don't know any better themselves, or because they don't want to open a can of worms. They figure the students have enough on their plate, so it's best to lie to them and get them to learn what they need to learn for now, and then if they go on in math they can learn something different. I think that the teaching of math and science should be the telling of a cohesive story, not one filled with half-truths, contradictions, and recantations. Whether a student goes on in math and science or not, he should come away with truths, not lies. When he learns that he was lied to, it makes the whole subject look suspect.
In short, I think Tyson's service outweighed any disservice if he got people to understand that the ToE isn't just a guess like I think Sally has a crush on Joe on a soap opera. Rather it is much, much stronger than that.
We were talking about gravity, not gravitons. You are always finding a way. We were also talking about evolution, not some ****ty details.
Humans-monkeys-apes. If you don't accept they are close to the same you shouldn't be able to believe in anything.
Lets just say that if it were Sagan he might have handled it a bit better. But i am sure Sagan would be seriously irritated if he knew many years after having died US still had politicians walking all over science issues like barbarians promoting ignorance to the general public and its more likely true that Sagan would not waste a moment nitpicking on NDGT given the landscape of idiots waiting outside to grab any space you give them and celebrate a victory of ignorance.
As for the (-1)^(1/3) teacher. I guess if the kids are trying to find a solution in reals -1 is just fine. All the teacher could do is ask them if someone told them that
there is an object that has the property i^2=-1 and i*y=0 =>y=0 for y reals, is there a number a+b*i that satisfies (a+bi)^3=-1 if a, b are real numbers?
He would then force the students to use the property i*i=-1 to deduce that the above means;
a^3-3a*b^2=-1 and 3a^2*b-b^3=0 with a,b reals meaning;
(3*a^2-b^2)*b=0 or (b=0 or 3a^2=b^2) and a*(a^2-3b^2)=-1
which means b=0 a^3=-1 but now a is a real number, leading to a=-1,b=0
and the rest simply leads to b^2=3a^2 and (a^2-3*3*a^2)*a=-1 or -8*a^3=-1 or a=1/2 and b=+-3^(1/2)/2
That only takes 10 min to show on a board and you have not only opened the door to complex numbers but helped kids with a system of equations problem.
However to say that (-1)^(1/3) has more solutions than -1 isnt doing anything for the student as obviously in reals it is the only solution.
Now if you want to call (-1)^(1/3)=-1 wrong what exactly is the crime you commit to students that are technically unable yet to do the above to establish that some other numbers not reals may satisfy this equation if you introduce objects like I.
I think a proper teacher gauges if the kids can do the above and if they can or are able to try it he could go for it otherwise to not call x^3=-1 as x=-1 in real numbers is weak. Come on now. What is the student going to lose by not getting told of complex numbers yet (without even calling them complex) if they are say very young and unable to do the above. I would go for telling them about complex numbers well in advance of the proper curriculum only if they could do the system above and find technically all the roots of x^3=-1. That would leave the good students extra satisfied and confident by the way (if they came on their own to the complex solutions above as extra credit thing) to the conclusion that there are numbers other than -1, not real with interesting properties, that satisfy that equation.
I think its a total fail to open the doors to the kids that there is something else and not show them what on earth you are talking about if it only takes 10 min. If it cant be done in 10 min its a fail to do that though unless you tell them in advance that they will understand it later and 2-3 kids in the class still exist that could get it. But why not help those good students to find more by telling them more. What is the point to open the door in total darkness and not even offer them a hint or a good question to try to think further.
So i would not tell them anything in class even then. I would simply ask them to think until next week if i told them there is an object that satisfies i^2=-1 whether they could then find a,b above so that (a+b*i)^3=-1 with a, b the usual real numbers they play with. Doesnt that help the students a lot better than some obscure reference?
And this is why also nitpicking on NDGT is waste of time really. (and this comes from a guy that without being the least racist or any bs like that thinks Sagan was so much better and i dont exactly like the style of NDGT in many interviews i have seen him where he has a less elegant/charismatic style of presenting things that could enjoy better approval across all levels of education. It might be true however that NDGT is better for kids, not sure if that is the case what do you guys think.)
As for the (-1)^(1/3) teacher. I guess if the kids are trying to find a solution in reals -1 is just fine. All the teacher could do is ask them if someone told them that
there is an object that has the property i^2=-1 and i*y=0 =>y=0 for y reals, is there a number a+b*i that satisfies (a+bi)^3=-1 if a, b are real numbers?
He would then force the students to use the property i*i=-1 to deduce that the above means;
a^3-3a*b^2=-1 and 3a^2*b-b^3=0 with a,b reals meaning;
(3*a^2-b^2)*b=0 or (b=0 or 3a^2=b^2) and a*(a^2-3b^2)=-1
which means b=0 a^3=-1 but now a is a real number, leading to a=-1,b=0
and the rest simply leads to b^2=3a^2 and (a^2-3*3*a^2)*a=-1 or -8*a^3=-1 or a=1/2 and b=+-3^(1/2)/2
That only takes 10 min to show on a board and you have not only opened the door to complex numbers but helped kids with a system of equations problem.
However to say that (-1)^(1/3) has more solutions than -1 isnt doing anything for the student as obviously in reals it is the only solution.
Now if you want to call (-1)^(1/3)=-1 wrong what exactly is the crime you commit to students that are technically unable yet to do the above to establish that some other numbers not reals may satisfy this equation if you introduce objects like I.
I think a proper teacher gauges if the kids can do the above and if they can or are able to try it he could go for it otherwise to not call x^3=-1 as x=-1 in real numbers is weak. Come on now. What is the student going to lose by not getting told of complex numbers yet (without even calling them complex) if they are say very young and unable to do the above. I would go for telling them about complex numbers well in advance of the proper curriculum only if they could do the system above and find technically all the roots of x^3=-1. That would leave the good students extra satisfied and confident by the way (if they came on their own to the complex solutions above as extra credit thing) to the conclusion that there are numbers other than -1, not real with interesting properties, that satisfy that equation.
I think its a total fail to open the doors to the kids that there is something else and not show them what on earth you are talking about if it only takes 10 min. If it cant be done in 10 min its a fail to do that though unless you tell them in advance that they will understand it later and 2-3 kids in the class still exist that could get it. But why not help those good students to find more by telling them more. What is the point to open the door in total darkness and not even offer them a hint or a good question to try to think further.
So i would not tell them anything in class even then. I would simply ask them to think until next week if i told them there is an object that satisfies i^2=-1 whether they could then find a,b above so that (a+b*i)^3=-1 with a, b the usual real numbers they play with. Doesnt that help the students a lot better than some obscure reference?
And this is why also nitpicking on NDGT is waste of time really. (and this comes from a guy that without being the least racist or any bs like that thinks Sagan was so much better and i dont exactly like the style of NDGT in many interviews i have seen him where he has a less elegant/charismatic style of presenting things that could enjoy better approval across all levels of education. It might be true however that NDGT is better for kids, not sure if that is the case what do you guys think.)
BS BS. How could anyone find any numbers if they had no concept of relativity or the curvature of space and time which predicts what to look for? Who would bother to go out during an eclipse to try and measure the sum of angles of a triangle with and without a big mass inside? Who would use cessium clocks to try and measure time difference in airplanes? It has been suggested that general relativity might not have been discovered for decades had it not been for Einstein. If a theory of quantum gravity had come along first and explained everything, it would have been hailed as something like Darwin's theory of natural selection. We might not have the gravitons yet, just like we didn't have genes yet when Darwin was around, but people would have believed in it anyway because it explained everything so well it just has to right. Tyson would have gone on TV and said it was fact. Then if I came on here and suggested that maybe it's not fact because maybe it works by time and space being curved, but I hadn't done the actual math yet to make the testable predictions, I would have been laughed at and argued with just as you are arguing now.
I think I clearly made the distinction between the aspects of gravity that are facts, and the theory of gravity which is not a fact. I was asked if the theory of gravity is a fact. That prompts the question of which one since there are so many to choose from. Most of them assume the force is mediated by a massless spin-2 boson which they call a graviton. But loop quantum gravity does not, and general relativity says it's caused by a curvature of space and time. String theory says that a graviton is a string and not a particle. These can't all be facts or you would have contradictory facts which would be completely illogical. The term fact is limited to what we observe. The term theory is used for the explanation of what we observe. A fact can never be a theory. They are different animals. We observe that we share 99.9% of the same DNA as orangutans. That's a fact. That we share a common ancestor is an explanation of that fact, so it's a theory. I'm trying to "find a way" to drill that into your thick skulls through about 10 feet of concrete. JFC.
We were talking about gravity, not gravitons. You are always finding a way. We were also talking about evolution, not some ****ty details.
Also is strange that Fitzhugh's opinion about what is a fact is somehow the 'correct one' even though the link I provided includes dozens of attributed sources from evolutionary biologists holding a contrary view. So then Tyson says the same thing. I guess he was not talking to the 1 correct evolutionary biologist out of all of the evolutionary biologists when he was scripting his show.
Originally Posted by wiki
Fact: In science, an observation that has been repeatedly confirmed and for all practical purposes is accepted as "true". Truth in science, however, is never final, and what is accepted as a fact today may be modified or even discarded tomorrow.
It's the observation that everyone accepts as true. It's a fact that we observe DNA commonality between humans and monkeys. It's a fact that we observe transitional figures in the fossil record. The term "evolution" encompasses these facts, and it also encompasses the theory of evolution which explains these facts. But the theory can never in itself be a fact. That's what Gould meant when he said evolution is fact and theory. He's not saying that facts can be theories. He's saying just the opposite:
Originally Posted by Stephen Jay Gould
Evolution is a theory. It is also a fact. And facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world's data. Theories are structures of ideas that explain and interpret facts. Facts do not go away when scientists debate rival theories to explain them. Einstein's theory of gravitation replaced Newton's, but apples did not suspend themselves in mid-air, pending the outcome. And humans evolved from ape-like ancestors whether they did so by Darwin's proposed mechanism or by some other yet to be discovered.
He's out there trying to educate people who think theory='guess' and needs to make it clear that a scientific theory is not a guess and is not a hypothesis. It is a model of a phenomenon which has been extensively supported by evidence and testing and has withstood decades of scrutiny and review. So... 'fact'...
Would you also dispute the germ theory of disease? The theory of gravity? The theory of relativity? Cell theory? Plate tectonics theory? Quantum theory? These are all equal in the eyes of science and have withstood rigorous testing and scrutiny. They all have mysteries about them, but to a non-scientist watching a tv show, they are all facts.
It is a fact that we observe certain microorganisms in the blood of diseased individuals but not healthy individuals. It is a fact that introducing said microorganisms into healthy individuals induces the disease. It's then a theory, not a fact, that these microorganisms cause the disease.
There is no single theory of gravity. There are many that contradict each other and contradict relativity. These are theories and not facts. They all explain observations which are facts.
It is a fact that living organisms are comprised of cells as we can confirm by direct observation. It is a fact that all cells we observe arise from living cells. It is a theory, not a fact, that all cells arise from living cells, and it is a theory not a fact, that the cell is the most basic unit of life.
It is a fact that the shape of the continents appear to fit together, and the rock formations on their edges are the same. It is a fact that the fossils found on the edges of these continents are the same. It is a fact that magnetization of rock layers from rocks said to be of different ages are different. We have facts about the sea floor obtained from geologic study. We have facts obtained from seismic data. The theory of plate tectonics is a theory, not a fact, that explains all of these facts.
It is a fact that individual electrons produce wavelike interference patterns when going through 2 slits. It is a fact that x-ray photons exhibit particle like properties when interacting with metals. It is a fact that the probability distribution of energy levels measured for photons is consistent with them behaving as indistinguishable particles. Quantum theory is a theory, not a fact, that explains these facts by postulating all particles to be a superposition of matter waves which determine the probability of finding a particle in a given region, that light is comprised of particles called photons of discrete energy, that certain physical variables have inherent uncertainties.
@masque
You of all people should know what I'm talking about with the odd roots of negative numbers because you agreed with me in the other thread:
http://forumserver.twoplustwo.com/sh...postcount=4199
Now the issue there was raising a negative number to the 2/3 power where the problem is that if we define (-1)^(1/3) = -1, then (-1)^(2/3) would be the square of that or 1, then
[(-1)^(2/3)]^(3/2) = 1
does not equal
(-1)^(2/3 * 3/2) = -1.
It is ugly to define the root this way and then suspend the power rule, or worse yet, not tell anyone about this issue with the power rule which I suspect is usually done. You don't need complex numbers to explain this to people. Of course once you have complex numbers, you define the principle cube root of a negative number as the complex root with positive imaginary part, and this preserves continuity of the analytic continuation. That's how Mathematica defines it, and those folks consider (-1)^(1/3) = -1 to be non-standard mathematics. Even an elementary student needs to have some understanding of this to make sense of their graphs. It's not just about finding the real root, it's about how exponentiation is defined. You could say that (-1)^(1/3) isn't a number but a multi-valued thing, and Wikipedia does that, but I think it's non-standard.
Referencing complex numbers is a separate issue. I wouldn't ask people just learning algebra for the first time to consider what you suggest as it's just too big a digression, unless perhaps they are really gifted. But I wouldn't lie to them and imply that you can't take the square root of a negative number as if that's some universal law that applies everywhere in the sphere of mathematics. It's too important of a thing to give people that false impression. There's nothing wrong with giving people a little preview of upcoming attractions and mentioning that there are other number systems besides the real numbers where this can be defined, and they are extremely useful and important. That's all you have to say. They won't understand it, but at least we're not lying to them, or leaving out something really important that they could at least be made aware exists. I think my algebra teacher said something like that, and he said it was useful in electrical engineering. I think by the time a student reaches what we call Intermediate and College Algebra (a high school course), complex numbers should be introduced because they are just so natural to the solution of polynomial equations. I mean look at the quadratic formula, the complex solutions are staring you right in the face. Why should we close our eyes and pretend they're not there? That's the kind of thing that might catch a student's interest and make him actually appreciate that mathematics is about interesting ideas, not just some kind of rote mechanical drudgery.
As for the NDGT thing, I think it's very important that people understand the difference between a theory and a fact. I mean, that's high school science for God sake. I agree with you that a scientist should never use these words interchangeably. I'm not so concerned with the idiots watching as I am with people here in SMP who should already know this. It's a definition, but it's not *just* a definition. The words we use and the meanings we attach to them affect the way we think.
You of all people should know what I'm talking about with the odd roots of negative numbers because you agreed with me in the other thread:
Originally Posted by masque de Z
No a^x (a<0) is not defined well without opening a whole set of problems. The cubic root of -1 is defined but negative numbers to powers in general is an issue requiring prescriptions now about how to evaluate things if you are restricted to real numbers.
Now the issue there was raising a negative number to the 2/3 power where the problem is that if we define (-1)^(1/3) = -1, then (-1)^(2/3) would be the square of that or 1, then
[(-1)^(2/3)]^(3/2) = 1
does not equal
(-1)^(2/3 * 3/2) = -1.
It is ugly to define the root this way and then suspend the power rule, or worse yet, not tell anyone about this issue with the power rule which I suspect is usually done. You don't need complex numbers to explain this to people. Of course once you have complex numbers, you define the principle cube root of a negative number as the complex root with positive imaginary part, and this preserves continuity of the analytic continuation. That's how Mathematica defines it, and those folks consider (-1)^(1/3) = -1 to be non-standard mathematics. Even an elementary student needs to have some understanding of this to make sense of their graphs. It's not just about finding the real root, it's about how exponentiation is defined. You could say that (-1)^(1/3) isn't a number but a multi-valued thing, and Wikipedia does that, but I think it's non-standard.
Referencing complex numbers is a separate issue. I wouldn't ask people just learning algebra for the first time to consider what you suggest as it's just too big a digression, unless perhaps they are really gifted. But I wouldn't lie to them and imply that you can't take the square root of a negative number as if that's some universal law that applies everywhere in the sphere of mathematics. It's too important of a thing to give people that false impression. There's nothing wrong with giving people a little preview of upcoming attractions and mentioning that there are other number systems besides the real numbers where this can be defined, and they are extremely useful and important. That's all you have to say. They won't understand it, but at least we're not lying to them, or leaving out something really important that they could at least be made aware exists. I think my algebra teacher said something like that, and he said it was useful in electrical engineering. I think by the time a student reaches what we call Intermediate and College Algebra (a high school course), complex numbers should be introduced because they are just so natural to the solution of polynomial equations. I mean look at the quadratic formula, the complex solutions are staring you right in the face. Why should we close our eyes and pretend they're not there? That's the kind of thing that might catch a student's interest and make him actually appreciate that mathematics is about interesting ideas, not just some kind of rote mechanical drudgery.
As for the NDGT thing, I think it's very important that people understand the difference between a theory and a fact. I mean, that's high school science for God sake. I agree with you that a scientist should never use these words interchangeably. I'm not so concerned with the idiots watching as I am with people here in SMP who should already know this. It's a definition, but it's not *just* a definition. The words we use and the meanings we attach to them affect the way we think.
Ok, science has its definition of facts, and that may be good. It's things that really can't be debated if not really stretching it. It has to be done by repeated experiments, and is about the very basic of our existence. If you measure a plant, and see it grow repeatedly, you really can't question it without totally throwing out the concept of reality. Saying: "the plant didn't grow" or "we can't say if the plant grows" are absurd statements.
Problem with everything that has happened in the past is you can't do the experiments. That's a lot of the science: evolution, cosmology (did a meteor hit earth? Did planets and sun form as supposed?), geology (were the continents moving back then?). Things explaining those phenomena are scientific theories, if they explain them really good (otherwise they are hypotheses or less). Problem is the term "theory" in normal everyday use of language indicates something vague, maybe pure speculation, while "fact" is something people know can be questioned. If you have a program directed to the general public, it's more correct to say evolution is a fact than to say it's a theory. You should avoid saying it's a scientific fact though.
Problem with everything that has happened in the past is you can't do the experiments. That's a lot of the science: evolution, cosmology (did a meteor hit earth? Did planets and sun form as supposed?), geology (were the continents moving back then?). Things explaining those phenomena are scientific theories, if they explain them really good (otherwise they are hypotheses or less). Problem is the term "theory" in normal everyday use of language indicates something vague, maybe pure speculation, while "fact" is something people know can be questioned. If you have a program directed to the general public, it's more correct to say evolution is a fact than to say it's a theory. You should avoid saying it's a scientific fact though.
Yeah i remember but my post was about a^x things (a<0). Not the real cubic root of -1.
"No a^x (a<0) is not defined well without opening a whole set of problems. The cubic root of -1 is defined but negative numbers to powers in general is an issue requiring prescriptions now about how to evaluate things if you are restricted to real numbers.
You see if you enter the slippery slope of seeing a^x for a<0 as a potential real number using prescriptions like when x is rational N/M not of the form (2n+1)/(2m) as first raising to N then taking the M root then what prevents me from asking you if you are so happy with
(-2)^(2/3) as conceivably 1.5875... what is (-2)^(2/3+q) with q an as small as you wish irrational number or what is (-2)^(3/4).
Basically you will not run into problems only if you raise to a rational power not of the form (2n+1)/(2m). (n,m integers)
Additionally you will see funny issues like if (-2)^(2/3) = ((-2)^(2/3Pi))^(Pi) what one earth is ((-2)^(1/Pi))^(2/3) any clue what (-2)^(1/Pi) is?
In an effort to stay within R when raising to powers we derived the identities like (x^(a))^b= x^(a*b) only for x>=0.
With complex numbers we of course can talk properly about things like (-2)^Pi etc.
So basically you will not see a function a^x for a<0 in reals with a real domain. It will be only possible to define it using prescriptions like above in some subset of R. "
In any case i agree that one cannot behave properly until they have the proper context in place like what the kids have done so far and what they can handle etc. I think you can teach kids about complex numbers very early way earlier than calculus earlier than 13 yrs even. To be honest i have forgotten when we learned to play with polynomials and identities, 1st and 2nd order equations and things that one can handle what is suggested and see where the problems appear (ie with (-2)^(Pi) type things). Maybe 12-13 years old? So complex numbers are pretty soon after doable up to a trivial simple level that can be enhanced from 15 to 18 in part of some math project team or on your own and at university even better at 19 anyway for all science sector majors. I mean you define powers when talking about reals so any time you try tricks like the above case you are getting in trouble when the number is not restricted to always be there in R during the entire process. You get in trouble with prescription processes like the quoted part examples.
I mean i can see no problem with asking to give me in reals (-1)^(1/3) but when you go to (-1)^(2/3) i have an issue right away because the answer will require a prescription how to obtain it, do you square and the cube root or cube root and then square etc vs de Moivre/Euler form or -1 then raising to 2/3) hence the worlfram answer "http://www.wolframalpha.com/input/?i=%28-1%29^%282%2F3%29" that is not real unlike the stupid prescriptions of square and cubic root things.
"No a^x (a<0) is not defined well without opening a whole set of problems. The cubic root of -1 is defined but negative numbers to powers in general is an issue requiring prescriptions now about how to evaluate things if you are restricted to real numbers.
You see if you enter the slippery slope of seeing a^x for a<0 as a potential real number using prescriptions like when x is rational N/M not of the form (2n+1)/(2m) as first raising to N then taking the M root then what prevents me from asking you if you are so happy with
(-2)^(2/3) as conceivably 1.5875... what is (-2)^(2/3+q) with q an as small as you wish irrational number or what is (-2)^(3/4).
Basically you will not run into problems only if you raise to a rational power not of the form (2n+1)/(2m). (n,m integers)
Additionally you will see funny issues like if (-2)^(2/3) = ((-2)^(2/3Pi))^(Pi) what one earth is ((-2)^(1/Pi))^(2/3) any clue what (-2)^(1/Pi) is?
In an effort to stay within R when raising to powers we derived the identities like (x^(a))^b= x^(a*b) only for x>=0.
With complex numbers we of course can talk properly about things like (-2)^Pi etc.
So basically you will not see a function a^x for a<0 in reals with a real domain. It will be only possible to define it using prescriptions like above in some subset of R. "
In any case i agree that one cannot behave properly until they have the proper context in place like what the kids have done so far and what they can handle etc. I think you can teach kids about complex numbers very early way earlier than calculus earlier than 13 yrs even. To be honest i have forgotten when we learned to play with polynomials and identities, 1st and 2nd order equations and things that one can handle what is suggested and see where the problems appear (ie with (-2)^(Pi) type things). Maybe 12-13 years old? So complex numbers are pretty soon after doable up to a trivial simple level that can be enhanced from 15 to 18 in part of some math project team or on your own and at university even better at 19 anyway for all science sector majors. I mean you define powers when talking about reals so any time you try tricks like the above case you are getting in trouble when the number is not restricted to always be there in R during the entire process. You get in trouble with prescription processes like the quoted part examples.
I mean i can see no problem with asking to give me in reals (-1)^(1/3) but when you go to (-1)^(2/3) i have an issue right away because the answer will require a prescription how to obtain it, do you square and the cube root or cube root and then square etc vs de Moivre/Euler form or -1 then raising to 2/3) hence the worlfram answer "http://www.wolframalpha.com/input/?i=%28-1%29^%282%2F3%29" that is not real unlike the stupid prescriptions of square and cubic root things.
But you can't have (-1)^(1/3) = -1 without causing the same problem with the power rule a^b^c = a^(bc) you mentioned because then certainly (-1)^(2/3) must be the same as [(-1)^(1/3)]^2 = 1. But then [(-1)^(2/3)]^(3/2) = 1 which isn't the same as (-1)^1 = -1. So either (-1)^(1/3) can't be -1, or we lose the power rule.
How about lose the power rule on things that are negative bases. And then open the door to real solutions or principal roots or all roots etc. Basically recover the power rule after complex numbers again. Apply also context in cases that you have negative numbers as needed.
Right, but it's ugly to lose the power rule for just the negative bases before you have complex numbers. They often do define it that way without even warning the students about this. Many calculators will return an undefined error in real mode.
Let's see how many diverse subjects we can jam into a single thread. COSMOS, piracy, philosophy of science, evolution, evidence for psi, superdeterminism, ancient astronauts, germ theory of disease, cell theory, plate tectonics, quantum gravity, relativity, complex arithmetic,.... Let's see if we can work in every subject ever discussed in SMP.
Let's see how many diverse subjects we can jam into a single thread. COSMOS, piracy, philosophy of science, evolution, evidence for psi, superdeterminism, ancient astronauts, germ theory of disease, cell theory, plate tectonics, quantum gravity, relativity, complex arithmetic,.... Let's see if we can work in every subject ever discussed in SMP.
Right, but it's ugly to lose the power rule for just the negative bases before you have complex numbers. They often do define it that way without even warning the students about this. Many calculators will return an undefined error in real mode.
Let's see how many diverse subjects we can jam into a single thread. COSMOS, piracy, philosophy of science, evolution, evidence for psi, superdeterminism, ancient astronauts, germ theory of disease, cell theory, plate tectonics, quantum gravity, relativity, complex arithmetic,.... Let's see if we can work in every subject ever discussed in SMP.
Let's see how many diverse subjects we can jam into a single thread. COSMOS, piracy, philosophy of science, evolution, evidence for psi, superdeterminism, ancient astronauts, germ theory of disease, cell theory, plate tectonics, quantum gravity, relativity, complex arithmetic,.... Let's see if we can work in every subject ever discussed in SMP.
Bruce will be happy to know that NDT now has a drink named after him:
http://gothamist.com/2014/03/25/neil...n_cocktail.php
Pacific Standard, a beer bar in Park Slope, created the "Neil deGrasserac Tyson" in honor of the country's Sexiest Astrophysicist on Saturday
Correction: The most recent study downgrades our DNA commonalities with chimpanzees to about 94%. I had said 99.9% for orangutans earlier, but even the older estimate was only about 99% for chimps, and orangutans would be even lower since they split off earlier.
Kudos to you finding out about that and self-correcting.
Did the cricket descend from Man or Man ascend from the cricket ?
No.
You don't know . Entertain the opposite thought, you don't have to believe, just consider it with an open mind.
We did. Twice. We moved on.
Series is continuing to impress me. Dis-aster = bad star = comet. Halley's important role in getting Newton to publish. And so on, and so on.
Will try not to miss any episode.
Will try not to miss any episode.
In the 4th one, he reveals what happens if you go through a black hole. You apparently end up in a parking lot.
But he didn't say it was a fact.
Everybody knows when you go through a black hole you find flight 377.
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