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#51 Re: Help Me ! » Integrate: » 2008-11-04 05:52:57
According to wolfram's integrator; the answer to this is pretty darn complex (although still analytical)
http://integrals.wolfram.com/index.jsp?expr=1%2F((Sin[x])^2+%2B+Sin[x]%2B+1)&random=false
Interestingly, my Ti-89 returned a real (i.e. ∈ R) answer.
#52 Re: Help Me ! » Elements of three dimensional geometry » 2008-11-04 05:40:39
Write down an equation or equations for each of the following.
a. A plane parallel to xy-plane at a distance of 4 units from origin on negative z-axis.
b. A curve produced due to the intersection of a circular cylinder and yz-plane. Center of this circular cylinder is x-axis (that is, cylinder is around x-axis) and its radius is one.
c. A curve produced due to the intersection of a sphere, center at origin and radius 3, and xy-plane.
b and c are just equations for circles on the intersecting planes, with appropriate radii and locations of center.
#53 Re: Help Me ! » Inequality » 2008-11-04 05:16:19
Apply the change of base formula:
From there, both expressions can be easily reduced to numerical values.
#54 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-11-04 05:10:17
Could you please explain how the endosymbiont theory could be reclassified as pure math?
There's nothing like a red herring when your argument can't stand on its own merit. 
Setting aside for a moment the fact that biology is still quite young in its development, and has very little mathematical reconciliation, something that was pointed out earlier in the discussion, I think you'll find that biology has not wholly abandoned endosymbiotic theory. There is simply not enough evidence to say conclusively whether the theory is correct or not.
Indeed, this is true with many subjects, science and pure mathematics included. But dynamic does not mean that demarcation is impossible. And I don't think I ever expected one.
Right. That demarcation is quite clear. Mathematics which are applied outside of the science of mathematics belong to applied mathematics. It's rather ironic a pure mathematician would be in denial about the implications of the very evidence that supports the assertion that studying mathematics for mathematics' sake is beneficial to society.
#55 Re: Help Me ! » TESTT on percentages » 2008-11-03 18:25:47
Does this help?
#56 Re: Dark Discussions at Cafe Infinity » love and feelings » 2008-11-03 18:22:14
Love stinks unless you're in it. Such is life.
#57 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-11-03 18:15:24
In mathematics, there is no concept of a hypothesis (which is what I believe you mean by assumption) being true or false. Whether the hypothesis leads to your conclusion or not depends upon your conclusion. In science however, there is a concept of a hypothesis (sciences call this an assumption) being wrong. That is, whether or not that hypothesis (assumption) holds true in the universe.
If you put it like that, scientific theories are never proven to be incorrect. They are just reclassified as pure mathematics, i.e. from one branch of science to another.
and the problem that arises when trying to find the precise distinction between the two.
Why would you expect a precise distinction between the two? The very nature of applied mathematics implies that it is a dynamic collection of mathematics, not a static one.
#58 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-11-02 16:56:51
What's wrong for a mathematician is making a (logically) false statement. Reaching a contradiction where none is expected, as well as just having something that doesn't follow.
Right. When that occurs, it is often, perhaps usually, due to an incorrect assumption, just like in any other branch of science.
And I disagree. I believe that understanding mathematics is having something to write about.
Your disagreeing with a claim that was not made. The analogy does not imply that understanding mathematics is not a worthy topic of writing. I wouldn't say such a thing. I like mathematics, I just recognize that they are a discipline of science.
Pure mathematics is not just mathematics that has not been applied
Yes, by the definition of applied, mathematics that have been applied in the real world are elements of applied mathematics.
That's weird. I had this thought come up before when reading a book by Hardy, but I have always thought that understanding the world we live in is useful. You and Hardy both seem to think that useful and economical are synonyms. Is this correct (at least for you...)?
Understanding the world we live in is useful. The utility of understanding the universe's origin, however, remains to be seen. Nonetheless, I favor the pursuit of more knowledge relevant to that endeavor.
#59 Re: Help Me ! » Can someone prove this please? » 2008-11-02 03:34:17
Thank you.
#60 Re: Formulas » Symbols » 2008-11-01 17:28:03
: Does not belong to, in set theory
#61 Re: Formulas » Symbols » 2008-11-01 17:18:25
= And, in symbolic logic = Or, in Symbolic logic
I believe those are reversed.
= Or (corresponding to )and
= And (corresponding to )#62 Re: Formulas » Science laws and theorems » 2008-11-01 16:43:44
Newton's laws of motion.
First law:- Every body continues in its state of rest or uniform motion unless compleeled by an external force to change its position.
Interestingly, this is not Newton's first law of motion. It is actually a restatement of his second law, F=ma.
Strangely enough, Newton made the oft repeated error when he stated the first law. However, in the pages of Principia immediately preceding his statement of the first law, he described the law as something substantially different.
The first law should be (and was described by Newton as):
There exists a frame of reference in which an object continues in its state of rest or uniform motion unless a net external force is applied to that object.
In fact, without this first law, the second law need not hold.
#63 Help Me ! » Can someone prove this please? » 2008-11-01 16:18:10
- All_Is_Number
- Replies: 2
I have to use this formula often in a couple Stat Methods classes, but haven't seen a proof (my text presents formulas, but does not prove them). My own attempts have been unsuccessful and messy. Thus far, I've just taken it on faith that it is true, but I would prefer to know why it's true. Thanks.
#64 Re: Puzzles and Games » How much would you need to invest? » 2008-11-01 15:55:59
My apologies.
No need to apologize. 
#65 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-31 12:34:14
If you agree, then of course mathematics makes no predictions about the universe in which we live (by itself), so "accuracy" has no meaning.
So you're saying that mathematicians can never be wrong because they make their own rules up as they go along?
A model makes a prediction, sure enough, but a model is separate from the mathematics that it uses.
I'm not sure I agree (or disagree) with that statement. Often, the mathematics are developed specifically to create models for the application.
Personally, I believe that pure mathematics is something worth writing about.
I think you misunderstood the analogy. Understanding maths is like being able to write. Being able to apply the maths is like being able to write and having something to write about.
And I would state this even ignoring the fact that pure mathematics is used throughout physics and applied mathematics.
That would make them applied mathematics.
Personally, it matters not to me whether this is "useful" in terms of physics or statistics or economics. It is meaningful to me without having a use.
I too recognize the allure of mathematics as their own means to an end. It's not the only branch of science that has limited usefulness. I also recognize the attraction to answering the age old question of where the universe came from. In both cases, I don't see any practical usefulness of such studies. And in both cases, it is highly probable that practical discoveries will be made.
But, in the end, pure mathematics consistently supplies the places where it's applied with new ideas and new ways to solve problems.
That statement is equally true the other way.
#66 Re: This is Cool » Zero !!! » 2008-10-31 12:12:07
In our case, when d = 0, what we are asking is "how much force does a particle exert on itself?" This is a nonsensical question, and therefore, it is a really really good thing that the mathematical model breaks down at this point.
On the macro-scale, that is indeed a non-sensical situation. It's not so much that it's a good thing the mathematics break down as it is non-consequential, since the model assumes d>0. There are other times when we are interested in that point, but don't have the tools to investigate.
I call it a physical failure because it is (typically) the physicst who comes up with the model.
That's a bit like saying inflation is the fault of the chocolate manufacturer because the price of chocolate rose with everything else. Mathematics are great, but they're nowhere near the perfection you seem to have attributed to them.
Case in point, the model breaks, not the mathematics behind the model.
The maths break down, even if the model breaks down also.
But as soon as you accept that you are working in a field (ring), you can prove that division by zero is impossible. If you remove the assumption that we are working in a field (ring), then you can possibly have a division by zero. But you also want it to represent reality, and I don't see how you could do that if what you consider to be a "number" didn't form a field.
Right. If we make certain assumptions then we can thereafter draw logical conclusions based on those assumptions. Progress, however, tends to come when we recognize that we are mistaken with our assumptions.
Correct me if I'm wrong, but we were talking about sufficiently small but positive d. In other words, the quantum level.
Okay, consider yourself corrected. BTW, at the quantum level, two particles are not prohibited from occupying the same space.
Maths may be capable of describing such a situation, but that does not imply that mathematicians know how to do it.
Is that the job of the mathematician or the physicst?
You seem to be under the impression that the two groups are mutually exclusive. Scientists often work in multiple disciplines of science. Perhaps you've heard of Calculus? Newton developed that to solve physics problems.
#67 Re: This is Cool » Zero !!! » 2008-10-31 09:59:14
I can agree with that, but I would argue in turn that the mathematical breakdown at the very specific distance of d = 0 is inconsequential because the model becomes wildly inaccurate before that point anyway. I'm more concerned about the fact that the answers that the model gives at very small values of d aren't even close to reality than I am by the fact that eventually it doesn't give an answer at all.
Again, on the scales Newton was modeling, the theory worked very well. It models point masses, not massive objects with non-zero volume. Once the radius of the objects become significant compared to [i]d[/d], calculus must be used to calculate the gravitational attraction. Don't think that because this is disregarded in the formula's simplest form that this has not been long known (it was likely understood by Newton himself).
#68 Re: Puzzles and Games » How much would you need to invest? » 2008-10-31 09:48:16
If my understanding of your projection is correct, then by your reckoning the perpetuitys balance at the end of the first year would be
This clearly cannot be the case.
Aha! There's the misunderstanding.
My assertion was that the balance after one year is $833,333.33 * 1.04 after the payment of $50,000 is made. Prior to the payment, the balance is $833,333 * 1.04 + $50,000.
This is because every year, 60% of the interest (i=0.1) earned during the previous year is used to make the annual payment, giving the perpetuity account a net growth rate of 4%.
* * * * *
Here is an excellent book covering annuities, perpetuities, and much more.
#69 Re: Puzzles and Games » How much would you need to invest? » 2008-10-31 09:11:29
Hmmm I entered the data into a spreadsheet, and calculated the account balance at the end of each year, both before and after payment is made. To simplify the initial amounts, I multiplied the initial balance by 3 to change $833,333.33 to $2,500,000, and did the same with the initial annual payment, which changed from $50,000 to $150,000. (This change will not affect the rate of growth.) I also calculated the annual growth of a $2,500,000 deposit, at 4% effective interest.
The balance of the perpetuity after each annual payment was identical to the value of the $2,500,000 deposit at the same time t. So, I have to stick with my original answer:
(I'll try to attach a pdf of the spreadsheet, but I've never been able to upload attachments on this site.)
Edit to add: the upload did not work.
#70 Re: This is Cool » Zero !!! » 2008-10-31 05:13:29
Yes, that's true, but it's not relevant to the specific example of Newtonian gravity because the model has been shown to be flawed. You can argue that we don't have sufficiently advanced knowledge of math to combine quantum mechanics and general relativity to handle situations like singularities (which, last I knew, is true), but in the very specific case of Newtonian gravity it is the model that is at fault, not math.
The flaw in your logic is your assumption that because the model isn't perfect, the mathematics must be. The flaw in the theory is additional to and independent of the mathematical breakdown, and make no mistake, the mathematics do break down.
Quantum gravity handles classical physics' singularities pretty well, btw, but that's a topic for a different thread.
#71 Re: Dark Discussions at Cafe Infinity » Debate #001: Is math a science? » 2008-10-31 05:07:52
Well, it doesn't mean we use them without recognizing their faults, and without knowing how they coincidentally discribes things right in some cases, but will definately fail in other cases.
I agree. We still use Newton's equations because in most cases, the difference between them and the more accurate equations associated with relativity theory is less than the precision of the measurements can show (i.e. √(1-v²/c²)≈1).
#72 Re: This is Cool » All Piecewise Defined Functions Unnecessary? » 2008-10-30 23:57:04
True, but you can get around that by using something like :sqrt(x²) instead.
Doesn't x=√(x²) have two solutions?
#73 Re: This is Cool » Zero !!! » 2008-10-30 11:47:50
Let's say I model gravitational strength by
. Can you call that a mathematical breakdown when my model doesn't represent reality?
Typically, scientific models are not chosen arbitrarily as you have done.
Like I said this is all semantics so I suppose you could, but I never would.
Nor would I, since your model wouldn't work for any distance. I would call it an incorrect and completely useless model, albeit one that does, in fact, highlight an aspect of mathematics in for which we currently lack understanding.
I never said it was, I was just referring to the example given many posts ago now. I fully agree that math can and does break down, my only point is that when you talk about Newton's gravitational formula I don't understand how you can say that math breaks down when the model is incorrect to begin with. If anything, the fact that you can't evaluate it when d = 0 is a signal that your model is wrong and needs to be fixed, not that math somehow needs to account for 1/0.
It seems you don't understand Newtonian gravity. The model doesn't break down; it was never intended to be used at d=0. Such a scenario cannot exist on the macro-scale for which Newton created his model. Of course that doesn't change the fact that mathematics, at our current level of understanding, breaks down when we attempt to divide by zero.
#74 Re: This is Cool » Zero !!! » 2008-10-30 04:41:34
Actually, the model never applies. It is never correct, at best it's a very accurate approximation.
Being correct and being applicable are two different things. If they were the same, it would be impossible for us to know anything quantitative about the world around us.
I would also say that it isn't the job of the mathematicians to come up with accurate models, that's the physicists' job.
It's the tool that doesn't work, not the physicists. It's in no way a physical breakdown. The breakdown is strictly mathematical.
My point is that math is (probably) capable of describing what happens at d = 0 (assuming such a situation is physically possible, which I'm not sure it is), it's just that the physicists haven't come up with an accurate model for that situation.
Newton's gravitational theory is not the only place in which the limitations of mathematical knowledge become apparent. Those limitations must be overcome in the science of mathematics before they can be overcome by the other disciplines of science.
If, on the other hand, math was not advanced enough to be capable of describing that kind of situation then I would call that a mathematical breakdown.
I eagerly await your explanation of the exact definition of division by zero. Until we have one, it is indeed a mathematical breakdown.
#75 Re: This is Cool » Zero !!! » 2008-10-30 03:55:45
All_Is_Number wrote:Ricky wrote:It is not a mathematical failure that occurs, but a physical one.
It's a mathematical failure, not a physical one. Your assertion requires us to believe that because we don't currently understand how to do something in mathematics, that it cannot possibly be done.
Please, tell us, from the perspective of the photon, what is the distance between The sun and the Earth? From the perspective of the photon, how long does it take to travel between the sun and the Earth?
It's simply a matter of semantics. When Ricky said "physical failure" he meant, I think, that our mathematical interpretation of the physical world breaks down. Of course it's the mathematics that break, since they are a man-made construction, while physics is just observation of a pre-existing universe.
I agree that it's the mathematics that break down, but I don't think I misinterpreted what Ricky said, based on his assertions in another thread.
What I mean to say is that math is just a tool. It is the physicist's job to use it correctly. When a physicist comes up with a mathematical model of the physical world and that model breaks down, as it does in the case of Newton's gravitational formula at distance = 0, it would be fair to call that a physical breakdown. It may not necessarily be true, since the physical world can't literally break down, but it also doesn't mean that math is incapable of describing such a situation. It's just that the mathematical model in question is inaccurate.
No, it's not fair to say that it's a physical breakdown. The model only applies when d > 0. On the scale Newton worked with, distance cannot equal zero. It isn't inaccurate because of that. Maths may be capable of describing such a situation, but that does not imply that mathematicians know how to do it.