Proving that 2=1 with Algebra: Say x = 1. Then x˛-1 = x-1. We then divide both sides by (x-1), and get x + 1= 1, which means 2 = 1.
The Hanging Paradox: A man, who is going to be hung, is told that it will be a surprise. But since he knows, it will not be a surprise, so it cannot happen. So when it does, it will be a surprise, allowing it to happen.
The Number Set Paradox: If a set of sets, which is a set containing sets of numbers contains those sets, does it contain itself?
The Exception Paradox: If every rule has an exception, is there an exception to the rule that states this?
The History Paradox: Man learns from history that man learns nothing from history.
The Ship Paradox: When every component of a ship is replaced, is it the same ship?
The Birthday Paradox: If there are 23 people in a room, there is a 50% chance that 2 of them will have the same birthday.
The Square Paradox: Though most numbers are not squares, there are no more regular numbers then squares.
The Room Paradox: If a hotel with and infinite amount of rooms is full, it can still hold more guests.
The Sphere Paradox: A sphere can be cut into 5 pieces, and reassembled to get 2 balls of equal size.
The Rational Paradox: How can a rational decision be made of two outcomes of equal value?
The Freezing Paradox: How can room temperature water freeze faster then cold water, even though it has the cold waters temperature?
The Truth Paradox: If truth does not exist, then the statement ‘truth does not exist’ is the truth, thereby proving it false, thereby proving it to be true, thereby…
The Halfway Paradox: You can never get from point A to point B, because you must always get halfway there, and half of that, and half of that. You must do so an infinite amount of times. But since it is impossible to live an infinite amount of time, it is impossible.
The Necessary Paradox: Why is diamond more valuable then water, though water is needed for survival, and thus a requisite for owning diamonds?
Proving that 2=1 with Algebra: Say x = 1. Then x˛-1 = x-1. We then divide both sides by (x-1), and get x + 1= 1, which means 2 = 1.
Does anyone nhave any other Paradoxes? Any that they have made up?
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I agree with Jon Boy, but they're still interesting to think about. I was confused about the sphere one though. I don't even know what it was talking about
Posts: 1789 | Registered: Jul 2003
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Was there really any point to this? Like Jon Boy said, most aren't really paradoxes as much as they are philosophical questions. Others aren't paradoxes in the sense that they are built on extremely faulty logic. I guess it was kind of amusing, but I just don't see the point.
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A paradox is a collection of purportedly true statements that are seem to be mutually exclusive. In a true paradox, they are actually mutually exclusive. The classic forumulation of this is "The next thing I say will be true. The last thing I said was false."
quote:Proving that 2=1 with Algebra: Say x = 1. Then x˛-1 = x-1. We then divide both sides by (x-1), and get x + 1= 1, which means 2 = 1.
Sort of a paradox, but really more bad math. If you're making x = 1, then you can't divide by x - 1, as that would then be 0.
quote:The Hanging Paradox: A man, who is going to be hung, is told that it will be a surprise. But since he knows, it will not be a surprise, so it cannot happen. So when it does, it will be a surprise, allowing it to happen.
This is a paradox.
quote:The Number Set Paradox: If a set of sets, which is a set containing sets of numbers contains those sets, does it contain itself?
I don't understand what this says. I'm voting for grammatical error, with a side of an attempt for Russel's paradox.
quote:The Exception Paradox: If every rule has an exception, is there an exception to the rule that states this?
Paradox.
quote:The History Paradox: Man learns from history that man learns nothing from history.
False paradox based on the hyperbole learning nothing.
quote:The Ship Paradox: When every component of a ship is replaced, is it the same ship?
This isn't really a paradox. It's a question of identity. In the classic formulation, it is a paradox though. In this situation, all the original parts of the ship have been replaced, but then someone took all the original parts and built a ship out of them, using the same name. Which ship is the real one then?
quote:The Birthday Paradox: If there are 23 people in a room, there is a 50% chance that 2 of them will have the same birthday.
Not a paradox at all, merely a function of probability.
quote:The Square Paradox: Though most numbers are not squares, there are no more regular numbers then squares.
Not a paradox.
quote:The Room Paradox: If a hotel with and infinite amount of rooms is full, it can still hold more guests.
A paradox, but not a true one.
quote:The Sphere Paradox: A sphere can be cut into 5 pieces, and reassembled to get 2 balls of equal size.
Not a paradox. Just a function of geometry.
quote:The Rational Paradox: How can a rational decision be made of two outcomes of equal value?
Not a paradox.
quote:The Freezing Paradox: How can room temperature water freeze faster then cold water, even though it has the cold waters temperature?
Not a paradox. It's just barely a riddle.
quote:The Truth Paradox: If truth does not exist, then the statement ‘truth does not exist’ is the truth, thereby proving it false, thereby proving it to be true, thereby…
True paradox.
quote:The Halfway Paradox: You can never get from point A to point B, because you must always get halfway there, and half of that, and half of that. You must do so an infinite amount of times. But since it is impossible to live an infinite amount of time, it is impossible.
Better known as Zeno's paradox.
quote:The Necessary Paradox: Why is diamond more valuable then water, though water is needed for survival, and thus a requisite for owning diamonds?
posted
Self-Reference Paradox - "This statement is false."
On the 1=2 thing, the way I learned it in high school, we had 0=1. Me and a couple friends tried formulating a system of math on that basis. I don't think we got past "0=1, therefore, every real numbers is equal to every other real number." When I learned there was a system of math based off of nonreal numbers (i), I became convinced that there will be another based off "0=1."
posted
While the proof that 1=2 is false, as Squick pointed out, it is possible (and sort of fun) to prove that 1.999... = 2.
Posts: 2926 | Registered: Sep 2005
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Juxtapose: there's a better analysis possible. Basically, realize that the demand curve is also a value curve. The marginal value (to consumers) of an additional unit is determined by the area under the part of the curve indicating that additional unit.
From this analysis, it becomes clear that water is far more valuable than diamonds, it is mostly the comparatively high demand for water (think how much water is demanded in the world) that results in equilibriums involving much higher prices for diamonds. The supply curve happens to be mostly irrelevant in this case.
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Actually, the math one is just bad math. The way you formulated the two premises, you assumed 2 premises, but one or the other is wrong, depending on which one is the First premise.
If x = 1 is the first premise:
x^2 = 1^2 => x^2 = 1
From here, you can't get to x^2 - 1 = x - 1. No matter how hard you try, therefore this assumption is incorrect.
If x^2 - 1 = x - 1 is the first premise:
Then x can be any value that satisfies the equation, which minimally includes 0 AND 1, possibly more.
He assumed x = 1, then multiplied both sides by x to get x^2 = x, then subtracted 1 from both sides to get x^2 - 1 = x - 1 .
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quote:Originally posted by Grim: The Sphere Paradox: A sphere can be cut into 5 pieces, and reassembled to get 2 balls of equal size.
I agree that many of these, including this one, are not paradoxes... but does anyone have a link visually explaining what this is talking about? I'm having a hard time visualizing whatever it's trying to describe.
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quote:The Room Paradox: If a hotel with and infinite amount of rooms is full, it can still hold more guests.
If I remember my 15 year old higher mathmatics correctly (and someone jump in if I'm wrong) This is where "countable infinities" come in represented by the letter "Aleph".
If the hotel has "Aleph(Null)" rooms and "Aleph(1)" guests try to check in, the hotel will be full becuase "Aleph(Null)" is a smaller infinity than "Aleph(1)"
Another example of this relationship would be the set of all positive integers vs the set of real numbers. Both are infinity. But there are infinately more real numbers than there are positive integers because, Not only do the set of real numbers extend toward negative infinity but there are an infinate number of real numbers between each integeter. Example: 1 and 2 are positive integers. But there are an infinate number of Real numbers between them (1.1, 1.9, 1.235423452454534, etc) Thus, if we assign the set of positive integers to Aleph(Null) and the set of all Real numbers to Aleph(1) we can see that both are infinity but Aleph(1) is larger.
posted
I'm a big fan of paradoxes, but I prefer them philisophical, ie: God is everywhere, God is nowhere. or: any given amount of time is infinitely long and infinitely short.
I'm not sure if this is a paradox, but i like thinking about it. We live only in the present, but there is no present. What we think is the present is actually the past because in the amount of time it takes to recognize 'time,' that time is gone. again, probably not a paradox, but I like it.
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The present is inherently paradoxical. If I recall correctly, this is one of the Zeno's lesser-known paradoxes. To wit, the present must be indivisible. It can't have one part happening before another. However, the present also separates the past and the future, both of which abut the present. The past and the future cannot touch each other and therefore must touch separate parts of the present. However, if this were true, then the present would have to be divisible into these separate parts.
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also, the 2=1 equation reminds me of another fun equation that isn't a paradox, but doesn't warrant a new thread, so I will post it here. I'm sure many of you have seen this before.
Women take time and money, so: Women = Time x Money
Everyone knows time is money, so : Time = Money, so: Women = Money^2
Everyone knows money is the root of all evil, so: money = square root of evil, so: women = (square root of evil)^2
the square root cancels out so: Women = evil.
I'm sure this isn't mathmatically correct, but I like it.
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quote:Originally posted by fugu13: Yes, of course, it does, I'm merely pointing out the algebra is otherwise correct.
If you want a fun mathematical one,
True: e^(i*pi) = -1
e^(2*i*pi) = 1
ln e^(2*i*pi) = 2*i*pi = ln 1 = 0
divide by the nonzero constants 2, i and pi
1 = 0
My college roommate once told me that a natural-born mathemetician is one to whom e^(i*pi) is obviously -1.
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And when you find yourself with a real paradox, they will both bill you, and your insurance will likely only pay for one.
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quote:Women take time and money, so: Women = Time x Money
I always was told in my word problems, and meand add, not multiply. Women take a lot of time and money, but the amounts of the two are not factors of each other. If you are giving money to women based on the time you spend with them, that would be prostitution.
So what you get is W = T + M W=M + M W = 2M
E = M*2 EW = E-2M EW = M*2-2M or Evil Women cost lots of money.
That is not a paradox. That is a fact.
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Bok, That's what I get for doing symbolic math with a head cold. I think you're right. I just wasn't thinking about the wider stuff. My bad.
Posts: 10177 | Registered: Apr 2001
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quote:Originally posted by fugu13: Yes, of course, it does, I'm merely pointing out the algebra is otherwise correct.
If you want a fun mathematical one,
True: e^(i*pi) = -1
e^(2*i*pi) = 1
ln e^(2*i*pi) = 2*i*pi = ln 1 = 0
divide by the nonzero constants 2, i and pi
1 = 0
I think that your last line of that is incorrect. Down to 2*i*pi = 0 is as far as I can tell, but the error comes when you divide by i, an imaginary number, to try to equate two real numbers. That is, I don't think 0/i is actually zero since you're treating i as any other real number (except 0) in saying that 0/i = 0. I'm not 100% on this, though, but obviously that equality can't hold true, and since your math up to that point is correct, my guess is that that's where the error lies.
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Its perfectly legal to divide i by i and get 1, or to divide 0 by i and get one. And of course, one could always not divide by i and still have i = 0, which is also very false.
The error is that ln 1 doesn't just equal zero. It has infinitely many values, one of which is (coincidentally), 2*i*pi. Similarly, ln e^(2*i*pi) has infinitely many values, I do believe, its just usually convenient to treat it as having one value.
Posts: 15770 | Registered: Dec 2001
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Tom Robbins is excellent at the paradox. I enjoy when he talks about them. And what was it the Nafai said about paradoxes? something along the lines of "you get a paradox when people don't want to think about the problem anymore." ok, thats not anything close to what he says, but you get the jist.
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quote:Originally posted by fugu13: Its perfectly legal to divide i by i and get 1, or to divide 0 by i and get one. And of course, one could always not divide by i and still have i = 0, which is also very false.
The error is that ln 1 doesn't just equal zero. It has infinitely many values, one of which is (coincidentally), 2*i*pi. Similarly, ln e^(2*i*pi) has infinitely many values, I do believe, its just usually convenient to treat it as having one value.
I know that you can divide i by i because you just have a ratio of something with itself which is (almost) always 1. I'm not sure where you're getting that the ln(1) equals numbers other than zero, though. The ln function is described as a log with a base of e, and the only power that you can raise e to and get 1 as your result is 0 (hence why ln(1)=0). Any other solution to that would say that e^(some power other than 0) = 1, which is false. There aren't infinitely many solutions to ln(1).
Posts: 143 | Registered: Sep 2005
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In the case of ln 1, ln 1 = ln (1 * e^(i*0)) = ln 1 + i* 0 + i*2*k*pi, so there are infinitely many solutions, one of which is 2*i*pi .
Posts: 15770 | Registered: Dec 2001
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(To understand why this is so, realize that e^(2*i*pi) is also one, as is e^(4*i*pi), and e^(6*i*pi), and so on. 0 is not the only thing you can raise e to the power of and get 1)
Posts: 15770 | Registered: Dec 2001
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Zeno's paradox or whatever about getting from point A to point B is rather stupid. You go half way, you go the rest of the way. I don't think it's a good basis for a paradox. Also the paradox about motion... It says that since we measure speed by distance/time, how can things move, because all time is made up of a series of single moments, and in each moment there is no passage of time. So how can you measure speed? I think this is stupid too. In a single moment, I'm in this spot, and in the next moment, I'm not in this spot. It's simple. Just because we can't measure speed doesn't mean things don't move. Those Greek guys must have been bored.
A paradox that we talked about (or made up, I can't remember) in math class last year that I rather like is this: My teacher tells us he's giving us a pop quiz next week. A pop quix means that we don't know which day it's going to be on. So then it can't be on Friday, because we'd know it was going to be on that day, since it's the last day of the week and we haven't had the test yet. So then it also can't be on Thursday, because we know it can't be on Friday, and Thursday comes and still no test, we know it'll be on that day. But then it also can't be on Wednesday...and so on. I like that one.
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Really, Soara? What, then, is the smallest fraction?
(And keep in mind that even though there comes a point in sequentially dividing by two that the remaining difference is negligible in practical terms, it never actually IS zero.)
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That's pretty tricksy, fugu. I didn't even think to use Euler's formula inside the ln function to manipulate it to still be 1.
Posts: 143 | Registered: Sep 2005
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As I understand it, Zeno's paradox is almost always misstated. It should be something more like "Before you can get all the way there, you have to get halfway there, and before you can get halfway there, you have to get a quarter of the way there, and before you get a quarter of the way there, you have to get an eighth of the way there . . ." and so on.
Since you can divide any finite distance into an infinite number of segments, and since any segment will take a finite time to move through, traveling any finite distance will take an infinite amount of time (or so they say).
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The flaw in Zeno's paradox is that the sum of an infinite number of finite time intervals isn't always infinite. In this case, the time intervals conveniently add up to distance / rate.
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The Mouth of Truth can signal unerringly whether a statement is true or false. You put your hand in it and say something. If it doesn't bite your hand off, it must have been true. Otherwise...
So, you put your hand in it and say, "The Mouth of Truth will not let me keep my hand."
A common one: There are no absolute truths.
A philosophy around 1900: logical positivism. Logical positivism states: all statements fall into one of these categories.
1. Tautology/contradiction. Something that is true or false simply by the way it's defined. "All bachelors are unmarried."
2. Statements verifiable by observation.
3. Nonsense.
So. What category does "Logical Positivism is true" fall into?
Posts: 1877 | Registered: Apr 2005
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quote:The flaw in Zeno's paradox is that the sum of an infinite number of finite time intervals isn't always infinite. In this case, the time intervals conveniently add up to distance / rate.
To expand slightly: If you have to go 60 yards in 60 seconds, you will go 30 yards in 30 seconds, then 15 yards in 15 seconds, 7.5 in 7.5, etc.
It's fairly intuitive that the time for all the intervals after the first will approach and not exceed 30 seconds.
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HONEY! Geet me ma' flame throwa'! We got us an enfestation o' NERDS!
oh... oh my god... OH. MY. GOD.!! HONEY! GET IN THE CAR! THEY'RE... THEY'RE MATH NERDS!!!!! WE'RE GEETTIN' OUTA' HERE......
Posts: 499 | Registered: Mar 2004
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It was named after a zebra who was named Al. Al Gebra.
"Algebra" don't have a Z in it, you say? Algebra ain't got nothing BUT Z's and X's and stuff like that.
Posts: 1877 | Registered: Apr 2005
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quote: The Ship Paradox: When every component of a ship is replaced, is it the same ship?
It's like that story where a man visted a museum devoted to George Washington. "This here's the ax young Washington used to chop down the cherry tree," the curator said.
"The very same?" the man asked. "After all this time?"
"Oh, we've had to take good care of it, for sure," the curator said. "But the blade's only been replaced twice, and the handle replaced three times."
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