as stated in the article is wrong. Try it - just fold a paper twice in random angles so that the creases meet. The angles will not add up to 180. The author forgot to indicate that n must be odd.
So you say that i/i is not 1? That's weird... and actually, 1/i is not i but -i; multiply and you'll see.
The real problem is that all nonzero numbers have 2 square roots; for -1 they are i and -i. We generally look at the positive root when we're dealing with real roots, but when we're not, it becomes ambiguous. In this case, 1/sqrt(-1)=sqrt(-1)/1 is only true if we're looking at the two different square roots of -1, which we later multiply to incorrectly get -1.
What do you mean? I never multiplied it out to a real and imaginary part except when I wrote that sqrt(-1)*sqrt(-1)=-1 and sqrt(1)*sqrt(1)=1. Which line, specifically, is wrong?
However, if the corporation owns a copy of a CD, then only one person can listen at once, otherwise it becomes a 'performance', which you haven't bought the rights to when you buy a CD.
Your argument implies that listening to music without headphones is a copyright violation.
Re:Who cares about decimal?
on
RMS Turns 50
·
· Score: 1
OK, our definitions seem very different. However, I use this stuff for math contests all the time. An integer must be a real number. It is either a natural number, its negative, or zero. A number is... well, a number. -5000 is a negative number. i is a complex number. The set of numbers is the broadest of the number sets. If -5000 is not a number, then what the heck is it? A non-negative real integer is... well, a non-negative integer. Some people call them natural numbers, but 0 is actually not a natural number according to many, so "non-negative integer" is the best name for them. (When I learned these sets then natural numbers were said to be positive integers and whole numbers were said to be non-negative integers, but since then, on contests, I have seen that they are usually called positive integers and non-negative integers to avoid confusion.) Maybe you think i is an integer because some programming language implementation thinks that complex numbers with both real and imaginary coefficients being integers are integers. They're not. You're right in that in number theory, only natural (positive integer) numbers are used (although sometimes zero is included). However, this must be explicitly stated.
(Score:_, Offtopic)
Re:Who cares about decimal?
on
RMS Turns 50
·
· Score: 1
He meant squares of integers. Otherwise, there is no smallest such number. -5000=(50i)^2+(50i)^2
Besides, even if the graininess of space somehow affected the physicists' pi, it could be expected to do no more than to make it rational; however, there would still probably not be a last digit. (Does 1/3 have a last digit?)
Your point is a good one, although believe you meant that we should be outraged, not outrageous. However, back then, with a smaller population, it was easier to discern who to support. Nowadays, if someone decides to start a band and play at concerts, they'll be almost invisible. This may be soluble if, instead of expecting the rich to find artists to support, we'd expect the artists to find rich people to support them. Of course, this doesn't fit very well into our capitalistic system. The only loophole with which it could be implemented without massive whining would be if the artists were paid by taxes. However, then someone has to decide who is and who is not an artist. At concerts, artists are already paid, but arranging these costs a significant sum of money. Maybe there should be tax-funded (or corporate sponsor-funded) venues where any artist can perform in the open; there would be a queue. Then, the artists could be paid based on how many people stop to listen to them. This would be hard but not impossible to arrange. However, the modern tax-paying public would complain.
It's not a very frequently used grammatical construct, but it is correct. Compare it to "I am dead", "I am killed", and "I am banned from elementary school contests". They can have two meanings; one meaning that the action is in progress and one describing the current state of the subject.
No, it is the other way around: on a plane, the circle encompasses the most area for its circumference/perimeter. However, as someone else has noted, the fence must be made of prefabricated sections anyway.
Yes, but if we go for minimal area, then a triangle with equal sides must always have the same area, while a quadrilateral with equal sides can have an arbitrarily small size (although it must be non-negative). A degenerate equilateral quadrilateral can have zero area. It is true, however, that if the segments are predetermined and reasonably short (less than a few thousand kilometers), then the area/perimeter is still higher with the triangle because the little area gained with the quadrilateral by no means matches the 1/3 gain in perimeter.
And you also want to use as little fence as possible. Thus, if you're not willing to accept a fence of length 0 enclosing the whole earth, then you'll have to resort to a wooden tube stuck in the ground.
The significant difference is that for it to be useful to learn esperanto, there need to be a lot of people who know esperanto (not that there aren't). If you need a domain name and it's already occupied under.com.org.net, then registering it under.us if you're American is useful even if (especially if) nobody else does it.
I ask you then, should running also be prohibited? It is certainly faster than walking. Maybe a sprinting midget does less harm. Should there then be a law limiting the product of your mass and speed?
Or: Conclude that modern operating systems aren't as good as they could be, join in developing the Hurd, work on it, port mozilla, integrate l4, fix up apt, and you're done! Easy as Fermat's Last Theorem.
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I thought I'd made my counterexample clear:
a1/a2
-------
a4/a3
Note that a1+a3 is not 180 and nor is a2+a4.
The theorem only holds when n is odd.
as stated in the article is wrong. Try it - just fold a paper twice in random angles so that the creases meet. The angles will not add up to 180. The author forgot to indicate that n must be odd.
Silly people... this is TOPOLOGY! It's not meant for people to USE it! It's just for mathematicians to RUMINATE UPON!
Has Fermat's Last Theorem actually been used in practical applications? I don't think so...
So you say that i/i is not 1? That's weird... and actually, 1/i is not i but -i; multiply and you'll see.
The real problem is that all nonzero numbers have 2 square roots; for -1 they are i and -i. We generally look at the positive root when we're dealing with real roots, but when we're not, it becomes ambiguous. In this case, 1/sqrt(-1)=sqrt(-1)/1 is only true if we're looking at the two different square roots of -1, which we later multiply to incorrectly get -1.
What do you mean? I never multiplied it out to a real and imaginary part except when I wrote that sqrt(-1)*sqrt(-1)=-1 and sqrt(1)*sqrt(1)=1. Which line, specifically, is wrong?
1*1=-1*-1( 1)/sqrt(-1)=sqrt(-1)/sqrt(1)t (-1)*sqrt(-1)
1/-1=-1/1
sqrt(1/-1)=sqrt(-1/1)
sqrt
sqrt(1)*sqrt(1)=sqr
1=-1
Your argument implies that listening to music without headphones is a copyright violation.
OK, our definitions seem very different. However, I use this stuff for math contests all the time. An integer must be a real number. It is either a natural number, its negative, or zero. A number is ... well, a number. -5000 is a negative number. i is a complex number. The set of numbers is the broadest of the number sets. If -5000 is not a number, then what the heck is it?
A non-negative real integer is... well, a non-negative integer. Some people call them natural numbers, but 0 is actually not a natural number according to many, so "non-negative integer" is the best name for them. (When I learned these sets then natural numbers were said to be positive integers and whole numbers were said to be non-negative integers, but since then, on contests, I have seen that they are usually called positive integers and non-negative integers to avoid confusion.) Maybe you think i is an integer because some programming language implementation thinks that complex numbers with both real and imaginary coefficients being integers are integers. They're not. You're right in that in number theory, only natural (positive integer) numbers are used (although sometimes zero is included). However, this must be explicitly stated.
(Score:_, Offtopic)
He meant squares of integers. Otherwise, there is no smallest such number.
-5000=(50i)^2+(50i)^2
Besides, even if the graininess of space somehow affected the physicists' pi, it could be expected to do no more than to make it rational; however, there would still probably not be a last digit. (Does 1/3 have a last digit?)
Your point is a good one, although believe you meant that we should be outraged, not outrageous. However, back then, with a smaller population, it was easier to discern who to support. Nowadays, if someone decides to start a band and play at concerts, they'll be almost invisible. This may be soluble if, instead of expecting the rich to find artists to support, we'd expect the artists to find rich people to support them. Of course, this doesn't fit very well into our capitalistic system. The only loophole with which it could be implemented without massive whining would be if the artists were paid by taxes. However, then someone has to decide who is and who is not an artist. At concerts, artists are already paid, but arranging these costs a significant sum of money. Maybe there should be tax-funded (or corporate sponsor-funded) venues where any artist can perform in the open; there would be a queue. Then, the artists could be paid based on how many people stop to listen to them. This would be hard but not impossible to arrange. However, the modern tax-paying public would complain.
It's not a very frequently used grammatical construct, but it is correct. Compare it to "I am dead", "I am killed", and "I am banned from elementary school contests". They can have two meanings; one meaning that the action is in progress and one describing the current state of the subject.
(Score:_, Offtopic)
No, it is the other way around: on a plane, the circle encompasses the most area for its circumference/perimeter. However, as someone else has noted, the fence must be made of prefabricated sections anyway.
Yes, but if we go for minimal area, then a triangle with equal sides must always have the same area, while a quadrilateral with equal sides can have an arbitrarily small size (although it must be non-negative). A degenerate equilateral quadrilateral can have zero area. It is true, however, that if the segments are predetermined and reasonably short (less than a few thousand kilometers), then the area/perimeter is still higher with the triangle because the little area gained with the quadrilateral by no means matches the 1/3 gain in perimeter.
(Score:_, Offtopic)
And you also want to use as little fence as possible. Thus, if you're not willing to accept a fence of length 0 enclosing the whole earth, then you'll have to resort to a wooden tube stuck in the ground.
(Score:_, Offtopic)
No, it wouldn't. An equilateral triangle of perimeter 12 has area 4sqrt(3). A square of perimeter 12 has area 9. 9>4sqrt(3)
(Score:_, Offtopic)
The significant difference is that for it to be useful to learn esperanto, there need to be a lot of people who know esperanto (not that there aren't). If you need a domain name and it's already occupied under .com .org .net, then registering it under .us if you're American is useful even if (especially if) nobody else does it.
I ask you then, should running also be prohibited? It is certainly faster than walking. Maybe a sprinting midget does less harm. Should there then be a law limiting the product of your mass and speed?
Or:
Conclude that modern operating systems aren't as good as they could be, join in developing the Hurd, work on it, port mozilla, integrate l4, fix up apt, and you're done! Easy as Fermat's Last Theorem.
Wow, times certainly have changed.
I have just one thing to say:
MATH!!!
Thus speaks he who has not used Galeon.
What's next? ... sorry, I couldn't resist the temptation.
Qt 9.5?
Qt 9.8?
Qt N.T?
Qt 200.0?
Qt X.P?
It's probably your prefs... your message hasn't been moderated yet, and it's at 2.
Pi is irrational. Pi does not repeat.