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timjones007
- 10
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why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?
Your question doesn't really make sense. If you know the proof, then what's your problem?timjones007 said:why do irrational numbers exist? I am well familiar with the proof that irrational numbers exist, but why do they?
timjones007 said:no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.
timjones007 said:no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.
So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself.
It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined.
In other words, sqrt(2) by definition is a number that you multiply by itself in order to get 2. However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.
I would be very surprised if he was actually asking interesting questions about formal language, computability theory, or anything like that. I think he simply doesn't have a clear understanding of what others (and he) means by 'number', and lacking such clarity, is flailing about with his intuition.csprof2000 said:It's actually a very interesting question the OP is getting at ...
A type of number system is defined by a list of properties. If a particular set* has those properties, it's a model of that number system, and we would call its elements numbers (of the appropriate type).What does everybody else think about what it takes to define a number?
timjones007 said:no, i don't think sqrt(2) exists. This is my reason: sqrt(2) is just a symbol for it's decimal representation which is 1.414213562..., and the decimal places continue on infinitely.
So, if we will never reach the last digit in the decimal places for sqrt(2), how can we multiply it by itself.
It's the same logic that goes with the fact that we can't multiply any number by infinity. For example, 0 x infinity is an ideterminate form, because although we know logically that you will get 0 if you keep multiplying 0s, we will never finish multiplying 0 infinitely many times so we say that it is undefined.
In other words, sqrt(2) by definition is a number that you multiply by itself in order to get 2. However, we will never be able to get that number so it should be undefined for the same reason that infinity is undefined.
But now, you're not doing mathematics anymore -- you've crossed over into physics, or possibly epistemology.csprof2000 said:No, you're right, it is sort of strange. Although, I challenge you to draw a line that's exactly 1 unit long, and prove that you got that right...
Basically, how do you get anything exactly right? How close is close enough to be exactly right?
What do you mean we can't measure them? We can construct and measure a length [itex]\sqrt{2}[/itex] as well as we can a length 1. It is true that we cannot write that out in terms of decimal numerals, but that is a problem with the numeration system, not the number.Dadface said:Yes csprof2000,yours is a good example,so numbers like root 2 come up but we can't measure them.It is a rabbit hole.
What exactly do you mean by "draw a line that's exactly 1 unit long"? In Euclidean geometry, we simply declare a segment to have length 1 and base everything else on that. I can then construct a segment that has length exactly [itex]\sqrt{2}[/itex]. (The physical "compasses" and "straight edge" represent the mathematics that is going on. Physical measurement is "approximate". Mathematical construction is not.) If you want to continue in this line, you should discuss [itex]\sqrt[3]{2}[/itex] which is not a "constructible" number!csprof2000 said:No, you're right, it is sort of strange. Although, I challenge you to draw a line that's exactly 1 unit long, and prove that you got that right...
Basically, how do you get anything exactly right? How close is close enough to be exactly right?
HallsofIvy said:If you want to continue in this line, you should discuss [itex]\sqrt[3]{2}[/itex] which is not a "constructible" number!
Do you understand the difference between the value of a number and the value as written in a particular number system? Certainly, in our standard decimal number system, "1" is already written to infinite precision, [itex]\sqrt{2}[/itex] is not. But, as I said before, that is an artifact of the numeration system, not the numbers themselves. If I were to use a place-value system, base [itex]\sqrt{2}[/itex], I can write [itex]\sqrt{2}[/itex] to "infinite precision": 10.csprof2000 said:"What exactly do you mean by "draw a line that's exactly 1 unit long"?"
Exactly what you said in your first post after me, if you think about what the meaning of that is. The point is both 1 and sqrt(2) are equally hard to nail down to infinite precision. I was just using the same language as Dad to say it, and you were using other language.
And what about the discussion of the value of numbers? Do numbers have to have value? And do you have to be able to find the value, at least in principle?
HallsofIvy said:If I were to use a place-value system, base [itex]\sqrt{2}[/itex], I can write [itex]\sqrt{2}[/itex] to "infinite precision": 10.
timjones007 said:Let me just say this in response to post number 17 by csproof2000.
Suppose there are two pieces of string forming the legs of an isosceles right triangle and each string measures 1 meter in length.
Now suppose that I ask you to cut a string of sqrt(2) meters to complete the triangle. Even if you use the most accurately calibrated meter stick that can be created, you wouldn't be able to do it because you would know where to stop cutting.
You might want to stop at 1.41 m, but that's to0 small. Then you'd try 1.414213562 m, but that is also too small.
(by the way, not that it matters, but I, the op, am a female...just thought I might clear that up since everyone keeps saying he)
csprof2000 said:However, you are committing a fallacy, Halls, and I think it's time to call you out on it. You keep saying that it's not valid to rely on the decimal representation of a number in order to talk about its value. My problem with this is, quite simply, that the decimal notation is one representation of a number, and I see no reason to exclude it as a way of talking about a number's value. Of course, if you have other notations which make the number's value easy to arrive at, fine, but it's all too easy to avoid talking about value by hand waving arguments related to their representation. Yes, the representation isn't the number. But we need a representation to talk about its value in a clear way. If you have another way, please, let me know. Otherwise, let's drop the whole "the representation isn't the number" thing. Ink isn't the same as the Declaration of Independence, but it makes it a lot easier to read.
I did not say any such thing!csprof2000 said:However, you are committing a fallacy, Halls, and I think it's time to call you out on it. You keep saying that it's not valid to rely on the decimal representation of a number in order to talk about its value.
If someone were to say that the Declaration of Independence is meaningless because the ink is too faded to be read, don't you think that talking about difference between the content and the representation in ink is relevant?My problem with this is, quite simply, that the decimal notation is one representation of a number, and I see no reason to exclude it as a way of talking about a number's value. Of course, if you have other notations which make the number's value easy to arrive at, fine, but it's all too easy to avoid talking about value by hand waving arguments related to their representation. Yes, the representation isn't the number. But we need a representation to talk about its value in a clear way. If you have another way, please, let me know. Otherwise, let's drop the whole "the representation isn't the number" thing. Ink isn't the same as the Declaration of Independence, but it makes it a lot easier to read.
csprof2000 said:But for sqrt(2)? How many should there be? 2? 1? 3? And what on Earth would you call them, and what does it mean? How would you right 2.5 in base sqrt(2)... I understand it's irrational in this base, if anything, but still...