Yep. I was very carefully going over the content of Sam Hughes' page that you linked to earlier. In any case where somebody is presenting a large volume of well-structured evidence supporting an idea that I have trouble accepting, I try to find the first point in their argument that I disagree with.
What I found was right near the top, where he says, "Different sets of numbers have different properties." (Which I don't disagree with, it's just from there I begin to follow a different line of reasoning.) I think -- without the necessary background in mathematics to give this notion any kind of legitimacy whatsoever -- that real numbers which necessarily have an infinitely repeating decimal value must be treated as a separate set of numbers, with some of their own properties. So, I see the case of 1 being represented as 1.000000... not any different from saying, 1 = 1 + 0 + 0 + ...; none of the decimals add any value at all to the number being operated on. However, in the case of something like .333..., each subsequent decimal adds some small value to the number in question.
Things begin to make some sense from there, although it's inconvenient. For example, that makes the precise value of the number somewhat difficult to deal with. It reminds me of the Koch snowflake; you have this thing with finite bounds but an infinitely long boundary.
So, let's say you wanted to add one Koch snowflake to another Koch snowflake. How would you do that? Well ... you sort of can't. You can get a very close approximation, close enough to do whatever it is you're trying to do. But you can't get the exact result, unless you first return the Koch snowflake you're working with to the original equation which described it. So, if I wanted to operate on .333..., the first thing I would do is return it to the equation that described it: 1/3. Then I could do useful things with it, like add another 1/3, and get the correct answer.
I think I'm making a distinction between the idea of a particular number, and its value, or its representation. I realize that that's something that would make a lot of mathematicians cringe. :-) But, again, there is a small but not insignificant difference between saying, "Koch snowflake of unit side four", and asking somebody to compare the drawing of such a snowflake to the drawing of another Koch snowflake. It's also better to say, "add a Koch snowflake of unit side 4 to a Koch snowflake of unit side 3 and tell me what you get", than it is to hand a person a drawing -- a representation -- of two Koch snowflakes, and say, "add these up for me".
no subject
Yep. I was very carefully going over the content of Sam Hughes' page that you linked to earlier. In any case where somebody is presenting a large volume of well-structured evidence supporting an idea that I have trouble accepting, I try to find the first point in their argument that I disagree with.
What I found was right near the top, where he says, "Different sets of numbers have different properties." (Which I don't disagree with, it's just from there I begin to follow a different line of reasoning.) I think -- without the necessary background in mathematics to give this notion any kind of legitimacy whatsoever -- that real numbers which necessarily have an infinitely repeating decimal value must be treated as a separate set of numbers, with some of their own properties. So, I see the case of 1 being represented as 1.000000... not any different from saying, 1 = 1 + 0 + 0 + ...; none of the decimals add any value at all to the number being operated on. However, in the case of something like .333..., each subsequent decimal adds some small value to the number in question.
Things begin to make some sense from there, although it's inconvenient. For example, that makes the precise value of the number somewhat difficult to deal with. It reminds me of the Koch snowflake; you have this thing with finite bounds but an infinitely long boundary.
So, let's say you wanted to add one Koch snowflake to another Koch snowflake. How would you do that? Well ... you sort of can't. You can get a very close approximation, close enough to do whatever it is you're trying to do. But you can't get the exact result, unless you first return the Koch snowflake you're working with to the original equation which described it. So, if I wanted to operate on .333..., the first thing I would do is return it to the equation that described it: 1/3. Then I could do useful things with it, like add another 1/3, and get the correct answer.
I think I'm making a distinction between the idea of a particular number, and its value, or its representation. I realize that that's something that would make a lot of mathematicians cringe. :-) But, again, there is a small but not insignificant difference between saying, "Koch snowflake of unit side four", and asking somebody to compare the drawing of such a snowflake to the drawing of another Koch snowflake. It's also better to say, "add a Koch snowflake of unit side 4 to a Koch snowflake of unit side 3 and tell me what you get", than it is to hand a person a drawing -- a representation -- of two Koch snowflakes, and say, "add these up for me".