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Actually, I have proven that .999... ==1, it does not suffer from round off error at all, and it has yet to be disproven.Quote:
originally posted by Gandalf_Grey_
i am not saying that the 8 is at any defined decimal place it is just at infinity. Maybe you aren't getting this specific reference however when i say .999 ... 8 i mean a number infitesimally smaller than .999 .... An infitesimally small number is possible (the difference between .99 ... and 1).
snakeeyes1000, it certainly does suffer from rounding error, that's the point:)!
Evil_Giraffe, I believe I have already explained my viewpoints clearly, so I will respond only that I certainly do think that infinity does exist, and that 1/infinity is not a real number. Infinity is not a real number, but a concept. It obviously makes no sense to use it in real math. However, 1/infinity is NOT an attempt to use infinity in real math, but rather the simplest and most accurate method of defining, literally, a value which is infinitely small. (0.9) = 1 - 1/infinity not due to real math, but conceptual math.
When I say that a number is not real, what I mean is that its true value is conceptual, and when we use it as a real we are only approximating it. I have personally come to the conclusion that all numbers with an infinite number of digits are not truly real, simply because it is impossible to USE them. You may disagree with me on that and many other points; that is my opinion. (Yes I know it is not exactly adhering to the definition of real numbers, I think the definition is flawed:)).
Sorry Alphanos, there isn't room for personal opinion here, since we're working within a common framework. Just because you disagree with the definition of a "real" number, doesn't mean you can disregard it.
And you're also spouting rubbish about "conceptual" values again. Mind telling me what's wrong with talking about numbers that can't be represented exactly in a non repeating decimal format? 1/7. Seems a fairly "real" number (using your definition of the term real number) to me. Then suddenly we look at it in decimal and it's a recurring decimal. How about we look at it in base 7? Oh look, it's 0.1 Seems fairly "real" to me.
Your concept of "real" numbers shouldn't depend on how you write them down. The notation used should have no bearing on the underlying maths. That's the whole point.
I guess you also disagree (personal opinion of course) with the summation to infinity argument?
Basically: i agree with Evil Giraffe
1) I think we all believe infinity "exists"Quote:
I certainly do think that infinity does exist
2) What, so we can't use Pi or e or 20.5 any more???Quote:
all numbers with an infinite number of digits are not truly real, simply because it is impossible to USE them
3) Oh, so 3^(0.9) = 3 and now you CAN use (0.9) to evaluate a real value, with (0.9) =1Quote:
You could say (9*(0.9)) / (3*(0.9)) = 3, and that is fine. You could not, however, evaluate 6*pi to a single real value without the second usage of such numbers, approximation
4) Sorry, i think i missed you proof, and can't find it. (There have been a lot of posts)Quote:
1 - (0.9) = 1/infinity
BTW: don't worry about sounding "rude or angry" ( it think you said that a few posts ago) Anyway, it is good to see someone standing up for what they think is right.
Famous quote of Einstein:
Only two things are infinite, the universe and human stupidity, and I'm not sure about the former.
I just thought I'd pop that quote in while we are debating about infinity and so on.
the concept exists, but number which has a value of infinity does not exist. but you can have infinitely many nines trailing (a concept here, not a value).Quote:
Originally posted by sql_lall
Basically: i agree with Evil Giraffe
1) I think we all believe infinity "exists"
actually, if we are going to disregard real and imaginary number definitions, is Sqrt(-10) really imaginary? I can read it just fine.
There is more to this than meets the eye!
When using the "=" sign it can be evaluated differently than what you normally would assume. The question you have to ask yourself is "what are the rules deciding that two numbers are equal". Mathematical rules like this are studied in subjects such as Group and Field theory and is normaly studied my MSc students.
Under the "normal" rules 1=2 or 1=4 is always false. This is so since the "counting rules" (the "identity object property" and reflexive property) imply that if a=b <=> (if and only if) a-b=0 (where a,b are real numbers), wich is not true in the case above.
However, to give an example when this can be true we turn to what is called "Modulo counting".
Assume that the "counting rules" belong to the field Z3:="integers modulo 3", then, as an example; 0=3,1=4, 5=11.
To see this you have to know that two integers are equal, modulo 3, iff the differece between them are divisible by 3. Ex:
a=b mod 3 <=> (a-b)/3 is an integer.
This "modul 3" counting rules stuff may seem like useless academia, however, the study of "counting rules" has important implications for advanced mathematics.
kimmy
Hey wow, if that's true I'm gonna start ripping people off by putting a "prices are in integer modulus 3 format" in the disclaimer.
Actually, you don't use the equal sign, nor do you say that they're equal when using modular arithmetic. When referring to numbers in this sense, you say that they're congruent, not equal. The symbol for this is the one similar to equal, but in that it has 3 horizontal lines, not 2.Quote:
Originally posted by kas
Assume that the "counting rules" belong to the field Z3:="integers modulo 3", then, as an example; 0=3,1=4, 5=11.
To see this you have to know that two integers are equal, modulo 3, ...
If I use -=- as the congruent sign (work with me :P), then your example would be:
0 -=- 3, 1-=- 4, 5 -=- 11 (modulo 3)
However, it can sometimes be thought of as being equal, since it's a very close definition. Modular arithmetic is definitely a very powerful tool.
Destined
It is true that when introducing modular arithmetics you use the "three line" equal sign and call it congruent.
However you can redefine the properties governing how the "=" can be used (this was ultimatly my point). If these properties follow certain rules you end up with a new type of arithmetics. (These rules are called the Groups, Rings or Fields properties, are a part of the study of Abstract Algebra.) "Normal" arithmetics or modular arithmetics are jus special cases. Another, common, arithmetic is matrix arithmetics where it is generally not true that AB=BA.
It is perfectly valid, in maths, to use the ordinary relation and operator signs even when using other (such as modular)than "normal" arithmetics . However, for the sake of clarity, you should clearly state wich space your numbers belong to (e.g. the "matrix" space).
kimmy
ok i think you can NOT have 1 = -1 or 1 = 2 becuase he said for just 1^2 = -1^2 that is not the same thing as 1 = -1 that is something totaly different.