- #211
terrabyte
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- 0
then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.
hence irrational
hence irrational
then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.
Hurkyl said:No... one of those is a number; the other two are gibberish.
Anyways, you've evaded my response. None of the approximations I intend to make have an infinite number of nonzero terms, however if you pick any positive value as an allowable tolerance for error, one of my approximations will be within this tolerance.
It seems you are trying to ask me to tell you what the "next" decimal number is, but there is no such thing. (If there was a "next" number, then what happens if I take their midpoint?)
And I'll ask again, what does all of this have to do with whether 1/3 = 0.333...?
the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
that distinction is obliterated by using an infinite string of digits to express a quantity.
Still don't understand the definition of infinity, very sad.terrabyte said:.333...2 and .333... and .333...4 are all the same number.
there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"
that's the purpose of the exercise
there is no longer any room for incrementation.
the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
that distinction is obliterated by using an infinite string of digits to express a quantity.
terrabyte said:how old are you to make comments that have no basis on what's being debated?
where is your facts to back up this "argument" that i don't understand anything?
real numbers DO have incrementation
no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.
terrabyte said:don't come back and say it ISN'T, hello3759 (i know that's going to be your next pithy argument) because it most certainly IS. if it isn't define me a number greater than it.
there isn't one is there?
terrabyte said:greater than infinity
Could you explain this?when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.
It has as many digits as there are natural numbers.terrabyte said:fine, you stick to that then it's absolutely impossible for you to utilize it as a number.
those are your guidelines.
now express the quantity of how many digits .333... has with values of 3?
ex-xian said:It has as many digits as there are natural numbers.
terrabyte said:fine, you stick to that then it's absolutely impossible for you to utilize it as a number.
those are your guidelines.
now express the quantity of how many digits .333... has with values of 3?
Countably infinitly many...this is going in a cirlce, and I don't see your point.terrabyte said:fine, how many natural numbers are there then?
you're just digging a deeper hole :D
how many natural numbers are there then?
ex-xian said:terrabyte, how much formal mathematics have you actually had? You seem to be a bright, but ignorant, person.
Could you explain this?
Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < 0.00000000000000003, right?
In the case of 0.333... and 0.5454... how is not clear that the latter is greater than the former?
If this isn't what you meant, could you elaborate?
Also, is 1/2 = .5 = .5000...?
How do you technically define rational and irrational numbers?
thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
ex-xian said:Countably infinitly many...this is going in a cirlce, and I don't see your point.
terrabyte said:rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.
thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
as it stands there IS NO rational decimal form for .333...
and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--
What do think a ratio is? 1/2 means 1 divided by 2 and 1/3 means 1 divided by 3. Why does the operation of division bother you? Why does division produce something "bad" when you have 1/3 and something "not bad" when you have 1/2? You seem to have a problem with infinity. Why? If you think there's something wrong with infinitly many digits, then the burden is upon you to demonstrate why.terrabyte said:rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.
Can you give a formal proof?thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.
as it stands there IS NO rational decimal form for .333...
Why not? What is special about the digit 0 there isn't there for 3?and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--