The Foundations of a Non-Naive Mathematics

[terrabyte]

then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.

hence irrational

 

[Hurkyl]

then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.

You said that 1.1 is slightly different than 1.2.

0.333 is quite a bit closer to 0.333... than 1.1 is to 1.2.

Were you wrong earlier when you said 1.1 is slightly different than 1.2?


And I'll ask again, what does this have to do with 0.333... = 1/3?

 

[zeronem]

With your logic terrabyte, you would equally assert that 1/7 = .142857... is a irrational number. Once again this proves your mis-understanding in Math.

You fail to realize that some rational numbers particularly 1/3 and 1/7 in there decimal notation run through a pattern of integers continuously. However irrational numbers have no significant pattern in there decimal notation.

1/3 = .333333333333333333333333333333333333333333333333333333333333333333

1/7 = .142857142857142857142857142857142857142857142857142857142857142857

Pi = 3.141592654,

There is no pattern in Pi, therefore it is irrational and such a number cannot be described as a ratio of two integers. We have only found fractions that closely resemble Pi that our rational. like 22/7 = 3 + 1/7

 

[terrabyte]

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.

case in point our little exercise to determine the closest numbers to .333... the exercize would have worked for any number, i was just using it to illustrate that when we take digits and extend them out to an arbitrar "infinity" we lose the distinction that determines WHAT numbers actually are.

without incrementation, the ability to add MORE digits to further define value, you hit a logical dead end.

this is completely due to the structuring of infinity

you ask how this has to do with .333... and 1/3 well it goes way beyond those two numbers and their properties and into the intrinsic values and quantities of ALL numbers, and the limitations of our mathematical and numerical system.

 

[Hurkyl]

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.

Would you care to prove your point? Give me two decimals representing different quantities, and I'll find a decimal representing a number between them. (One way is I could simply add them and divide by 2)

 

[terrabyte]

that's the point.

when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.

we just did the exercise expanding .32 <.333... <.34 | .332 < .333... < .334

you ran into the same roadblock

 

[hello3719]

how old are you terrabyte ? It seems you don't understand anything.

 

[hello3719]

.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

Still don't understand the definition of infinity, very sad.

 

[terrabyte]

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?

 

[Hurkyl]

there is no longer any room for incrementation.

Which is fine, because the real numbers don't have incrementation.

You claimed:

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.

I ask you to present two decimals representing different quantities for which I cannot find a decimal between them.

 

[terrabyte]

instead of hollow "very sad" remarks how about you explain what YOU understand infinity to be and how it fits into the picture.

if you have nothing relevant to say, by all means, keep quiet

 

[hello3719]

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?

You can't even accept the definition of irrational neither infinity. Second of all mathematics isn't a question of "debate". This is not a politics forum.

 

[chroot]

We sure seem to have a lot of arrogant, ignorant kids on the forum all of a sudden. Are you all friends? Why are you here?

- Warren

 

[hello3719]

you think that .333...4 is a number where there's an infinite of 3's. Tell me how can you add a 4 if there is a NON-ENDING quantity of 3's?

 

[terrabyte]

real numbers DO have incrementation AND distinction.

every digit 0-9 that is different from one in the same location of another number DEFINES difference.

this is intrinsic AND integral to the structure and utilization of the number system.

when you put a limit on the ability of that system to function of course discrepancies will arise. which is exactly what "Infinity" is. a Limit.

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?

 

[Hurkyl]

real numbers DO have incrementation

On what grounds do you suggest that? Remember your own statement:

no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.

If there was incrementation, then what comes between a number and its increment?

 

[hello3719]

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?

Greater than .333... ? no problem
why not 0.4?

 

[terrabyte]

a smaller increment.

always

 

[terrabyte]

greater than infinity

 

[hello3719]

greater than infinity

infinity isn't a number

 

[terrabyte]

semantics

say something that has meaning, Hello3719, PLEASE

 

[hello3719]

why doesn't it has meaning ?
it's true it isn't a number unless you can prove it

 

[terrabyte]

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?

 

[sparkster]

terrabyte, how much formal mathematics have you actually had? You seem to be a bright, but ignorant, person.

when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.

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?

 

[sparkster]

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?

It has as many digits as there are natural numbers.

 

[terrabyte]

It has as many digits as there are natural numbers.

fine, how many natural numbers are there then?

you're just digging a deeper hole :D

 

[hello3719]

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?

by notation i am using " ... " to mean never ending quantity of 3's.
why would you say that "never ending" is a number ?
Keep focus.

 

[sparkster]

fine, how many natural numbers are there then?

you're just digging a deeper hole :D

Countably infinitly many...this is going in a cirlce, and I don't see your point.

 

[Hurkyl]

how many natural numbers are there then?

The natural numbers have cardinality \aleph_0 and order type \omega.

 

[terrabyte]

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?

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 <--