Your question: "Why is one third plus one third plus one third not equal to 0.333.... plus 0.333... plus 0.333...??? HELP.?"
Errata. My first answer did not even address your question.
I think this one does.
________________________________
You want to know why 1/3 is not equal to (0.333...) + (0.333...) + (0.333...) + . . .
The first two terms add to 0.666...
That plus the third term adds to 0.999...
That plus a fourth term adds to 1.333...
If you set n = 0.333... and multiply by increasingly larger positive integers, the product gets larger and larger, or as is said in calculus, increases without bound. That would be the same as adding the term repeatedly.
1/3 however is just 1/3 so long as you take it to be a representation of a number, not an indication to begin dividing one by three. If it were the latter you would get the result described below.
______________________________
First I wrote the following, which in hindsight demonstrates how my mind got so caught up in automated behavior without resort to objective analysis that the first principle of rational inquiry is completely missing (i.e., there is a real world, it is not necessarily the same as the perception of reality that the mind creates).
One third is this: 1/3
But the quotient 0.333... is a number found by dividing one by three and approaches 1/3 but never really gets there. It is an approximation to as many decimal places of accuracy as you care to calculate.
Another way of thinking about it is that 1/3 is an exact number.
But 0.333... represents the continuation of this process:
3x10^-1 + 3x10^-2 + 3x10^-3 . . .
In calculus it is postulated that the limit of this process of adding the terms in question is equal to 1/3, an entirely different meaning than saying that the two things are numerically equal.
For practical applications to measurement of physical phenomena, a choice is made as to what will be the most appropriate degree of accuracy needed. But for purely numeric considerations, the two things in question are not equal.
______________________________
It is not a hopeful sign to me that others responded in like manner:
Lee wrote:
"Because changing fractions to decimals is never equal for repeating decimals unless you carry the decimal point to infinity
1/3 + 1/3 + 1/3 = 3/3 = 1
1/3 = .3333333333333333333333333333333 + .3333333333333333333333333333333+.333333...
Get the idea? 2/4 = . 50 exactly because it is not repeating"
Starybaby 1981 wrote:
" The decimal form of 1/3 is .333333333 but the 3's go on forever. When you punch it into a calculator, you can't make the 3's go on forever, so the calculator says the sum is 0.9999999 instead of 1."
scott1848 wrote:
"it is equal if the infinitely repeating decimals are used
0.999... approaches one as the number of digits approaches infinity
let x = 0.999... ___ 10x = 9.999... ___ 10x - x = 9.999... - 0.999... ___ 9x = 9 ___ x = 1"
All of these answers including my own were attempts to rationalize, i.e., to supposedly explain an objective phenomenon (your question in this case) via a preconceived idea without regard to the actual details of the phenomenon itself.
That is one of the many common errors, I think, among human beings that makes us so prone to stupidity. I do not think that other creatures have quite enough brain power to be as stupid in such ways. Interestingly, the answers including mine all are working from culturally acquired cognitive and perceptual schemas with regard to the appropriateness of those schemas.
The same erroneous use of one's cognitive and perceptual capacity is a source of a very common source of unnecessary suffering among human beings.
A history of human thought shows that although people may believe they have acquired insight into the nature of reality and the nature of sentient experience, often what they report to be reality is a reflection of their conformity to unwarranted opinion or randomly constructed delusional schemas without any objective reality checks.
It is our natural inclination to acquire behaviors including cognitive and perceptual schemas by doing what we observe others doing, even from what we can construct mentally as models of what others might do. There is a neural basis for this, but even though such behavior is observable, until quite recently it was not known why such behavior occurs.
Neuropsychology | Mirror Neuron System:
• http://www.pbs.org/wgbh/nova/body/mirror-neurons.html
Social Psychology | Conformity:
• http://en.wikipedia.org/wiki/Asch_conformity_experiments
• http://en.wikipedia.org/wiki/Milgram_experiment
Cognitive Psychology | Schemas:
• http://en.wikipedia.org/wiki/List_of_cognitive_biases
• http://en.wikipedia.org/wiki/List_of_fallacies
Cognitive Behavioral Psychology | Observational Learning:
• http://en.wikipedia.org/wiki/Observational_learning
It is not lost on me that the same error underlies religion, a common source of unnecessary suffering.
See:
"God made the integers, all else is the work of man." • http://en.wikipedia.org/wiki/Leopold_Kronecker
and
"Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light." • http://en.wikipedia.org/wiki/Isaac_Newton
We do not necessarily perceive the real world as it is but rather as our brains' best guess and any particular time depending on sensory data and existing mental content. Often the guess is dreadfully wrong:
"The Fifty Years War with Israel and the Arabs" • http://www.youtube.com/watch?v=fSAD9pS8NIw
It is now a much longer war since that documentary was made.
Similar stupidity based on preconceived, erroneous perceptions of the nature of reality have driven mass human behavior all through history and all over the world.
_______________________________________________
Educate yourself. You will be pleased with the results.
_______________________________________________
• Where we live:
Christopherson, Robert. “Geosystems: An Introduction to Physical Geography” (Prentice Hall, several editions available)
• Where we came from:
Jurmain, Kilgore, Trevathan, Ciochon. “Introduction to Physical Anthropology” (Cengage Learning, several editions available)
• What we have been doing:
Roberts, J.M.; Westad, O.A. “History of the World” (Oxford University Press: 2013)
Hourani, Albert “A History of the Arab Peoples” (Actually the history of Islam, per the author.)
(Harvard University Press: 2002)
• Philosophy of Science:
Sagan, Carl “The Demon-Haunted World: Science as a Candle in the Dark” (Ballantine Books: 1997) You may be able to locate it posted online.