How can 1/3+1/3+1/3=1 but .3333…+.3333…+.3333… never truly equal 1? I believe a recent thread tried to explain this, but truthfully I didn’t understand the answer. Can someone explain this in plain English. Pretend I’m your six year old.
“Why Daddy, Why?” 
because 1/3 as a decimal is a recurring decimal.
Now eat your greens.
A six year old ain’t gonna get it.
Try this explanation: The question is why .99999… is the same as 1. Not “close to” 1, but exactly the same as 1.
OK, if two numbers are different, like 6 and 7, you can find numbers in between them, like 6.5. So, if a and b are different positive numbers, then you can always find (a + b)/2 to be halfway between them.
OK, so take .999999… and 1, and try to find a number halfway between (or anywhere in between, for that matter.) There isn’t. You can’t. If there’s nothing separating .9999… from 1, then they must be the same number, but written in different format (just as 1/2 and .5 are the same number written in different format.)
QED.
Th
1/3 is a rational number but .33333333 is irrational…they’re not the same numbers! No matter how you put it.
jeel, 0.3333333… is rational. Infinite periodic decimals are rational. Irrational numbers have infinite aperiodic decimal expansions.
It’s probably easier to explain to a six-year-old than to an adult, because a six-year-old is more prone to accept truth on faith, such as the truth found in the Transitive Law of Implication: [[[S => T ] ^ [T => U ]] => [S => U ]].
Just tell your six-year-old, “If this thing is identical to that thing, and that thing is identical to the other thing, then this thing is identical to the other thing. If 1/3 + 1/3 + 1/3 is equal to 1 and 1 is equal to 0.99999…, then 1/3 + 1/3 + 1/3 is equal to 0.99999…”
Get it?
Libertarian: While the transitive law of implication is very true, I think what you’re more specifically looking for is the transitive property of equality (a = b and b = c implies a = c). No big deal, but just enough…
C K Dexter Haven: There’s actually a whole class of numbers which has members between .9999… and 1; namely, the infinitesimals. The reason you don’t find them is that they don’t have decimal representations (I think).
Consider this: .3333… is equal to the sum over i from 1 to infinity of 3/(10^i). If you use the formula for the sum of a geometric series, you get that this is equal to 1/3, so by the transitive property of equality, .3333… = 1/3. It turns out that by using the series representations of .3333… and .9999…, you can show that 3 * (.3333…) = .9999…; it follows that .9999… = 3 * (1/3) = 1.
Well, this is the way I saw .999… = 1 proved:
Say n = .999…
Then, 10*n = 9.999…
so, (10*n - n) = (9.999… - .999…)
or,
9*n = 9.
Ergo, n = 1
This is a matter of definitions. What does
.99999… mean? It is intended to describe the
point on the real line that has the following location:
If you chop the interval [0,1] up into 10 equal length
segments [0,1/10], [1/10, 2/10], … , [9/10, 10/10],
and call them the 0th segment, the 1st segment, …,
the 9th segment, then it is in the 9th segment,
so the first digit is a 9, .9???.. To get the second
digit, chop up the 9th segment, [9/10, 10/10] into
10 equal length segments. It will again be in the
9th segment (remember we call the first one the 0th
segment, …, and the tenth one the 9th segment).
Thus the next digit is also a 9. So .99???..
And so on, so you get it has base 10 expansion .99999…
But wait you say don’t the intervals intersect? For
instance don’t the 0th and 1st intervals above share the
number 1/10? That is precisely the problem motivating the
thread. For if we had chopped up the interval [0,10] into
ten equal length segments, the 0th and 1st would be [0,1]
and [1,2] respectively. The number 1 is in both of these
so the first digit can be 0 or 1. Then chopping up [0,1]
or [1,2], you get either 0.99999… as above or
you get 1.??.. (since [1,2] was labelled the 1st segment
when we chopped up [0,10]). The chopping up [1,2] into
[1+0/10, 1+1/10], [1+1/10, 1+2/10], … , [1+9/10, 1+10/10],
we see 1 is in the 0th of these, [1+0/10, …, 1+1/10].
Then as we keep chopping up subsequent intervals 1 is always
in the 0th one so we get expansion 1.00000…
1/3 is different from .3333 etc. In real world terms all measurements will expand into infinity as your capacity to accurately measure anything increases. So to say 1/3 + 1/3 + 1/3 of any “real” thing = 1 is a only a concept.
Think about it. If you cut a pie into thirds they are not “perfect” 1/3’rds in absolute real world terms thay are only as equal as you can reasonably make them. In this sense 1/3 is an “irrational” mathematical concept vs the “rational” or real .3333 etc measurement. In real world terms it is impossible to get an absolutely perfect 1/3 of anything. 1/3 and .3333 are apples and oranges.
Nobody is talking about cutting up pies. We’re talking about numbers - This is mathematics, not engineering. In the abstract. .333…(repeating to infinity) is equal to 1/3, and .999… is equal to one. The misunderstanding comes from the facts that our brains aren’t equipped to deal with infinities, and that positional number systems aren’t equipped to deal with fractional numbers whose denominators are relatively prime to the base in question. But the fact that it can’t be represented as a terminal fraction doesn’t mean that it is a different number.
And curiousgeorgeordeadcat:
Sorry to nitpick, but there’s no such thing as a 0th segment. You have a first segment at index zero.
Yeah I did screw up that explanation. Sorry.
That’s why. You’ll understand when you’re older.
<< There’s actually a whole class of numbers which has members between .9999… and 1; namely, the infinitesimals. The reason you don’t find them is that they don’t have decimal representations (I think). >>
Sorry, Ultra, this is dead wrong. “Infinitesimals” aint real numbers. Every real number has a decimal representation as an infinite decimal; if the decimal is repeating (periodic), then the number is rational; if the decimal is not ever repeating, then the number is irrational. The union of the set of rationals and irrationals is the real number line. Period. And on the real number line, there ain’t nuthin’ between 0.9999… and 1.
There’s been a lot of interesting new stuff since I left math with my Ph. D., but that ain’t changed.
help me out here,
I dont undrestand why 1/3 is different than 0.333…, the why I recall it being taught in public schools 25 years ago was that 1/3 represented the number one devided by the number three. So 1/3 is equil to 0.333… The reason 1/3 + 1/3 + 1/3 = 3/3 = 1, but 0.333… + 0.333… + 0.333… = 0.999… is because the fractons represents the infanite repeating number that the “…” represents in the decimal.
OK, I stand corrected. Guess I need to go back and brush up on my non-standard analysis.
However, I came up with the same explanation you used and offered it to one of my professors, and he said something about how I might get into trouble assuming that every real number has a decimal representation. Any idea what he might’ve meant, or did I just misunderstand him?
Sorry for the hijack, everybody else.
The answer to the OP is of course, 1.000… = 0.999…
I can suggest a couple of things he might have meant. The first thing that he could have meant that you get into trouble assuming that every real number has a unique decimal representation. Since 1.000… = 0.999…, We have two decimal representations for the number 1. In fact, every number expressible in the form p/q where p and q are integers and q is of the form 2[sup]n[/sup]*5[sup]m[/sup] will have two decimal representations; one ending (or is it unending?) with a string of zeros and one ending with a string of nines.
Or possibly, if you are including infinitesimals in your numbers, you cannot assign a decimal representation to every number. An infinitesimal does not have a decimal representation.
Your UserID reminded me of an article I read on non-standard analysis that constructed pseudo-reals using functions from integers into the reals. Then an ultrafilter is built from a filter on the integers. The filter used as the basis for the ultrafilter is the set of all co-finite sets of integers. (A set is co-finite if its compliment is finite.) Two functions represent the equal (less than, etc) pseudo-reals if the two functions are equal (less than, etc) on a member of the ultrafilter. The pseudo-reals include the reals as a subset, but also include infinitesimals and infinities.
John Conway also built a system of Numbers that include the reals together with infinitesimals and infinities that is not a set, but rather a proper class.
Virtually yours,
DrMatrix - Zorn is pro-Choice.
ultrafilter:
Whether every real number has a decimal expansion depends
on your definition of real numbers and your definition of
decimal expansion. Using the standard definition of real
numbers and of decimal expansion, you can prove that each
real number has at least one decimal expansion. Maybe
your prof. was trying to get you thinking about such issues.
Nonstandard analysis does not use the standard real number
system. So your prof. may have been pointing out that it
depends on how you define the real numbers.
Right answers, to the wrong question. What furryman asked in the OP, and ?authority reiterated later, was why 1/3 was not equal to 0.33333… The answer to that, of course, is that’s it’s the wrong premise: One might as well ask why the sky is green.
Chronos, I still don’t get it. What do you mean 1/3 is not equal to 0.333…? Was I wrong in my recollection that the fractional expression of 1/3 represents the number one devided by the number three? Had the fraction been 3/5 the answer would have been 0.6 without debate, so it follows that 3/5 is the fractional representation of the decimal 0.6. Why is it any different that 1/3 is the fracional representation of the decimal 0.333…? Going back to the OP the reason we don’t recognise 0.999… as being equal to 1 is because 1 is the reduced fraction of 3/3 and 0.999… is the sum of 0.333… + 0.333… +0.333… in decimal form. In other words one third (be it 1/3 or 0.333…) three times is one. The other side of this (the way I remember it), is that decimals are no more than fractions whos denominator is one. So 0.333… is equal to 0.333…/1.0, to bring this to a fraction with a whole number in the numerator (rational fraction I think, but it has been years) you must devide/multiply it by 0.333… over 0.333… (real value of 1) witch leaves you with 0.333…/0.999… witch reduses to 1/3. So I think that means 1/3 is equal to 0.333… But like I said earlier it has been years and I was no math wiz.