This post is regarding the article concerning the following paradox:
If
1/3 = 0.333…
and
2/3 = 0.666…
then
3/3 = 0.999…
but
3/3 is equal to 1
so
1 = 0.999…
How can that be?
OK, that’s the problem, here’s my explanation. This is a problem of nomenclature. Actually 1/3 is NOT = 0.333…, but 1/3 is APPROXIMATELY = 0.333…
So, if we substitute APPROXIMATELY = for the equal sign in the original problem, then everything will make sense.
Previous threads on the subject:
June 2003
April 2003
October 2001
August 2001
May 2001
Just don’t ask us about the third word that ends in -gry or whether a duck’s quack will echo.
By the way, I don’t think that Cecil wrote a column about this question, so it doesn’t really belong in this forum. GQ is where all those other threads were found.
When I first heard the -gry riddle, I wasn’t given the whole what’s the thrid word in the english language part. Just asked what the third word that ends in -gry was. I spent four years until a friend of mine looked up the riddle and found the proper way to tell it.
You just ain’t got the word yet: An infinite question: Why doesn’t .999~ = 1? is today’s column. That should put to rest those other threads but ccwaterback seems to disagree.
However, I just want to repeat: “there’s just no substitute for the pleasures of an infinite series.” 
x = .999~
10x = 9.999~
subtract both sides
10x - x = 9.999~ - .999~
9x = 9
x = 1
Wrong. 1/3 is EXACTLY equal to 0.333…, where 0.333… represents an infinite number of threes following the decimal point.
See the first linked thread above of June 2003 for a near infinite discussion of the topic.
There’s no paradox to it. The reason thirds have no exact decimal equivalents it that three is not a factor of 10 (the number base we usually use). Any denominator whose factors are (like those of 10) 2 and/or 5, will work.
To illustrate this, all we have to do is start working in base 3. Voila, 1/3 is precisely 0.1 base 3. 2/3 is 0.2 base 3. But 1/2 works out to 0.111~ base 3.
Likewise 2/7 (which is 0.285714~ base 10) is a contented 0.2 in base 7. Und so weiter.
[figured this out a few years back without any help from Cecil; will also show on request how Zeno cheated in his paradoxes]
Again, it’s all nomenclature. The series .3 + .03 + .003 + … CONVERGES to 1/3 but does NOT EQUAL 1/3. The problem is in the definition of the = sign. In the nomenclature of infinite series they use the = sign as a symbol to represent the limit of a convergent series, NOT the exact number the series represents.
ccwaterback, I notice you write “the exact number the series represents” - surely if there is any number that .333… represents, then that number is 1/3.
Or if you meant to say that such a series does not represent any real number, then, well, why not? As a representation of a number, a series is well defined and unambiguous (if a bit unwieldy) and I’d say that’s good enough.
So 1/3 does not actually EQUAL anything, correct?
When I was at the U, one of our Engineering instructors had this joke… I think this was professor Gold or perhaps Choma, it has been a long time, almost 22 and 3/3 years.
One year there were too many applicants for the Math program. So the Dean of Engineering and the Dean of Mathematics got together and devised this method to determine who became mathematicians and who would become engineers.
They had all the male candidates line up on one side of the auditorium. They had the school cheerleading squad line up on the otherside.
The two deans then announced that they would blow a whistle every minute. At the sound of the whistle, each male student was to cross half the distance of the auditorium between them and the cheerleading squad. Immediately half of the male students left the auditorium.
The Dean of Mathematics said “those boys will become Mathematicians since they know they will never get to the other side”. The Dean of Engineering said, “and the remainder will become engineers, since they know they will get close enough”.
I think this kind of describes the mathematical question.
Jim Henderson
You’re very close, but not quite. It’s not a matter of redefining the = sign. It’s a matter of defining what the infinite series means in the first place. In the nomenclature of infinite series, a representation of an infinite series is defined to have the value of its limit.
If you understand this stuff, then you’ll know that:
SUM(2[sup]-p[/sup], p = 1 to infinity)
is by definition shorthand for:
lim(n gose to infinity) of SUM(2[sup]-p[/sup], p = 1 to n)
which has a value 1.0, of course. No need for an equal sign; the limit is implicitly there to begin with.
Hey guys, mathematical calculations are cool and all, but I think we should delve in the the heart of the matter – is Rebecca-Romijn Stamos really only wearing body paint in the new X-Men movie :eek:
The problem is not with the definition of the equals sign. It’s with the understanding of what .333… means. It means the limit of that series, which does equal 1/3.
1/3 = .333… is a common way of expressing that, and it is not a misuse of the equals sign.
Fair enough. A series either converges to a number or it is divergent. If the series converges to a number we say the series is = to that number, but in fact it only converges to the number. LOL … I give up, even I am getting confused.
So I think what we can conclude is that normal laws of arithmetic on repeating decimals is well defined ONLY if the repeating decimals are first transformed in some way to their rational counterparts or their abstract algebraic definition (infinite series). In conclusion, arithmetic on repeating decimals is UNDEFINED!!!
But, a can of Budweiser is a well defined concept I am about to prove to myself
“Only converges”?
Perhaps you are confusing the sequence of partial sums with the series itself? Series link. A series either has one value, or none at all. .333… represents a series, is convergent, and is well-defined.
I mean, it has a well-defined value.
Note in the site you link to that … “the number s is called the sum of the series” … and NOT … the series is equal to the number s. That is, the number s is the number that satisfies the rigorous definition of this particular convergent series. So the = sign in the case of convergent series is actually defined by the epsilon argument for its partial sums. I know it’s a subtle difference, but it’s all in the nomenclature.
The version of this story that I heard was an illustration of the meaning of “for all practical purposes.” The two sides would theoretically never reach each other, but they would soon be close enough for all practical purposes.