When you say that it is normal practice to use the expression 0.3 to represent 1/3, what exactly do you mean by represent? To everyone else responding to your original post, the “expression” 0.3 means exactly the result of dividing 1 by 3, and it “represents” an infinite string of 3’s after the decimal point. If you are proposing an alternate definition of the notation, please share it with us.
In the same post quoted above you also define a function, f(n) = [sum i = 1 to n](3 x 10^(-i)) and speak of the limit of this as n tends to infinity. Please share with us what you think this limit is equal to.
Sorry, 0.3 does mean 0. followed by an infinite number of zeros to everyone but you, and it is not nonsense, it is 1/3.
It’s an exactly specified number (except to some people who don’t agree…)
It doesn’t have to “stop.” It’s an infinite number of 3s. This thread has dealt with the problem of thinking of it as an algorithm.
10 add another 3
20 goto 10
But that isn’t how it works – and, worse, even if that was how it works, it goes to 1/3 anyway, because it can’t go anywhere else! It gets arbitrarily close to 1/3.
Another challenge: how would you express 1/3 in decimal notation? If 0.333~ isn’t it, what is? Are you really ready to declare that the number 1/3 does not have a decimal expansion?
(Repeating decimals are fun. As a specific case, easily generalizable, abcdef/999999 = .abcdefabcdefabcdef… This leads to a rather charming – and somewhat sexist – number puzzle. Each letter stands for a decimal digit. Solve…
I think there is an easier way to look at all this. 0.333333… is a sequence of digits; it is a sequence that extends infinitely in one direction.
The real numbers can be defined as the set of such sequences. Without going into details (sign and decimal point), there will be, with one class of exceptions, a one-to-one correspondence between the set of reals and the set of such infinite sequences.
The exceptions are sequences that end in 000… or 999… There are two sequences which map to reals of that form. There are many such numbers, e.g. 1/2 = .500… = .499999… but 1/3 = .33333… is not one of the exceptions. (This leads to a simple proof that 1 = .999…: 3/3 = 3 * .33333… = .99999…)
I tried to point out that the issue is “=”, but no one pays attention.
As I see it, engjs is essentially saying that, if you cannot carry out the decimal representation of 0.333… to the point that it exactly has the same value as 1/3, the two are not equal, and since you can never write enough threes to get there, there’s always some infinitesimally small difference, so the two are not equal.
Everyone else is saying that, if you cannot identify the difference between the two, then they are equal. Since you are using an infinitely long string of 3s, there’s no point where you can identify the difference between 0.333… and 1/3. Or, if you will, that 1/3 - 0.333… is not equal to some value distinct from zero. Thus, they are equal.
So long as the two “sides” to the argument cannot agree upon the definition of “=”, there’s really no point to the discussion, is there?
The more difficult aspect in my mind was always accepting that “close enough” was the same as “equals.” But that was back in my youth, when I didn’t stop to wonder about the fact that negative integers didn’t really “exist”, but were merely constructs of the human brain. Liberating mathematics from “reality” tends to make dealing with math much easier.
As long as you don’t write enough 3s (only a finite number of them) the difference is not even infinitesimally small (whatever infinitesimal means in this context).
But we can write enough 3s:
t = 0: write 0.3
t = 0.5: append another 3 (0.33)
t = 0.75: append another 3 (0.333)
…
at t = 1s we have written 0.333…
(taking into account that we have liberated mathematics from “reality”)
I understand this point. But I think what engjs is trying to say (I’ll let him actually SAY it, if he’ll ever respond to my questions) is that, to quantify that difference requires that you identify exactly where the two are not matched up. Can’t do that. But conceptually, you know it exists, because you can’t match them up. Hence there is a difference, even if it cannot be quantified (thus, is infinitesimally small).
I think that asserting that a point exists on the number line, which resides between 0.333… and 1/3, without being able to identify it is unhelpful at best. I conceive that would be the inherent result of there being an infinitesimally small difference between the two. Unless it’s intended to mean something smaller in dimension than even a single number…
