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Pjpic
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Why is 1/3 considered a number line point when the decimal expansion never seems to stop at a point (if this is indeed the case)?
The sequence of numbers {.3, .33, .333, ...} converges to 1/3. Each number in this sequence represents a point on the number line.Pjpic said:Why is 1/3 considered a number line point when the decimal expansion never seems to stop at a point (if this is indeed the case)?
The real number line is considered to also contain a point at the limit to which the sequence converges. This is one of the ways in which the real number line can be defined -- as the set of all limit points to "convergent" sequences. To be more technically correct it can be defined as a set of equivalence classes of Cauchy sequences of rational numbers.Mark44 said:The sequence of numbers {.3, .33, .333, ...} converges to 1/3. Each number in this sequence represents a point on the number line.
jbriggs444 said:The real number line is considered to also contain a point at the limit to which the sequence converges.
A sequence is not considered to be the same as its limit. What we are saying is that the limit exists.Pjpic said:I wonder why a sequence is considered to be the same as its limit.
I don't know why you would say that. For the number in this thread, 1/3, the sequence is {.3, .33, .333, ...}. 1/3 is not an element of this sequence, but the farther you go in the sequence, the closer to 1/3 that element is.Pjpic said:I wonder why a sequence is considered to be the same as its limit.
Because the number line models a continuum of points.Pjpic said:Why is 1/3 considered a number line point when the decimal expansion never seems to stop at a point (if this is indeed the case)?
jbriggs444 said:The real number line is considered to also contain a point at the limit to which the sequence converges. This is one of the ways in which the real number line can be defined -- as the set of all limit points to "convergent" sequences. To be more technically correct it can be defined as a set of equivalence classes of Cauchy sequences of rational numbers.
A Cauchy sequence is one where all of the terms eventually get arbitrarily close to each other. Two Cauchy sequences are considered "equivalent" if both of their terms get arbitrarily close to each other. So you can interleave their terms and still have a Cauchy sequence.
You learned intermediate algebra and the Pjpic did not.Krylov said:To be honest, I don't really understand the difficulty that the OP seems to encounter? I agree with what was written above, but ##\frac{1}{3}## is just an innocent rational number. Yes, it happens to have an infinite decimal expansion, but that's just an artifact of our common choice of base.
symbolipoint said:You learned intermediate algebra and the Pjpic did not.
Because in order to understand the definition of ##0.33333\ldots## one needs to understand convergence of a geometric series? In my opinion, this does not immediately quality it as a "much more difficult problem", but it does probably explain why here it would appear in a calculus course, rather than in "intermediate algebra".micromass said:I wouldn't be so harsh, since this is really a much more difficult problem than at first sight.
micromass said:I wouldn't be so harsh, since this is really a much more difficult problem than at first sight. Sure, people learn in intermediate algebra that ##1/3 = 0.33333...##, but they never learn why.
The reason 1/3 is considered a number line point is because it is a finite decimal. This means it has a finite number of digits after the decimal point. In contrast, the decimal representation of .333... is infinite, meaning it goes on forever without repeating.
On a number line, 1/3 is represented as a point on the line between the whole numbers 0 and 1. This point is located exactly one-third of the way between 0 and 1.
.333... cannot be represented on a number line because it is an infinite decimal. This means that it cannot be pinpointed to a specific point on the number line, as it goes on forever without repeating.
The concept of rounding affects the representation of numbers on a number line by allowing us to approximate the location of a number. For example, if we round .333... to the nearest hundredth, we get .33, which can be represented as a point on the number line between 0.3 and 0.4.
No, there is not a way to accurately represent .333... on a number line. As mentioned before, it is an infinite decimal and cannot be pinpointed to a specific location on the number line. However, we can use rounding or other mathematical techniques to approximate its location.