21 Jul 18
2 edits
Originally posted by @kazetnagorraI don't see your intended meaning;
I wouldn't say that an "infinitely large circle" is the same as an "infinitely large square."
For example, consider a square with sides 2R and a circle with radius R. The ratio of the circumference of both shapes is 4 over pi. Now take the limit R -> \infty. The ratio of the circumference remains the same, but the ratio of circumference of the circle with respect to the circle is 1. So in this limiting procedure, the two aren't the same.
When you say "ratio of the circumference" in this context, you mean the ratio between which two numbers?
21 Jul 18
Originally posted by @humyThe ratio between the circumference of the square and the circumference of the circle.
I don't see your intended meaning;
When you say "ratio of the circumference" in this context, you mean the ratio between which two numbers?
21 Jul 18
7 edits
Originally posted by @kazetnagorraOK. But you said
The ratio between the circumference of the square and the circumference of the circle.
"Now take the limit R -> \infty. The ratio of the circumference remains the same, "
but how can that be true if both circumferences become +infinity? ("+infinity" means infinity in the positive direction)
I mean;
+infinity (the length of first circumference) = +infinity (the length of second circumference)
so now there is nothing to distinguish between the lengths of the two circumferences because they are now both become the same length (of +infinity), right?
And I am also unsure if (+infinity)/(+infinity) can be still meaningfully called a "ratio" because (+infinity)/(+infinity) can be ANY positive number!
ANYONE: can you call (+infinity)/(+infinity) a "ratio"?
21 Jul 18
Originally posted by @humyWelcome to limits!
OK. But you said
"Now take the limit R -> \infty. The ratio of the circumference remains the same, "
but how can that be true if both circumferences become +infinity? ("+infinity" means infinity in the positive direction)
I mean;
+infinity (the length of first circumference) = +infinity (the length of second circumference)
so now there is nothing to di ...[text shortened]... (+infinity) can be ANY positive number!
ANYONE: can you call (+infinity)/(+infinity) a "ratio"?
The following limits are all "infinity/infinity":
1. lim x-> \infty x/x
2. lim x-> \infty x²/x
3. lim x-> \infty x/x²
(the answers are 1, infinity and 0 respectively)
21 Jul 18
2 edits
Originally posted by @kazetnagorraSo the two infinitely long circumferences (one of the infinitely large circle and the other of the infinitely large square) are of the same length, right?
Welcome to limits!
The following limits are all "infinity/infinity":
1. lim x-> \infty x/x
2. lim x-> \infty x²/x
3. lim x-> \infty x/x²
(the answers are 1, infinity and 0 respectively)
If finite in size, the circumference of one to the other is of the same ratio so, to calculate what happens as their radius r tends to infinity;
lim r-> \infty c*r/r
where c is some constant which has no effect on this limit and thus the above infinite limit equals 1 and the two circumferences tend to the same +infinity thus they are the same at infinitely large, right?
21 Jul 18
Originally posted by @humyThey are both infinitely long. The ratio of their respective circumferences, in the limiting procedure I outlined, is 4/pi.
so the two infinitely long circumferences (one of the infinitely large circle and the other of the infinitely large square) are of the same length, right?
21 Jul 18
9 edits
Originally posted by @kazetnagorraWhere did you get that "4/pi" figure from?
They are both infinitely long. The ratio of their respective circumferences, in the limiting procedure I outlined, is 4/pi.
(+infinity)/(+infinity) = any_positive_number
and so I don't see how it could be a meaningful ratio.
The limit of c*r/r as r tends to infinity is always
(+infinity)/(+infinity) = any_positive_number
But I suppose, if you are to represent it by any particular real number and you insist it is still a ratio, you can call it "1" just to stick to the convention of saying that if the two numbers are finite and equal then the ratio of one to the other is 1 so say that is the same if both the same infinity. But I have a suspicion that isn't sound maths no matter how you look at it. My suspicion is that it cannot meaningfully be defined as a ratio.
21 Jul 18
Originally posted by @humyJust write down the ratio of the circumferences, and compute the limit.
Where did you get that "4/pi" figure from?
(+infinity)/(+infinity) = any_positive_number
and so I don't see how it could be a meaningful ratio.
The limit of c*r/r as r tends to infinity is always
(+infinity)/(+infinity) = any_positive_number
But I suppose, if you are to represent it by any particular real number and you insist it is still a rati ...[text shortened]... no matter how you look at it. My suspicion is that it cannot meaningfully be defined as a ratio.
lim R-> \infty 8R/(2*pi*R) = 4/pi.
Originally posted by @kazetnagorraThat isn't 'the' limit because it is only necessarily 4/pi while it is finite.
Just write down the ratio of the circumferences, and compute the limit.
lim R-> \infty 8R/(2*pi*R) = 4/pi.
At infinity it can be any positive real (or +infinity) number you like (which makes me also think it isn't even meaningful as a ratio) because
(+infinity)/(+infinity) = any_positive_number.
21 Jul 18
3 edits
Originally posted by @kazetnagorradoesn't that confirm that
1. lim x-> \infty x/x
2. lim x-> \infty x²/x
3. lim x-> \infty x/x²
(the answers are 1, infinity and 0 respectively)
(+infinity)/(+infinity) = any_positive_number (or +infinity)
?
21 Jul 18
5 edits
ANYONE;
Is it meaningful to call infinity/infinity a "ratio"? Or is that nonsense?
Note infinity/infinity can equal any number!
So perhaps infinity/infinity itself is nonsense? because, for example, can have;
infinity/infinity = 1
and
infinity/infinity = 2
which seems to imply the nonsense of 1 = 2.
21 Jul 18
Originally posted by @humyNot quite, all shapes all of whose points are on the points at infinity of the plane formed by the product of the extended real line with itself are circles because they fulfil the defining property of a circle. However, it doesn't necessarily follow that a circle is also a square. A square is defined by some property and a set of points either has it or does not. So you'd have to show that an infinite circle has the defining property of being a square.
OK. Thanks for all that.
From our conversations I think I can now conclude that;
A said infinitely large circle or a said infinitely large sphere or any other said infinitely large 'shape' of a kind that has a said center point and has all of its point on its 'circumference' or 'surface' defined as being an infinite distance from this center point;
(1) ha ...[text shortened]... pe" at all!?
Couldn't you just as legitimately call it an infinitely large shapeless construct?
Let P(S) mean that the set of points S has property P, similarly for Q(S).
Let C and D be sets of points. That P(C) & P(D) is true does not entail that Q(D) follows from Q(C).
21 Jul 18
Originally posted by @kazetnagorraSince we are using the extended real number line (there being no other way to produce shapes of infinite size) it's not clear that this limiting procedure will produce the correct answer.
Just write down the ratio of the circumferences, and compute the limit.
lim R-> \infty 8R/(2*pi*R) = 4/pi.
An infinite square is a circle because all its points are points at infinity and so the same distance from its centre.
Whether an infinite circle is a square depends on whether one can show that an infinite circle has the property that defines squares. Taking limits won't help here, since we never leave the normal real number line during the limiting procedure.