Zeno’s Paradox, even though it’s pretty much resolved. If you fire an arrow at an apple, before it can get all the way there, it must get halfway there. But before it can get halfway there, it’s gotta get a quarter of the way there. But before it can get a fourth of the way, it’s gotta get an eighth… etc, etc. The arrow never runs out of new subdivisions it must cross. Therefore motion is actually impossible QED lol.
Obviously motion is possible, but it’s neat to see what ways people intuitively try to counter this, because it’s not super obvious. The tortoise race one is better but seemed more tedious to try and get across.
So the resolution lies in the secret that a decreasing trend up to infinity adds up to a finite value. This is well explained by Gabriel’s horn area and volume paradox:
https://www.youtube.com/watch?v=yZOi9HH5ueU
If I remember my series analysis math classes correctly: technically, summing a decreasing trend up to infinity will give you a finite value if and only if the trend decreases faster than the function/curve x -> 1/x.
Great. Can you give me example of decreasing trend slower than that function curve?, where summation doesn’t give finite value? A simple example please, I am not math scholar.
So, for starters, any exponentiation “greater than 1” is a valid candidate, in the sense that 1/(n^2), 1/(n^3), etc will all give a finite sum over infinite values of n.
From that, inverting the exponentiation “rule” gives us the “simple” examples you are looking for: 1/√n, 1/√(√n), etc.
Knowing that √n = n^(1/2), and so that 1/√n can be written as 1/(n^(1/2)), might help make these examples more obvious.
Zeno’s Paradox, even though it’s pretty much resolved. If you fire an arrow at an apple, before it can get all the way there, it must get halfway there. But before it can get halfway there, it’s gotta get a quarter of the way there. But before it can get a fourth of the way, it’s gotta get an eighth… etc, etc. The arrow never runs out of new subdivisions it must cross. Therefore motion is actually impossible QED lol.
Obviously motion is possible, but it’s neat to see what ways people intuitively try to counter this, because it’s not super obvious. The tortoise race one is better but seemed more tedious to try and get across.
So the resolution lies in the secret that a decreasing trend up to infinity adds up to a finite value. This is well explained by Gabriel’s horn area and volume paradox: https://www.youtube.com/watch?v=yZOi9HH5ueU
If I remember my series analysis math classes correctly: technically, summing a decreasing trend up to infinity will give you a finite value if and only if the trend decreases faster than the function/curve
x -> 1/x.Great. Can you give me example of decreasing trend slower than that function curve?, where summation doesn’t give finite value? A simple example please, I am not math scholar.
So, for starters, any exponentiation “greater than 1” is a valid candidate, in the sense that 1/(n^2), 1/(n^3), etc will all give a finite sum over infinite values of n.
From that, inverting the exponentiation “rule” gives us the “simple” examples you are looking for: 1/√n, 1/√(√n), etc.
Knowing that
√n = n^(1/2), and so that 1/√n can be written as 1/(n^(1/2)), might help make these examples more obvious.Hang on, that’s not a decreasing trend. 1/√4 is not smaller, but larger than 1/4…?
From 1/√3 to 1/√4 is less of a decrease than from 1/3 to 1/4, just as from 1/3 to 1/4 is less of a decrease than from 1/(3²) to 1/(4²).
The curve here is not mapping 1/4 -> 1/√4, but rather 4 -> 1/√4 (and 3 -> 1/√3, and so on).