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Mathematical Paradox

Ever heard of this problem?

Let us a assume a frog jumps half a distance to it's destination. Therefore, its jumps would account as: 1/2 the distance + 1/4 the distance + 1/8 the distance and so forth.

Essentially, 1/2n

Would it ever reach its destination? Well I have computed that formula numerous amount of time and mathematically, it always results as impossible to reach.

Or perhaps not...

Some theoretical mathematicians have argued theoretically, after a googleplex of times, the destination would be reach as the resulting step is a positive integer. Infinitely small, yes, but nevertheless a positive integer.

Do the same laws of mathematics still apply when one reaches an enormous, unfathomable value?

Please discuss.

_________________

Sebastian

"Don't forget to floss." - Darkwing Duck

I'd be curious to see their theory or how they came to that conclusion. Logically, it's not a paradox and could only equal the final distance if the .000000000015 were rounded up to .000000000002. No matter the number you start with, and no matter how many hops, the frog will never get closer than a sliver away.

_________________

"When does the human cost become too high for the building of a better machine?"

If you use the limit test, than yes, you will not ever reach 1.

But as mentioned,

As long as the value is a

**positive integer**, XeR, every subsequent leap by the frog would be closer to 1, theoretically.

_________________

Sebastian

"Don't forget to floss." - Darkwing Duck

However at what point might the frog be splined back into tadpoles? I suppose I might never know how many cubits a frog needs to hop before it becomes a poisonous frog, ultimately it's almost like this question suggests the transitive property is all that's necessary for the existence of life.

_________________

"Standing on a well-chilled cinder, we see the fading of the suns, and try to recall the vanished brilliance of the origin of the worlds."

-Georges Lemaitre

"I fly through hyperspace, in my green computer interface"

-Gem Tos

I sneezed my Pepsi when I read this.

_________________

Sebastian

"Don't forget to floss." - Darkwing Duck

This is just a variant on the 2500 year old "Zeno's Paradox" from ancient Greece.

Forget about frogs hopping. Just use yourself walking, or driving a car. The same thing applies.

Lets say you wanna get from your sofa to the fridge: you first have to reach the point halfway between your sofa and the fridge. But before that you have to reach the point halfway between your sofa and that halfway point, but before that you hafta reach the point halfway between your sofa and that second halfway point (the quarterway point), but before that...and so on. You would have to hit every point between the sofa and the fridge. And since there are an infinite number of geometric points along the line between your sofa and the fridge (just like there are an infinite number of points on any line) it would take an infinite time to accomplish that. Which means you cant move from your sofa to the fridge. Indeed ALL motion is an illusion. There is no motion. Nor time itself. That according to the ancient Greek philosopher Zeno in 490 BCE.

If you use the limit test, than yes, you will not ever reach 1.

But as mentioned,

As long as the value is a

**positive integer**, XeR, every subsequent leap by the frog would be closer to 1, theoretically.

My original statement remains -i'd be curious to see a publication on this or something since positive integer or not, though the destination would be reached if you generalized close enough, technically, there would always remain an unmeasurable (.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000infinite000001) between the frog and destination. So i don't see how they are theorizing it ever would be mission complete.

_________________

"When does the human cost become too high for the building of a better machine?"

According to theoretical physics the smallest finite length is the Plank length which is approximately 1.616199x10^(-35) metres. That is the smallest finite distance possible, according to theoretical physics.

Let 0 be the starting line and 1 be the finish line. Let,

s_n=1+1/2+(1/2)^2+...+(1/2)^n

There exists a cardinal number n such that the frog, theoretically, has to touch the finish line. I'll work out this number.

Using the partial geometric sum formula (which is easy to prove by mathematical induction or other methods),

The cardinal number n such that the frog would theoretically have to touch the finish line is abotu 116.

_________________

"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider

Let 0 be the starting line and 1 be the finish line. Let,

s_n=1+1/2+(1/2)^2+...+(1/2)^n

There exists a cardinal number n such that the frog, theoretically, has to touch the finish line. I'll work out this number.

Using the partial geometric sum formula (which is easy to prove by mathematical induction or other methods),

The cardinal number n such that the frog would theoretically have to touch the finish line is abotu 116.

This is not correct.

_________________

Sebastian

"Don't forget to floss." - Darkwing Duck

Phenomena portraying absolute series strikes me as the computational singularity, anything can be in-vivo vivisected to Planck depth provided the scientists with the funding manage to stay there rather than using all that same equipment for cryptographic brute-force exploits. Are we examining the sum wave state of Zeno & Euclid?

_________________

"Standing on a well-chilled cinder, we see the fading of the suns, and try to recall the vanished brilliance of the origin of the worlds."

-Georges Lemaitre

"I fly through hyperspace, in my green computer interface"

-Gem Tos

Planck's length is only relevant if you only wish to discuss quantum foam. This was my specialty when I did physics, it is irrelevant to this problem, as a real number can get much smaller than Planck's length.

Actually, strings are even more elementary than quantum mechanics.

_________________

Sebastian

"Don't forget to floss." - Darkwing Duck

I sneezed my Pepsi when I read this.

I aim to sneeze...

_________________

"Standing on a well-chilled cinder, we see the fading of the suns, and try to recall the vanished brilliance of the origin of the worlds."

-Georges Lemaitre

"I fly through hyperspace, in my green computer interface"

-Gem Tos

Planck's length is only relevant if you only wish to discuss quantum foam. This was my specialty when I did physics, it is irrelevant to this problem, as a real number can get much smaller than Planck's length.

Actually, strings are even more elementary than quantum mechanics.

I suppose if the frog never trips, the plank length isn't closed since the frog's got his own whole dimensional plane. So then my question shifts to; can frog feet compress quantum foam in the absence of spatial relativity?

_________________

"Standing on a well-chilled cinder, we see the fading of the suns, and try to recall the vanished brilliance of the origin of the worlds."

-Georges Lemaitre

"I fly through hyperspace, in my green computer interface"

-Gem Tos

Planck's length is only relevant if you only wish to discuss quantum foam. This was my specialty when I did physics, it is irrelevant to this problem, as a real number can get much smaller than Planck's length.

Actually, strings are even more elementary than quantum mechanics.

It is not mathematically proven that the fabric of the universe is made out of subatomic strings or quantum foam.

The smallest

**known**finite length is the Planck length.

_________________

"God may not play dice with the universe, but something strange is going on with prime numbers."

-Paul Erdos

"There are two types of cryptography in this world: cryptography that will stop your kid sister from looking at your files, and cryptography that will stop major governments from reading your files."

-Bruce Schneider