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#1 2006-07-13 07:34:48

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

Infinite lightbulb.

A light bulb blinks faster and faster until it is blinking infinitely fast at two seconds time.
The light bulb starts on from zero to one second.
From 1.0 to 1.5 seconds the bulb is off.
From 1.5 to 1.75, the bulb is on.
Keep cutting the time in half and switch on and off.
After two seconds completes, the bulb is left either off or on.
Is the bulb off or on?
Someone asked me this question in 1986 or so and I still don't know the answer.


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#2 2006-07-13 07:39:54

luca-deltodesco
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Re: Infinite lightbulb.

it cant be determined, because halving the difference, you can never actually reach an on or off at two seconds, if you try to work it out, youll just get on/off infinately, until you reach 2 seconds, which you cannot do, you can only get infintisamely close


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#3 2006-07-13 07:56:39

John E. Franklin
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Re: Infinite lightbulb.

But if you could determine if infinity is even or odd, you would know the answer.
Also if a hare runs 9 times faster than turtle and they are in a race.
And we give the turtle a 100 meter lead.
All speeds are constant, zero acceleration.  The turtle travels 1 meter per second.
The hare travels 9 meters per second.
When or where does the hare overtake the turtle if the race is long enough?
Before the race begins the turtle places a marker to mark his 100 meter spot.
When the hare reaches the 100 meter spot, the turtle is at perhaps the 111.1111 meter spot and
the turtle drops another marker.  When the hare reaches each marker, the turtle drops another
marker.  We already figured out when and where the hare overtakes the turtle, but were an even or odd number
of markers dropped when the hare overtakes the turtle??


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#4 2006-07-13 08:22:00

luca-deltodesco
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Re: Infinite lightbulb.

well that one again is impossible to find out, because the are an infinate number of markers dropped, and infinity cant be defined as odd or even


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#5 2006-07-13 08:59:59

Ricky
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Re: Infinite lightbulb.

You could never do something, anything, infinitely fast.  To do something infinitely fast, you have to be moving infinitely fast.  But you can't break the speed limit, the speed of light.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2006-07-13 13:07:14

Zhylliolom
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Registered: 2005-09-05
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Re: Infinite lightbulb.

1986? I wasn't even born yet!

Anyway, I think the problem here is that people are treating infinity as an integer by trying to assign parity to it. You should be considering infinity a concept, not a number. Infinity isn't some definite quantity. Consider this: the sum of the reciprocals of the integers (the harmonic series) is divergent, and the sum of the reciprocals of the primes is as well. You can be comfortable in saying that the integers are a much denser set of numbers than the primes, so the sum of the reciprocals of the integers add up to something greater than the sum of the reciprocals of the primes, even though both sums are infinite. I suppose what I am trying to say is that you can't say you're alternating something an infinite amount of times and ask what it will end on because there's no way to have a definite value for infinity that we could assign parity to, and an infinite process would of course never be completed.

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#7 2006-07-13 13:30:33

MathsIsFun
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Re: Infinite lightbulb.

Maybe infinity is evodd or odven.

If you go around a circle and each time around you flick a switch (off to on, or on to off), is the switch on or off when you reach the end of the circle?

But you never reach the end of the circle ...


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#8 2006-07-13 13:56:21

John E. Franklin
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Registered: 2005-08-29
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Re: Infinite lightbulb.

Thanks for all the input, guys.  Funny you said evodd and odven because I was thinking the answer might be that infinity is even and odd both just a few hours ago.  So anyway an infinite number of ideas can pass in two seconds however.
Can ideas exceed the speed of light?


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#9 2006-07-13 14:06:13

Ricky
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Re: Infinite lightbulb.

Can ideas exceed the speed of light?

A better question is "Can ideas have a speed?" or even "what the heck is an idea?"


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#10 2006-07-13 14:44:34

John E. Franklin
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Registered: 2005-08-29
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Re: Infinite lightbulb.

Right on!  What the heck is an idea?  Is it chemical, nuclear, gravitational, electrical, magnetic?
And can the same idea be exactly duplicated in two persons brains, or just similar enough that they jive.

And besides, you can think up scenarios of recursion, but you probably don't really iterate all the possibilities to infinity in the 2 seconds, you just imagine the possibility of thinking these things.

Last edited by John E. Franklin (2006-07-13 14:46:13)


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#11 2006-07-13 15:02:53

Zhylliolom
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Re: Infinite lightbulb.

You could say that "ideas" have a speed... we could assign the speed at which electrochemical impulses travel through the brain as the speed of an idea. Of course, these impulses cannot exceed the speed of light, no matter how myelinated our neurons are.

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#12 2006-07-13 15:05:19

John E. Franklin
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Re: Infinite lightbulb.

What's myelinated mean?

Last edited by John E. Franklin (2006-07-13 15:05:43)


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#13 2006-07-13 15:13:00

Zhylliolom
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Re: Infinite lightbulb.

Something that is myelinated has a myelin sheath. The myelin on the axon of our brain's neurons helps to speed up electrochemical impulses in the brain. I'm not actually sure if more myelin produces more of a speed-up though, I was just using the term to look cool.

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#14 2006-07-13 15:25:04

John E. Franklin
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Registered: 2005-08-29
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Re: Infinite lightbulb.

Interesting, I just read that metal poisoning can wear away the myelin layer doing bad stuff to the brain.
And a rare disease in boys may go blind or lose control of movement due to myelin layer reduction.
Something about Multiple Scerosis too, but didn't catch that.


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#15 2006-07-13 15:53:38

mikau
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Registered: 2005-08-22
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Re: Infinite lightbulb.

you were asked this in 1986? I was born that year. Creepy.


A logarithm is just a misspelled algorithm.

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#16 2006-07-13 17:31:54

luca-deltodesco
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Re: Infinite lightbulb.

Zhylliolom wrote:

Something that is myelinated has a myelin sheath. The myelin on the axon of our brain's neurons helps to speed up electrochemical impulses in the brain. I'm not actually sure if more myelin produces more of a speed-up though, I was just using the term to look cool.

they are also stop electrical inteferance with other neurones to keep the signal pure and to stop it passing to other neurones, i.e. you want to move your left finger, and the other fingers in your hand move, kind of like if you have a bunch of audio cables to speakers, and you get a strange buzzing noise if they dont have a layer of protection to stop inteferance, and you can get a sound being sent down one cable to one speaker, and you can hear it slightly on the other speaker aswell

mikau wrote:

you were asked this in 1986? I was born that year. Creepy.

i was -5 years old :X


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#17 2006-07-13 17:41:17

Ricky
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Re: Infinite lightbulb.

To answer an ealiert question:

myelinated:

The way nerves work is that they attach to one another.  One "fires" (meaning builds up enough of a positive ion charge), and it sends this singal to another nerve, which may fire as well.

So how does it send this signal?  Well, there is a protien which exchanges potassium for sodium.  potassium has a negative charge, sodium a positive.  So when you get rid of potassium and take in sodium, you are increasing the amount of positive charge in your cell.  And interestingly enough, it "fires" when there is enough of a positive charge.  So image you have a bunch of these protients at a, b, c, d, and e inbetween two nerves.

The nerve becomes positively charged, for various reasons including influence from other nerves.  So station a has a high enough positive charge to fire.  When station a fires, now b has a high enough positive charge to fire.  Then c, then d, and finally e, where e can (but doesn't have to) act on another nerve, making it "fire".  And so the process continues.

In comes the myelinated sheath.  Imagine such a sheath as a length of wire that connects a and e.  As soon as a becomes positive, this charge travels all the way down the line to e, where e fires because of it.  We just skipped b, c, and d, and since these all take time, we just vastly increased the speed of our nerve.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#18 2006-07-14 02:45:21

George,Y
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Registered: 2006-03-12
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Re: Infinite lightbulb.

Many years ago a Chinese mathematician said one statement:
" Let there be a pole, and cut the pole by half each day, you'll not get it eliminated even after millions of years."

This is used as the proof that we Chinese had the concept of limit.

But actually it is more like a statement discussing the flaw of limit, that you can approach the limit as close as you want, but you may never reach it in finite steps.

Zeno(n) suggests the same paradox, and he put it further, he said a moving arrow cannot move.

If you cannot add the series 0.5+0.25+0.125+...to zero, you will not get a 1, but the zero term does not show in term N when N is a natural number.

So does the series 0.5+0.25+0.125+...(infinite terms) make sense? does the notation 0.999...(infinite digits) make sense?

Why are we so addicted to get a perfect and precise result? Why don't we simply admit that infinity only exists in imaginary and reality is compatable with small residual errors?

By the way, none of a physic law envolving Calculus is universal, especially in small world, as far as I know.


X'(y-Xβ)=0

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#19 2006-07-14 03:35:14

Ricky
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Re: Infinite lightbulb.

By the way, none of a physic law envolving Calculus is universal, especially in small world, as far as I know.

Every time we study something that has a change in rate, we must use calculus.  That's pretty much what physics has been ever since Newton.  Is it just a cosmic coincidence that Calculus works so well in the real word?  I think not.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#20 2006-07-14 13:01:32

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

Re: Infinite lightbulb.

So we can imagine dividing distances and time into smaller and smaller pieces forever, but in reality maybe time has a smallest unit like an atom of time.  But distance probably can be infinitely small, even in reality.  Who knows?  I don't think that I know, at the conscious level of thinking.


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#21 2006-07-14 16:07:35

Ricky
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Re: Infinite lightbulb.

I don't think that I know, at the conscious level of thinking.

Rather than the unconscious level of thinking? big_smile

We can study "things" really easily in science.  That is because "thing" interact with other "things".  We know that energy can be thought of as in packets, and that there is a smallest sized packet that you can have.

Space (which includes time as well) on the other hand, is something completely different.  Space doesn't really interact with things.  All things require it.  We can't test in an enviornment where we can manipulate space.  In fact, the only thing we know how to do really is bend it, ever so slightly.

This makes space really hard to probe (which would probably be done with a... space probe).  We can't test it like we can matter, and because of this, we have a hard time learning about it.  As far as I know, John, no one really knows.  But it does seem to answer a whole lot of paradoxes if there is a smallest unit of space.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#22 2006-07-14 19:12:05

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: Infinite lightbulb.

Ricky wrote:

By the way, none of a physic law envolving Calculus is universal, especially in small world, as far as I know.

Every time we study something that has a change in rate, we must use calculus.  That's pretty much what physics has been ever since Newton.  Is it just a cosmic coincidence that Calculus works so well in the real word?  I think not.

Yes you got it! If you mean the rate of a water flow, or an electric current, or a ray of light, I can definitely tell you they are concrete and the rate  calculated from calculus is just an approximation.

Can you draw a real circle on the paper? I guess you cannot. Nevertheless, the formula with π discribes its area well. And it's not coincidence because a mass polynomials is just similar to an ideal circle.

The only two subjects so far people haven't discovered its discountinousity dispite their previous thought are time and space. But I don't know if space know a real axis.


X'(y-Xβ)=0

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#23 2006-07-14 23:22:32

RickyOswaldIOW
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Registered: 2005-11-18
Posts: 212

Re: Infinite lightbulb.

In my A-level maths I learn about sequences and series.  In geometric progression (a series of number where each previous number is multiplied by a ratio) we have what's called "Sums to infinity".  For example, take the series:
1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...

We can see the sum of the series:
1st term, s1 = 1
2nd, s2 = 1 1/2
3rd, s3 = 1 3/4
4th, s4 = 1 7/8
5th, s5 = 1 15/16
6th, s6 = 1 31/32
And so on.

The sum will get ever close to 2.  If we use the formula for the sum S to n terms, first term a = 1, ratio r = 1/2

Sn = a(1-r^n)/1-r

Sn = (1(1-(1/2)^n)) / (1 - 1/2) = (1-(1/2)^n) / (1/2) = 2 - (1/2)^n-1

The difference between Sn and 2 is the term (1/2)^n-1.  As n gets very large, this term gets very small.  We can make Sn as close to 2 as we like by making the value of n very large.  Thus the limit of the sum of the series of n tends to infinity 2, we write:
S∞ = 2

Now this really confuses me.  When I try to apply this rule anywhere else it is wrong yet we have a whole module on sums to infinity that states that you do eventally reach S = 2!

Last edited by rickyoswaldiow (2006-07-14 23:25:16)


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#24 2006-07-22 21:07:15

krassi_holmz
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Registered: 2005-12-02
Posts: 1,905

Re: Infinite lightbulb.

Very interesting question about the infinity.
My opinion is that infinity is not a number, so it can't be even or odd.
It doesn't have (even or odd) property.
An illustration of what happens:

Last edited by krassi_holmz (2006-07-22 21:07:57)


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