Zeno’s Paradox
Most recent answer: 10/22/2007
Q:
Hi, Two Questions... First one is about matter,light and energy. If there is such thing as anti-matter, could that mean the energy would be negative. And since light is a form of energy, could there be anti-light? For example, could there be such a thing as a "Flashdark" or Anti-FlashLight that would cast a shadow, or beam of anti-light? Next question is about math in respect to time. Say we were to throw something at a wall. And the time it took to reach the wall (from my hand) was 1 second. It would then make sence to say that at about .5 seconds, the ball was half way there. And at .25 seconds it was approximately a quarter of the way there, if you kept dividing in half, time will have never reached zero, therefore the ball will have never been in my hand. This seems weird, but mathematically it doesn’t seem to make sence.. Thanks :)
- Micah
Canada
- Micah
Canada
A:
Hi Micah,
1a) Yes, there is such a thing as antimatter, and it is routinely produced and studied in physics laboratories around the world. Take a look at our , or use the search function to look for answers containing the word "antimatter".
1b) No, the energy of antimatter is positive, just like the energy of matter. In fact, to make antimatter, you usually have to make an equal amount of matter to keep constant other things (like the total electrical charge), and you need enough energy to make 2x of what you want. But some particles are their own antiparticles, like photons, the particles of light, so you can get away with making a single photon at a time.
1c) Light travels in waves, and if you have two waves of the same frequency, you can arrange them so that their crests line up and their troughs line up, making the combined wave stronger ("constructive interference"), or you can get them to cancel each other out ("destructive interference"). The total energy in all of space is still the sum of the energies of the two light beams. You cannot get light to cancel everywhere. Here’s .
2) This situation was first publicised by the Greek Philosopher Zeno of Elea who lived from 495 to 435 BC. His complaint was that an infinite sum of positive time intervals could not possibly be finite. This assertion (stated without proof) took quite a long time to reject; approximately 2000 years had to pass for the theory of infinite series summations to develop in order for people to be comfortable with Zeno being wrong.
Each term in the infinite sum is half the size of the one before it, and they get small very quickly. Here’s a proof (found just about anywhere):
sum(n=1 to infinity) 1/2^n =
1/2 + sum(n=2 to infinity) 1/2^n =
1/2 + sum(n=1 to infinity) 1/2^(n+1) =
1/2 + 1/2(sum(n=1 to infinity) 1/2^n.
If we say sum(n=1 to infinity) 1/2^n = x, this last relationship says that 1/2 + x/2 = x. This can be solved s that x=1.
This kind of trick only works if the series converges (there’s a whole theory behind infinite series). For example, we cannot add up 1+2+4+8+... there is no upper limit to this one. But summing powers of 1/2 converges nicely. Have a look at a calculus book for more details.
Tom
1a) Yes, there is such a thing as antimatter, and it is routinely produced and studied in physics laboratories around the world. Take a look at our , or use the search function to look for answers containing the word "antimatter".
1b) No, the energy of antimatter is positive, just like the energy of matter. In fact, to make antimatter, you usually have to make an equal amount of matter to keep constant other things (like the total electrical charge), and you need enough energy to make 2x of what you want. But some particles are their own antiparticles, like photons, the particles of light, so you can get away with making a single photon at a time.
1c) Light travels in waves, and if you have two waves of the same frequency, you can arrange them so that their crests line up and their troughs line up, making the combined wave stronger ("constructive interference"), or you can get them to cancel each other out ("destructive interference"). The total energy in all of space is still the sum of the energies of the two light beams. You cannot get light to cancel everywhere. Here’s .
2) This situation was first publicised by the Greek Philosopher Zeno of Elea who lived from 495 to 435 BC. His complaint was that an infinite sum of positive time intervals could not possibly be finite. This assertion (stated without proof) took quite a long time to reject; approximately 2000 years had to pass for the theory of infinite series summations to develop in order for people to be comfortable with Zeno being wrong.
Each term in the infinite sum is half the size of the one before it, and they get small very quickly. Here’s a proof (found just about anywhere):
sum(n=1 to infinity) 1/2^n =
1/2 + sum(n=2 to infinity) 1/2^n =
1/2 + sum(n=1 to infinity) 1/2^(n+1) =
1/2 + 1/2(sum(n=1 to infinity) 1/2^n.
If we say sum(n=1 to infinity) 1/2^n = x, this last relationship says that 1/2 + x/2 = x. This can be solved s that x=1.
This kind of trick only works if the series converges (there’s a whole theory behind infinite series). For example, we cannot add up 1+2+4+8+... there is no upper limit to this one. But summing powers of 1/2 converges nicely. Have a look at a calculus book for more details.
Tom
(published on 10/22/2007)