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
antimatter answers, 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
our answer to that question.
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)
