Point your telescope at the "bottom" of the two pointers (the one furtherest from the Southern Cross, known as alpha Centauri, also as Rigil Kent), and increase the magnification. What do you see?
Our sun is in a minority of stars, living all by itself, no star closer than 4 light years. The majority of stars have companions, one, two, maybe more? Bound together by gravity they orbit around each other. Hence "double stars". "Splitting them" means resolving the two stars in your binoculars or telescope, to see each of them clearly. Some are easy to do, some hard, some impossible with our basic technology (yours and my 8" reflectors on earth's surface). Depends on the distance they are apart, and that is from our point of view (If they are orbiting each other, their apparent separation distance (usually measured as an angle) will vary from a maximum to where they appear to be "on top" of each other. Also depends on the relative magnitude of the stars. Much the same magnitude (as are alpha Centauri A and alpha Centauri B - well close enough anyway) and its easier to see each of the pair than one very bright and one very faint in a double.
Have a look here:-
http://www.iceinspace.com.au/forum/s...ad.php?t=21264
and some other recent threads that Dennis has started.