In addition, IIRC, if you decide to split an atom in half and keep doing it, you would find that there is no end to the process. You could keep doing it forever. There is no point in which an atom simply disappears as you keep cutting it in half. It is simply reshaped or maybe becomes smaller and smaller.
I think you might have the atom confused with something else. An atom is made up of a number of particles: protons, neutrons, and is orbited by electrons. If you need a mental image, think of a bunch of grapes being an atom. You could split an atom into two smaller atoms (break a branch off the grape stalk). Once you have an atom with only one particle left, there's nothing left to split (just one lone grape). If you could split a particle, on the other hand, you might be onto something (wine!). I've heard nothing on splitting particles, though.
