How can antimatter exist at all? What amount of math had to be done until someone realized they can create it?
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Antimatter is a poorly understood name. It’s really just “less common”. You’re used to a positive proton and a negative electron but there’s nothing inherent to physics that says those charges and masses have to go together. Antimatter basically just flips those charges so that you have a positive electron and negative proton. Anything you can do with a proton and electron you can do with their antiparticles, such as make atoms, molecules, even whole macroscopic objects and star systems.
As to how we realized it could exist and we could make it, Dirac was thinking about how electrons made sense with relativity. He came up with a useful equation (in that it explained some stuff that was this far observed but not explained and made sense starting from very basic principles) from his thoughts but there was a “problem” with his solution. It worked for negative energies. Working for electrons (the positive solution) could have been enough, but Dirac thought about these solutions and in collaboration with other scientists, concluded that there could be a particle that was like an electron but with positive charge. A few years later Carl David Anderson observed positrons in high energy cosmic rays using a bubble chamber and that was it, we knew they existed and how they were made.
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