Why Do Computers use Binary, Anyway?

There are three new things that I learned after watching the video, Why do computer's use binary, anyway?.  One thing that I learned is that an analog signal is a signal that's proportional to the value it represents.  I also learned that a computer's ultimate job is to do math and come up with a result and or information.  The last thing that I learned was that binary is a two base number system; base two is composed of 0 and 1, base 5 is composed of 0 and 4, and base 10 is compose of 0 and 9.  One question I have is how does 0 and 1 equal base 2 and so on?  Another question I have is why he chose to select nine sections for the adder to encode first?  On thing that I would like to learn more about is how computers use binary numbers when I am completing daily activities on the computer; for example checking my email or typing on Word.