That's how additional bases work. You can go above 10 (most commonly base-16) by finding additional symbols too.

If you want to have it be anything more than a mathematical curiosity, you would have to find a practical application for it.

Binary's largest practical application is in digital systems, where every signal can ultimately be broken down into series discrete "on" or "off" switches on a circuit board.

Consider a byte of data. Instead of thinking of it as a number between 0 and 255 (inclusive), think of it as an electrical circuit with eight switches. Each bit desugnates one of those switches as on or off. 00000000 might mean that they are all in the off position, while 11111111 would likewise mean that they are in the on position.

If you were working with a system with three-position switches, a trinary number system like you describe would fit it well, but I don't know of any systems with "off," "on," and "more" as the switch positions.