Sometimes there is no symbol "X" in the computer, so "*" is used instead of multiplication sign, so "*" in mathematics means multiplication sign.

3.* In your question, an operation symbol is defined here. According to you, there may be two situations: P*Q=(P+Q)/2 means that the operation of specifying * is the average of two numbers p and q, and P*Q=(P/2)+Q means that the operation of specifying * is the sum of half of p and q.