A mathematical investigation might be demonstrated by a situation where someone gets a math problem and tries to sort it out. Normally a mathematical investigation has a question to answer or a problem to solve. The field of statistics is populated largely by mathematical investigation. And a link is provided to the excellent Wikipedia on that topic. Surf on over and read just the first paragraph....
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Mathematical rules refers to several things within mathematics. It could stand for mathematical equations, or formulas. It could also mean mathematical laws.