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Unary system:Â Draw lines in the sand -- as simple as it gets. Great for keeping score in games; you can add to a number without erasing and rewriting.
Roman Numerals: More advanced unary, with shortcuts for large numbers. You can find full article on Roman numerals at How to type Roman Numerals ?
Decimals: Huge realization that numbers can use a "positional" system with place and zero.
Binary:Â Simplest positional system (two digits, on vs off) so it's great for mechanical devices.
Scientific Notation:Â Extremely compact, can easily gauge a number's size and precision (1E3 vs 1.000E3).