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To the best of my knowledge (which is not completely inconsiderable), there is only one binary number system. That is the system in which each place is worth twice that of the preceding place, starting at 1. A 1 will count the place, a 0 will not.

You may have meant how many number system are there 'like' the binary system, that is to say, how many number system are there which, like the binary system, do not use 10 digits. The answer to that is an infinite number. Binary is only special in that it uses the least possible number of digits. Our normal system (decimal) uses 10 digits (0-9). A number system can be made which uses any number of digits.

The only three that I know of which are commonly used are the following. Decimal is our normal day to day number system with the digits 0-9. Binary is the number system used by computers and has only two digits, 0 and 1. Hexadecimal is a number system used by programs and has 16 digits, 0-f (that is, it uses 0-9 like normal, but continues by replacing "10" with "a", "11" with "b" and so on, up to "15" with "f").

To look into this more fully, they're called bases. Binary is base 2, decimal is base 10, and hexadecimal is base 16.

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Q: How many types of binary number system are there?
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How many digits are used in a binary number system and what are they?

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