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No this would not work. In a base number system, each place value position is a multiple of the previous one by a factor of the base [ for example: 100, 10, 1, 0.1, etc. in the decimal or 10 system].

If there was a one-base system, then each place value would be multiplied (or divided by) one: [1, 1, 1, 1, ....]. Also, in a number system: there are the same number of digits as the base [in base 10, we have 0-9 which is ten digits]. The highest digit is one less than the base. So for base one you would have 1 digit, which would be 0.

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