How does microprocessor differentiate between a positive and negative number?

Microprocessors use binary, so all they have is ones and zeros (♪♫...and sometimes we ran out of ones...♪♫) Sorry.

The chip notes negative numbers by the leading bit

If the chip can handle 8 bits if information, they use the first for the sign (0 is positive, 1 for negative) and the remaining 7 bits for the number from 0 to 127.