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Latest Updated:04/01/2006




The numbers used in digital circuits are usually n-bit-length binary numbers. For example, in the case of 8 bits, the numbers used are from 00000000 to 11111111 (0 to FF in hexadecimal notation), and in the case of 16 bits, the numbers used are from 0000000000000000 to 1111111111111111 (0 to FFFF in hexadecimal notation). When numbers are expressed in binary notation or hexadecimal notation, which can be broken down into 4-bit units, there is no problem. However, caution is required when expressing numbers in decimal notation.
A problem is the numerical range. Specifically, the problem is how to express negative numbers. In binary notation, the concept of positive and negative numbers doesn't exist. However, there are cases where things that exist in the natural world must be expressed as a negative number.
The expression generally used for negative numbers is the 2's complement (of the binary number). With this expression, calculations can be performed by subtracting the value you want to express from 0.
For example, -100, -10, and -1 are expressed as follows.

2's complement expressions can be used for addition and subtraction, but not for multiplication and division. When multiplication and division use a 2's complement expression, the signs of the numbers must be calculated separately from the absolute values of the numbers; i.e., signs are calculated with other signs and absolute values with other absolute values. In this case, numbers are expressed as a combination of a sign bit and an absolute value.
The method of expressing negative numbers differs depending on the situation. When converting analog and digital signals in particular, the negative numbers are defined as difference expressions depending on the converter used.
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