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This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Two%27s_complement


00:02:12 1 History
00:04:16 2 Potential ambiguities of terminology
00:04:50 3 Converting from two's complement representation
00:05:23 4 Converting to two's complement representation
00:07:53 4.1 From the ones' complement
00:09:06 4.2 Subtraction from 2supiN/i/sup
00:11:16 4.3 Working from LSB towards MSB
00:12:11 5 Sign extension
00:12:29 6 Most negative number
00:12:42 7 Why it works
00:13:08 7.1 Example
00:13:22 8 Arithmetic operations
00:13:36 8.1 Addition
00:15:07 8.2 Subtraction
00:16:24 8.3 Multiplication
00:19:39 8.4 Comparison (ordering)
00:21:00 9 Two's complement and 2-adic numbers
00:21:11 10 Fractions conversion
00:21:35 11 See also
00:21:53 12 References
00:22:16 13 Further reading
00:22:31 14 External links
00:23:19 8) is too large to represent. For example, an 8-bit number can only represent every integer from −128 to 127 (28 − 1
00:25:02 Arithmetic operations
00:25:13 Addition
00:27:59 5 two's complement can represent values in the range −16 to 15) so overflow will never occur. It is then possible, if desired, to 'truncate' the result back to N bits while preserving the value if and only if the discarded bit is a proper sign extension of the retained result bits. This provides another method of detecting overflow—which is equivalent to the method of comparing the carry bits—but which may be easier to implement in some situations, because it does not require access to the internals of the addition.
00:28:39 Subtraction
00:30:40 Multiplication
00:35:30 Comparison (ordering)
00:36:40 Two's complement and 2-adic numbers
00:37:34 ...111 − 1", i.e., "2X
00:37:45 ...111
00:38:51 Fractions conversion
00:40:12 16 reduces to 3/8). So the denominator is 8. So, the final result is 3/8.
00:40:24 See also



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SUMMARY
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Two's complement is a mathematical operation on binary numbers, best known for its role in computing as a method of signed number representation. For this reason, it is the most important example of a radix complement.
The two's complement of an N-bit number is defined as its complement with respect to 2N. For instance, for the three-bit number 010, the two's complement is 110, because 010 + 110 = 1000.

Two's complement is the most common method of representing signed integers on computers. In this scheme, if the binary number 0102 encodes the signed integer 210, then its two's complement, 1102, encodes the inverse: -210. In other words, to reverse the sign of any integer in this scheme, you can take the two's complement of its binary representation. The tables at right illustrate this property.
Compared to other systems for representing signed numbers (e.g., ones' complement), two's complement has the advantage that the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers (as long as the inputs are represented in the same number of bits, and any overflow beyond those bits is discarded from the result). This property makes the system simpler to implement, especially for higher-precision arithmetic. Unlike ones' complement systems, two's complement has no representation for negative zero, and thus does not suffer from its associated difficulties.
Conveniently, another way of finding the two's complement of a number is to take its ones' complement and add one: the sum of a number and its ones' complement is all '1' bits, or 2N − 1; and by definition, the sum of a number and its two's complement is 2N.