2.2 Signed Magnitude versus Two s-Complement | Java Number Cruncher: The Java Programmers Guide to Numerical Computing

	Java Number Cruncher: The Java Programmer's Guide to Numerical Computing By Ronald Mak
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	Chapter 2. How Wholesome Are the Integers?

2.2 Signed Magnitude versus Two's-Complement

How does the Java virtual machine encode integer values internally in binary?

Positive values are straightforward. If we imagine that Java has a primitive integer type "nybble" (half a byte) of 4-bit integer values, then the bit patterns for the values 0 through 7 are

 0  0000 1  0001 2  0010 3  0011 4  0100 5  0101 6  0110 7  0111

We reserve the leftmost bit to represent the sign: 0 for positive and 1 for negative.

Now, what about the negative numbers ? One encoding format, signed magnitude, simply sets the sign bit to 1:

 C0  1000 C1  1001 C2  1010 C3  1011 C4  1100 C5  1101 C6  1110 C7  1111

But signed magnitude has two undesirable features. First, there are two zeros, positive (bit pattern 0000) and negative (bit pattern 1000). Second, we can't use the normal binary addition logic when we have negative values for example,

 6  0110        C3  1011  C3   1011   C2   1010  1  0001         5  0101

Clearly, the preceding sums are wrong. Special negative number logic would be required to get the correct answers.

Another encoding format is two's-complement. It encodes the positive values the same way as signed magnitude, with the leftmost bit as the sign. However, it forms each negative value by complementing each bit of the positive value and then adding 1. So, for example, to form the two's-complement representation of -6,

Start with positive 6:	`0110`
Complement each bit:	`1001`
Add 1:	`1`
Gives negative 6:	`1010`

All of the negative values are encoded as follows :

 C1  1111 C2  1110 C3  1101 C4  1100 C5  1011 C6  1010 C7  1001 C8  1000

There is only a single 0, and there is room for another negative value, -8. Now, we can also use the normal binary addition logic when we have negative values and still get the correct answers:

 6  0110        C3  1101  C3   1101   C2   1110  3  0011        C5  1011

The Java virtual machine uses the two's-complement format for its byte, short, int, and long values. A value of the char type does not have a sign bit all 16 bits encode zero or a positive value.

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