Two’s component

Two’s component is one of the Signed number representations. Its principles is described as following. (refer to original at http://en.wikipedia.org/wiki/Two%27s_complement )

The 2_n_ possible values of n bits actually form a ring of equivalence classes, namely the integers modulo 2_n_, Z/(2_n_)Z . Each class represents a set {j + k 2_n_ | k is an integer} for some integer j, 0 ≤ j ≤ 2 n − 1. There are 2_n_ such sets, and addition and multiplication are well-defined on them.

If the classes are taken to represent the numbers 0 to 2_n_ − 1, and overflow ignored, then these are the unsigned integers. But each of these numbers is equivalent to itself minus 2 n. So the classes could be understood to represent −2_n_−1 to 2_n_−1 − 1, by subtracting 2_n_ from half of them (specifically [2_n_−1, 2 /n/−1]).

For example, with eight bits, the unsigned bytes are 0 to 255. Subtracting 256 from the top half (128 to 255) yields the signed bytes −128 to 127.

The relationship to two’s complement is realized by noting that 256 = 255 + 1, and (255 − x) is the ones’ complement of x.

−95 modulo 256 is equivalent to 161 since

−95 + 256

\= −95 + 255 + 1

\= 255 − 95 + 1

\= 160 + 1

\= 161

And I have to say I can’t understand the content well. The mathmatics concept is somehow complex to me. Why I got the idea to find out what’s two’s component is just because the Java puzzler below.

public class JoyOfHex { 
  public static void main(String\[\] args) { 
    System.out.println( Long.toHexString(0x100000000L + 0xcafebabe)); 
  } 
}

What’s the result? Obviously it should print 1cafebabe. Right? Wrong.During the computation, is promoted to the long value 0xffffffffcafebabeL.(I got missed here. Why?)So the result is: 0x00000000cafebabeL. Fix this problem as simple as using a long hex literal to represent the right operand. Long.toHexString(0x100000000L + 0xcafebabe L));

• Java™ Puzzlers: Traps, Pitfalls, and Corner Cases By Joshua Bloch, Neal Gafter
• www.wikipedia.org