For one’s complement, (Java unary ~ operator) take the bit pattern for the equivalent positive number, invert all bits 1⇒0 and 0⇒1, e.g. ~5 is 00000101 ⇒ 11111010.
For two’s complement, (Java unary - operator) take the bit pattern for the equivalent positive number, invert all bits 1⇒0 and 0⇒1, then add one, e.g. -5 is 00000101 ⇒ 11111010 ⇒ 11111011.
Binary is rather bulky to write out, so instead it is often written in terser hexadecimal (hex) or sometimes octal. Unfortunately, there is no way in Java prior to 1.7 to include binary literals in programs other than by encoding them in hex or the more old fashioned octal. In version 1.7+ you can write them with a lead 0b, the same way you can write hex with a lead 0x. Further, you can use _ to group digits e.g. 0b1010_1100.
Common boolean operators include & (bitwise AND for masking), | (bitwise OR for combining), >> (right signed shift), >>> (right unsigned shift), << (left signed/unsigned shift), ~ logical not and ^ bitwise xor. Unfortunately Java does not support binary literals 0b111 or underscores as shown in the examples. You must use hexadecimal notation without underscores instead.
|0b0101_0101 & 0b0001_1100||0b0001_0100||& logical AND, logical carryless bitwise multiply, used for masking
(getting rid of parts of a word you don’t want), 1s where both operands have a 1 otherwise 0.
Don’t confuse this with &&.
0 & 0 →
|0b1110_0000 | 0b1000_0001||0b1110_0001||| logical OR, logical carryless bitwise addition, 1s where either operand
has a 1, otherwise 0. Useful for combining bit masks. Don’t confuse this with ||. Given that you
have a mask to describe the bit (all zeros, one one), to turn on a bit, use |
0 | 0 →
|0b0000_0000_1001_0001 << 2||0b0000_0010_0100_0100||<< signed/unsigned shift left. Slide to left 2-bit places, dropping high order bits, shifting into the sign bit, filling on right with 0s. Shifting left by one bit is equivalent to multiplying by 2. You can calculate 2 to the nth power with 1 <<n .|
|0b1101_0000_1001_0001 << 2||0b0100_0010_0100_0100||ditto. Example with sign bit on.|
|0b0000_0000_1001_0001 >> 2||0b000_0000_0001_00100||>> signed shift right. Slide to right 2 bit places, drop low order bits, filling on left with the 0 sign bit. Shifting right by one bit is equivalent to dividing by 2, with the following exception -1 >> 1 = -1 because of sign extension. -1 / 2 should be 0. Other negative numbers work fine.|
|0b1111_1111_1001_0001 >> 2||0b1111_1111_1110_0100||ditto. Example with sign bit on.|
|0b0000_0000_1001_0001 >>> 2||0b000_0000_0001_00100||>>> unsigned shift right. Slide to right 2-bit places, drop low order bits, filling on left with 0 bit. Shifting right by one bit is equivalent to dividing by 2, however this won’t work for negative numbers because the sign bit gets converted to 0 with an unsigned shift.|
|0b1111_1111_1001_0001 >>> 2||0b0111_1111_1110_0100||ditto. Example with sign bit on.|
|~ 0b1111_1111_1001_0001||0b0000_0000_0110_1110||~ logical not, bitwise 1’s complement. 0s become 1s and 1s become 0s. Given that you have a mask to describe the bit (all zeros, one one), to turn off a bit, use & ~mask .|
|- 0b1111_1111_1001_0001||0b0000_0000_0110_1111||- negation, 2’s complement. 0s become 1s and 1s become 0s, then you add 1.|
|0b1100_0000 ^ 0b1010_0001||0b0110_0001||^ logical XOR (exclusive OR), exclusive or, bitwise difference. 1s where operands differ, 0 where they are the same. Useful in cryptography because xor has a magic symmetry: encrypted = plain ^ key; plain = encrypted ^ key;|
|0b0000_0000_0000_1010 - 0b0000_0000_0000_0011||0b0000_0000_0000_0111||- subtraction. 10 - 3 = 7. Works much like decimal subtraction, with borrowing.|
|0b0000_0000_0000_1010 + 0b0000_0000_0000_0011||0b0000_0000_0000_1101||+ addition. 10 + 3 = 13. Works much like decimal addition, with carrying. Since + works like | when there are no carries, programmers sometimes get into the dangerous habit of using + to combine masks instead of | .|
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