Bitwise math/Operators

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Note: all examples in this section will be using hexadecimal and binary.

NOT

NOT is a bitwise operator that takes only ONE operand. It inverts or reverses the binary value.

Example:

 A = 1010 in binary or 10 in decimal. NOT A results in the inversion of 1010 which is 0101. Therefore NOT A = 5.

AND

AND returns “TRUE” per true bit in both of the required two operands. True is 1 and false is 0. It operates bit by bit, just like NOT.

AND rules

AND compares each bit and if both bits per placeholder are true, then it returns a true for that placeholder, all else gets turned into 0.

• 1 and 1 = 1
• 1 and 0 = 0
• 0 and 0 = 0

AND properties

• anything AND'd by itself results in itself
• anything AND'd with 0xF results in itself
• anything AND'd with 0x0 results in 0x0

AND example

Example: 0x6 AND 0x5 = 0x4

Operation	Hexadecimal	Binary	comment
and	6 5	0110 0101	The second and third bits are true. The second and fourth bits are true.
=	4	0100	The second bit is the only one true in both instances (5 and 6).

AND logic table

AND	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
2	0	0	2	2	0	0	2	2	0	0	2	2	0	0	2	2
3	0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3
4	0	0	0	0	4	4	4	4	0	0	0	0	4	4	4	4
5	0	1	0	1	4	5	4	5	0	1	0	1	4	5	4	5
6	0	0	2	2	4	4	6	6	0	0	2	2	4	4	6	6
7	0	1	2	3	4	5	6	7	0	1	2	3	4	5	6	7
8	0	0	0	0	0	0	0	0	8	8	8	8	8	8	8	8
9	0	1	0	1	0	1	0	1	8	9	8	9	8	9	8	9
A	0	0	2	2	0	0	2	2	8	8	A	A	8	8	A	A
B	0	1	2	3	0	1	2	3	8	9	A	B	8	9	A	B
C	0	0	0	0	4	4	4	4	8	8	8	8	C	C	C	C
D	0	1	0	1	4	5	4	5	8	9	8	9	C	D	C	D
E	0	0	2	2	4	4	6	6	8	8	A	A	C	C	E	E
F	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

OR

OR will return true bits for the respective placeholder if any of the bits are true.

OR rules

OR determines if any bits are true from the given binary operands

• 1 or 1 = 1
• 1 or 0 = 1
• 0 or 0 = 0

OR properties

• Anything OR'd with itself results in itself
• Anything OR'd with 0xF results in 0xF
• Anything OR'd with 0x0 results in itself

OR example

Example: 5 OR C = D

Operation	Hexadecimal	Binary	comment
or	5 C	0101 1100	The second and fourth bits are true. The first and second bits are true.
=	D	1101	The first, second and fourth bits are true in at least one instance.

OR logic table

OR	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
0	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
1	1	1	3	3	5	5	7	7	9	9	B	B	D	D	F	F
2	2	3	2	3	6	7	6	7	A	B	A	B	E	F	E	F
3	3	3	3	3	7	7	7	7	B	B	B	B	F	F	F	F
4	4	5	6	7	4	5	6	7	C	D	E	F	C	D	E	F
5	5	5	7	7	5	5	7	7	D	D	F	F	D	D	F	F
6	6	7	6	7	6	7	6	7	E	F	E	F	E	F	E	F
7	7	7	7	7	7	7	7	7	F	F	F	F	F	F	F	F
8	8	9	A	B	C	D	E	F	8	9	A	B	C	D	E	F
9	9	9	B	B	D	D	F	F	9	9	B	B	D	D	F	F
A	A	B	A	B	E	F	E	F	A	B	A	B	E	F	E	F
B	B	B	B	B	F	F	F	F	B	B	B	B	F	F	F	F
C	C	D	E	F	C	D	E	F	C	D	E	F	C	D	E	F
D	D	D	F	F	D	D	F	F	D	D	F	F	D	D	F	F
E	E	F	E	F	E	F	E	F	E	F	E	F	E	F	E	F
F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F	F

XOR

XOR is a bitwise comparison operator that returns a true bit if the compared bits in question are different. If they are the same, it returns a false bit.

XOR rules

XOR determines which bits differ in the two binary numbers used as operands.

• 1 xor 1 = 0
• 1 xor 0 = 1
• 0 xor 0 = 0

XOR properties

• Anything xor'd with itself results in 0
• Anything xor'd with 0xF is the same as a "not"
• Anything xor'd with zero results in itself

XOR example

Example: A xor F = 5

Operation	Hexadecimal	Binary	comment
xor	A F	1010 1111	The first and third bits are true. The first, second, third, and fourth bits are true.
=	5	0101	The second and fourth bits are true in ONLY one instance as opposed to two.

• The 8’s and 2's placeholders are the same so they return 0.
• The 4’s and 1’s placeholders are different, therefore return true.

XOR logic table

XOR	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
0	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
1	1	0	3	2	5	4	7	6	9	8	B	A	D	C	F	E
2	2	3	0	1	6	7	4	5	A	B	8	9	E	F	C	D
3	3	2	1	0	7	6	5	4	B	A	9	8	F	E	D	C
4	4	5	6	7	0	1	2	3	C	D	E	F	8	9	A	B
5	5	4	7	6	1	0	3	2	D	C	F	E	9	8	B	A
6	6	7	4	5	2	3	0	1	E	F	C	D	A	B	8	9
7	7	6	5	4	3	2	1	0	F	E	D	C	B	A	9	8
8	8	9	A	B	C	D	E	F	0	1	2	3	4	5	6	7
9	9	8	B	A	D	C	F	E	1	0	3	2	5	4	7	6
A	A	B	8	9	E	F	C	D	2	3	0	1	6	7	4	5
B	B	A	9	8	F	E	D	C	3	2	1	0	7	6	5	4
C	C	D	E	F	8	9	A	B	4	5	6	7	0	1	2	3
D	D	C	F	E	9	8	B	A	5	4	7	6	1	0	3	2
E	E	F	C	D	A	B	8	9	6	7	4	5	2	3	0	1
F	F	E	D	C	B	A	9	8	7	6	5	4	3	2	1	0