Binary numbers are a base 2 numeral system.
A binary number is a number expressed in the base-2 numeral system, which uses only two symbols: 0 and 1. Each digit is referred to as a bit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices.
Basics
- Bit: The smallest unit in computers. It is either a 1 or a 0.
- Nibble: Half a byte, or 4 bites.
- Byte: 8 bits together form one byte, a number from 0 to 255. Bits in the byte are numbered starting from the right at 0.
- Short: Two bytes put together is 16 bits, forming a number from 0 to 65535. The low byte is the rightmost eight bits.
- Hex Number: A HEX number consisting of 4 numbers is 16-bit.
Binary to Hexadecimal Conversion
Break down the binary value in chunks of 4, multiply each 1 by its equivalent value, either 8, 4, 2 or 1. Add the resulting numbers together to get the result. For example, the value 1100, or (8*1 + 4*1), is equal to C(decimal 12).
| 0 | 1 | 0 | 1 | Binary |
| 8 | 4 | 2 | 1 | Values |
| 4 | 1 | Result: 5 |
Hexadecimal to Binary Conversion
| 0 | 0000 | 4 | 0100 | 8 | 1000 | C | 1100 |
| 1 | 0001 | 5 | 0101 | 9 | 1001 | D | 1101 |
| 2 | 0010 | 6 | 0110 | A | 1010 | E | 1110 |
| 3 | 0011 | 7 | 0111 | B | 1011 | F | 1111 |
Arithmetic
Addition
| input | ADD | output | carry |
| 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 |
Subtraction
| input | SUB | output | borrow |
| 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 |
Multiplication
| input | MUL | output |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
Bit Masks
AND, or "both", sets individual bits to 0. AND is useful for masking bits, for example, to mask the high order bits of a value AND with $0F: $36 AND $0F = $06.
| input | AND | output |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 1 | 0 | 0 |
ORA(OR), or "either one or both", sets individual bits to 1. OR is useful for setting a particular bit, for example, $80 OR $08 = $88
| input | ORA | output |
| 0 | 1 | 1 |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 1 | 0 | 1 |
EOR(XOR), or "one or the other but not both", inverts individual bits.
| input | EOR | output |
| 0 | 1 | 1 |
| 0 | 0 | 0 |
| 1 | 1 | 0 |
| 1 | 0 | 1 |
Rotate
ROL rotate one bit left to multiply by 2, and ROR rotates one bit right to divided by 2.
| input | op | output |
| 0110(6) | ROL | 1100(C) |
| 0110(6) | ROR | 0011(3) |
Signed Integers
If Bit 7 is not set (as in the first example) the representation of signed and unsigned numbers is the same. However, when Bit 7 is set, the number is always negative. For this reason Bit 7 is sometimes called the sign bit.
| Binary | Unsigned | Signed |
| 0010 0011 | 35 | 35 |
| 1010 0011 | 163 | -93 |
| 1111 1111 | 255 | -1 |
| 1000 0000 | 128 | -128 |
Add with bitwise in C
int Add(int x, int y)
{
/* Iterate till there is no carry */
while (y != 0)
{
/* carry now contains common */
/* set bits of x and y */
int carry = x & y;
/* Sum of bits of x and y where at */
/* least one of the bits is not set */
x = x ^ y;
/* Carry is shifted by one so that adding */
/* it to x gives the required sum */
y = carry << 1;
}
return x;
}
incoming(1): assembly