03 Number Systems
Number Systems¶
\(d_{n-1} \dots d_2 \ d_1 \ d_0 . d_{-1} \ d_{-2}\)
- \(d_{n-1} \to d_0\) = integer part
- \(d_{-1} \to \dots\) = fraction part
-
\(d_{n-1}\) = MSD (most significant digit)
-
\(d_0\) = LSD (least significant digit)
| Decimal | Binary | Octal | Hexa |
|---|---|---|---|
| 10 | 2 | 8 | 16 |
| Groups of 3bits | Groups of 4bits |
| Decimal | Binary | Octal | Hexa |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 3 | 3 |
| 4 | 100 | 4 | 4 |
| 5 | 101 | 5 | 5 |
| 6 | 110 | 6 | 6 |
| 7 | 111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A |
| 11 | 1011 | 13 | B |
| 12 | 1100 | 14 | C |
| 13 | 1101 | 15 | D |
| 14 | 1110 | 16 | E |
| 15 | 1111 | 17 | F |
Binary¶
Bit = each digit
Nibble = group of 4bits
Byte = group of 8bits
| Unsigned | Signed Magnitude | 1's comp | 2's comp | |
|---|---|---|---|---|
| Range | \(+(2^n - 1)\) | \(-(2^{n-1} - 1) \longleftrightarrow +(2^{n-1} - 1)\) | \(-(2^{n-1} - 1) \longleftrightarrow +(2^{n-1} - 1)\) | \(-(2^{n-1} - 1) \longleftrightarrow +2^{n-1}\) |
| +ve | regular | same as unsigned | same as unsigned | same as unsigned |
| -ve | - | invert MSD | Bit-by-bit complement of unsigned | 1. Bit-by-bit complement of unsigned 2. Add 1 |
| +ve MSD | 0 | 0 | 0 | 0 |
| -ve MSD | - | 1 | 1 | 1 |
// 1s comp
module ones(a,y);
input[3:0] a; // 4bits (0 to 3)
output[3:0] y;
assign y = ~a;
endmodule
// 2s comp
module twos(a,y);
input[3:0] a; // 4bits (0 to 3)
output[3:0] y;
assign y = (~a) + 1;
endmodule