- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0, and carry 1 to the next more significant bit
For example,
| 00011010 + 00001100 = 00100110 |
|
1 1 |
| carries |
| 0 0 0 1 1 0 1 0 |
= | 26(base 10) |
+ 0 0 0 0 1 1 0 0
|
= | 12(base 10) |
| 0 0 1 0 0 1 1 0 |
= | 38(base 10) |
|
| 00010011 + 00111110 = 01010001 |
|
1 1 1 1 1 |
| carries |
| 0 0 0 1 0 0 1 1 |
= | 19(base 10) |
+ 0 0 1 1 1 1 1 0
|
= | 62(base 10) |
| 0 1 0 1 0 0 0 1 |
= | 81(base 10) |
Note: The rules of binary addition (without carries) are the same as the truths of the XOR gate.
- 0 - 0 = 0
- 0 - 1 = 1, and borrow 1 from the next more significant bit
- 1 - 0 = 1
- 1 - 1 = 0
For example,
| 00100101 - 00010001 = 00010100 |
|
0 |
| borrows |
0 0 1 10 0 1 0 1 |
= | 37(base 10) |
- 0 0 0 1 0 0 0 1
|
= | 17(base 10) |
| 0 0 0 1 0 1 0 0 |
= | 20(base 10) |
|
| 00110011 - 00010110 = 00011101 |
|
0 10 1 |
| borrows |
0 0 1 1 0 10 1 1 |
= | 51(base 10) |
- 0 0 0 1 0 1 1 0
|
= | 22(base 10) |
| 0 0 0 1 1 1 0 1 |
= | 29(base 10) |
- 0 x 0 = 0
- 0 x 1 = 0
- 1 x 0 = 0
- 1 x 1 = 1, and no carry or borrow bits
For example,
| 00101001 × 00000110 = 11110110 |
|
0 0 1 0 1 0 0 1 |
= | 41(base 10) |
× 0 0 0 0 0 1 1 0
|
= | 6(base 10) |
| 0 0 0 0 0 0 0 0 |
|
| 0 0 1 0 1 0 0 1 |
|
0 0 1 0 1 0 0 1
|
|
| 0 0 1 1 1 1 0 1 1 0 |
= | 246(base 10) |
|
| 00010111 × 00000011 = 01000101 |
|
0 0 0 1 0 1 1 1 |
= | 23(base 10) |
× 0 0 0 0 0 0 1 1
|
= | 3(base 10) |
| 1 1 1 1 1 |
| carries |
| 0 0 0 1 0 1 1 1 |
|
0 0 0 1 0 1 1 1
|
|
| 0 0 1 0 0 0 1 0 1 |
= | 69(base 10) |
Note: The rules of binary multiplication are the same as the truths of the AND gate.
Another Method: Binary multiplication is the same as repeated binary addition; add the multicand to itself the multiplier number of times.
For example,
| 00001000 × 00000011 = 00011000 |
|
1 |
| carries |
| 0 0 0 0 1 0 0 0 |
= | 8(base 10) |
| 0 0 0 0 1 0 0 0 |
= | 8(base 10) |
+ 0 0 0 0 1 0 0 0
|
= | 8(base 10) |
| 0 0 0 1 1 0 0 0 |
= | 24(base 10) |
Binary division is the repeated process of subtraction, just as in decimal division.
For example,
| 00101010 ÷ 00000110 = 00000111 |
|
|
| | | | | |
1 | 1 | 1 |
= | 7(base 10) |
|
| 1 1 0 |
) |
0 | 0 | 1 | 10 | 1 | 0 | 1 | 0 |
= | 42(base 10) |
| | |
| |
- | 1 | 1 | 0 |
| |
= | 6(base 10) |
| |
|
|
| | | | |
1 |
| | |
| borrows |
| | | |
1 | 0 | 10 | 1 |
|
| | | |
- | 1 | 1 | 0 |
|
| |
|
|
| | | | |
| 1 | 1 | 0 |
| | | | |
- | 1 | 1 | 0 |
| |
|
| | | | | | | |
0 |
|
| 10000111 ÷ 00000101 = 00011011 |
|
|
| | | |
1 | 1 | 0 | 1 | 1 |
= | 27(base 10) |
|
| 1 0 1 |
) |
1 | 0 | 0 | 10 | 0 | 1 | 1 | 1 |
= | 135(base 10) |
| |
|
- | 1 | 0 | 1 |
| | | |
= | 5(base 10) |
|
|
| | |
1 | 1 | 10 |
| | |
| |
- | 1 | 0 | 1 |
| | |
| |
|
|
| | | | |
1 | 1 |
| |
| | | |
- | | 0 |
| |
| |
|
|
| | | | |
1 | 1 | 1 |
|
| | | |
- | 1 | 0 | 1 |
|
| |
|
|
| | | | | |
1 | 0 | 1 |
| | | | |
- | 1 | 0 | 1 |
| |
|
| | | | | | | |
0 |
- Binary Number System
- System Digits: 0 and 1
- Bit (short for binary digit): A single binary digit
- LSB (least significant bit): The rightmost bit
- MSB (most significant bit): The leftmost bit
- Upper Byte (or nybble): The right-hand byte (or nybble) of a pair
- Lower Byte (or nybble): The left-hand byte (or nybble) of a pair
-
- Binary Equivalents
- 1 Nybble (or nibble) = 4 bits
- 1 Byte = 2 nybbles = 8 bits
- 1 Kilobyte (KB) = 1024 bytes
- 1 Megabyte (MB) = 1024 kilobytes = 1,048,576 bytes
- 1 Gigabyte (GB) = 1024 megabytes = 1,073,741,824 bytes
[ Index | Technical Notes ]
DISCLAIMER
Page author: Dawn Rorvik (rorvikd@evergreen.edu)
Last modified: 05/20/2003
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