Binary
From Compsci1
Revision as of 14:15, 1 October 2006
Rules of Binary Addition 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.
Rules of Binary Subtraction 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)
Rules of Binary Multiplication 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 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
Notes 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
