# Binary arithmetic

## Miscellaneous

 Binary Arithmetic

• 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