Binary Arithmetic

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