Binary

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-	Rules of Binary Addition	+	a good resource on [http://en.wikipedia.org/wiki/Binary_numeral_system  Binary]
		+
		+	==Rules of Binary Addition==
		+
	0 + 0 = 0		0 + 0 = 0
		+
	0 + 1 = 1		0 + 1 = 1
		+
	1 + 0 = 1		1 + 0 = 1
		+
	1 + 1 = 0, and carry 1 to the next more significant bit		1 + 1 = 0, and carry 1 to the next more significant bit
		+
	For example,		For example,
-	00011010 + 00001100 = 00100110                 1  1  carries	+	00011010 + 00001100 = 00100110
-	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)	+
-		+	1  1                     carries
-	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  1  1  0  1  0    =    26<sub>(base 10)</sub>
		+	0  0  0  0  1  1  0  0    =    12<sub>(base 10) </sub>
		+	+
-	Rules of Binary Subtraction	+	--------------------------------------------------------------------------------
-	0 - 0 = 0	+
-	0 - 1 = 1, and borrow 1 from the next more significant bit	+
-	1 - 0 = 1	+	0 0 1  0 0 1 1 0   =   38<sub>(base 10) </sub>
-	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	+	00010011 + 00111110 = 01010001
-	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)	+
		+	1  1  1  1  1              carries
		+	0  0  0  1  0  0  1  1    =    19<sub>(base 10) </sub>
-	Rules of Binary Multiplication	+	0 0 1 1  1 1  1  0   =   62<sub>(base 10) </sub>
-	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)	+	0  1  0  1  0  0  0  1    =    81<sub>(base 10)</sub>
-		+
-		+
-	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.	+	==Rules of Binary Subtraction==
		+	0 - 0 = 0
-	Another Method:  Binary multiplication is the same as repeated binary addition; add the multicand to itself the multiplier number of times.	+	0 - 1 = 1, and borrow 1 from the next more significant bit
		+
		+	1 - 0 = 1
		+
		+	1 - 1 = 0
	For example,		For example,
-	00001000 × 00000011 = 00011000                    1  carries	+	00100101 - 00010001 = 00010100
-	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)	+
		+	0                        borrows
		+	0  0  <strike>1</strike> <sup>1</sup>0  0  1  0  1    =    37<sub>(base 10) </sub>
-	Binary Division	+	0  0  0  1  0  0  0  1    =    17<sub>(base 10)</sub>
-	Binary division is the repeated process of subtraction, just as in decimal division.	+
-	For example,	+	-
-	00101010 ÷ 00000110 = 00000111                        1  1  1    =    7(base 10)	+	-------------------------------------------------------------
-	--------------------------------------------------------------------------------	+	0  0  0  1  0  1  0  0    =    20<sub>(base 10)</sub>
-	1  1  0  )  0  0  1  10  1  0  1  0    =    42(base 10)
-	-    1  1  0        =    6(base 10)
-	--------------------------------------------------------------------------------	+	00110011 - 00010110 = 00011101
-		+	0 <sup>1</sup>0  1                   borrows
-	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)	+
-	--------------------------------------------------------------------------------	+	0  0  <strike>1  1  0 </strike><sup>1</sup>0  1  1    =    51<sub>(base 10)</sub>
-	1 0  1 ) 1   0  0  10  0  1   1   1    =    135(base 10)	+	0  0 0 1 0  1 1 0    =    22<sub>(base 10)</sub>
-	-   1  0  1            =    5(base 10)	+
		+	-
	--------------------------------------------------------------------------------		--------------------------------------------------------------------------------
-		+
-	1  1 10	+	0 0 0  1 1 1 0 1    =   29<sub>(base 10)</sub>
-	-    1  0   1	+
-		+
-	--------------------------------------------------------------------------------	+
-		+
-	1  1	+
-	-      0	+
-		+
-	--------------------------------------------------------------------------------	+
-		+
-	1   1   1	+
-	-    1  0   1	+
-		+
-	--------------------------------------------------------------------------------	+
-		+
-	1  0  1	+
-	-    1  0  1	+
-		+
-	--------------------------------------------------------------------------------	+
-		+
-	0	+
	Notes		Notes
		+
	Binary Number System		Binary Number System
		+
	System Digits:  0 and 1		System Digits:  0 and 1
		+
	Bit (short for binary digit):  A single binary digit		Bit (short for binary digit):  A single binary digit
		+
	LSB (least significant bit):  The rightmost bit		LSB (least significant bit):  The rightmost bit
		+
	MSB (most significant bit):  The leftmost bit		MSB (most significant bit):  The leftmost bit
		+
	Upper Byte (or nybble):  The right-hand byte (or nybble) of a pair		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		Lower Byte (or nybble):  The left-hand byte (or nybble) of a pair
	Binary Equivalents		Binary Equivalents
		+
	1 Nybble (or nibble)  =  4 bits		1 Nybble (or nibble)  =  4 bits
-	1 Byte  =  2 nybbles  =  8 bits	+
		+	1 Byte  =  2 nybbles  =  8 bits
		+
	1 Kilobyte (KB)  =  1024 bytes		1 Kilobyte (KB)  =  1024 bytes
		+
	1 Megabyte (MB)  =  1024 kilobytes  =  1,048,576 bytes		1 Megabyte (MB)  =  1024 kilobytes  =  1,048,576 bytes
		+
	1 Gigabyte (GB)  =  1024 megabytes  =  1,073,741,824 bytes		1 Gigabyte (GB)  =  1024 megabytes  =  1,073,741,824 bytes

---

Current revision as of 20:31, 1 October 2006

a good resource on Binary

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)

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)

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