Binary number operations

Binary Number Operations

Addition

0 + 0 → 0
0 + 1 → 1
1 + 0 → 1
1 + 1 → 0, carry 1

For e.g

  1 1 1 1 1    (carried digits)
    0 1 1 0 1
+   1 0 1 1 1
-------------
= 1 0 0 1 0 0 = 36

Subtraction

0 − 0 → 0
0 − 1 → 1, borrow 1
1 − 0 → 1
1 − 1 → 0

Subtracting a “1" digit from a "0" digit produces the digit "1", while
1 will have to be subtracted from the next column. This is known as
borrowing.

    *   * * *   (starred columns are borrowed from)
  1 1 0 1 1 1 0
−     1 0 1 1 1
----------------
= 1 0 1 0 1 1 1

Multiplication

           1 0 1 1   (A)
         × 1 0 1 0   (B)
         ---------
           0 0 0 0   ← Corresponds to the rightmost 'zero' in B
   +     1 0 1 1     ← Corresponds to the next 'one' in B
   +   0 0 0 0
   + 1 0 1 1
   ---------------
   = 1 1 0 1 1 1 0

Binary Multiplication for binary point
                 1 0 1 . 1 0 1       A (5.625 in decimal)
                 × 1 1 0 . 0 1       B (6.25  in decimal)
             -------------------
                   1 0 1 1 0 1    ← Corresponds to a 'one' in B
     +           0 0 0 0 0 0      ← Corresponds to a 'zero' in B
     +         0 0 0 0 0 0
     +       1 0 1 1 0 1
     +     1 0 1 1 0 1
     ---------------------------
     = 1 0 0 0 1 1 . 0 0 1 0 1  (35.15625 in decimal)

Negative Binary number

How can we represent a negative number? We cannot use a ‘-‘ sign because all we can store in the computer is zeros and ones.

There are three methods

• Signed Magnitude
• 1’s Complement
• 2’s complement

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Credits
http://turing.cs.camosun.bc.ca/COMP166/resources/binary.html

My Links

World of Bits and Bytes

Recently, I have realized that no matter how much C, C++ or other high level languages you know it all comes down to bit and bytes view of program. 
I started realizing that after all it’s not that easy to actually think and write in bit/byte manipulation. This is my attempt to learn and be more comfortable about thinking in bit/bytes.

Table of Contents

• Basics
• Bit Fields
• Problems
• How set a single bit in a byte?
• How to unset single bit in a byte?
• One function to set and unset
• How to unset range of bits?
• How to set range of bits?

Basics

Inroduction link

 0 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0
||              |               |               |              ||
|+- bit 31      |               |               |       bit 0 -+|
|               |               |               |               |
+-- BYTE 3 -----+--- BYTE 2 ----+--- BYTE 1 ----+-- BYTE 0 -----+
|                               |                               |
+----------- WORD 1 ------------+----------- WORD 0 ------------+
|                                                               |
+--------------------------- DWORD -----------------------------+

Hexadecimal Numbers
0 1 2 3 4 5 6 7 8 9 A B C D E F

Bitwise Operators
The & operator (AND)
1   &   1   ==   1
1   &   0   ==   0
0   &   1   ==   0
0   &   0   ==   0

The | operator (OR)
1  |   1   ==   1
1  |   0   ==   1
0  |   1   ==   1
0  |   0   ==   0

The ^ operator (XOR)
1  ^   1   ==   0
1  ^   0   ==   1
0  ^   1   ==   1
0  ^   0   ==   0

The ~ operator
The ~ (Ones Complement or inversion) operator acts only on one value and it inverts it.

The >> (Right Shift)
00001100  - b 
00110000  - b << 2

The << (Left Shift)
00001100  - b
00000011  - b >> 2

Another example is
1<<4; 0001 0000

Bit Fields 

struct date_struct {
    BYTE day   : 5,   // 1 to 31
         month : 4,   // 1 to 12
         year  : 14;  // 0 to 9999
    } date

|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|0 0 0 0 0 0 0 0|
  |                           |       |         |
  +------ year ---------------+ month +-- day --+

date.day = 12;
dateptr = &date;
dateptr->year = 1852;

Basics of Binary numbers and operations (link) 

Problems 

How set a single bit in a byte?

 For e.g In byte 0000 1000 set bit no 6 will produce 0100 1000
 (Remember bit number starts with 0-7)
         

//For problems where certain bit values needs to be changed, first we
//need to create a bit mask.
//Bit mask is a temporary variable with some value. Using this value
//we will access and change specific bits in a byte of data.

//For e.g.
//To set 6th bit in a byte 0000 1000
//We have             MASK 0100 0000  (OR)
//                    ——————————————
//                         0100 1000

//To turn on certain bit in a byte (OR) is used.

int set_bit(int val, int num, bool bitval)
{
    return (val | (bitval << num));
}
//Here, val    = 0000 1000 = 8
//      num    = 6 (set 6th bit)
//      bitval = 1 (set to 1)

//      0000 1000
//(OR)  0100 0000  (1 << 6)
//      ————————-
//      0100 0000

How to unset single bit in a byte?

For e.g In byte 0100 1000 unset bit no 6 will produce 0000 1000

(Remember bit number starts with 0-7)        

//To unset specific bit we will use (AND) operation.
//Mask value need to be ‘0’ for the bit to unset but rest of the bits
//need to ‘1’. The reason for rest of the bits to set as ‘1’ is as
//we are doing (AND), we don’t want to unset other bits which are
//already set.
//
//For e.g.
//To unset 6th bit 0100 1000
//            MASK 1011 1111 (AND)
//                ——————————-
//                 0000 1000   (Result)
int unset_bit(int val, int num, bool bitval)
{
    return (val & ~(bitval << num));
}

//Here, val    = 0100 1000
//      num    = 6
//      bitval = 0
//      (bitval << num) = 0100 0000
//     ~(bitval << num) = 1011 1111
//
//      0100 1000
//      1011 1111 (AND)
//      —————————-
//      0000 1000    (Result)

One function to set and unset

int change_bit(int val, int num, bool bitval)
{
    return (((val & ~(bitval << num)) | (bitval << num));
}

Unset range of bits

Set range of bits

Links

• Gamedev
• Vipan
• Stanford
• Stack overflow