2.5 Binary Arithmetic

Binary Addition

Binary addition follows similar rules to decimal addition but is simpler since it only involves two digits (0 and 1).

Rules of Binary Addition

A B Sum Carry
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1

Example: Add 10112 and 11012

  1111    (carry)
   1011
 + 1101
  -----
  11000

Verification:
10112 = 1110
11012 = 1310
11 + 13 = 24
110002 = 2410

Binary Subtraction

Binary subtraction can be performed using two methods: direct subtraction and two's complement method.

Method 1: Direct Subtraction

A B Difference Borrow
0 0 0 0
0 1 1 1
1 0 1 0
1 1 0 0

Example: Subtract 10012 from 11012

  1101
 -1001
  ----
   100

Verification:
11012 = 1310
10012 = 910
13 - 9 = 4
1002 = 410

Method 2: Two's Complement

For computers, subtraction is often performed using two's complement:

  1. Find the two's complement of the subtrahend (number being subtracted)
  2. Add this to the minuend (number being subtracted from)
  3. Discard any carry out of the leftmost bit

Binary Multiplication

Binary multiplication is simpler than decimal multiplication since it only involves multiplying by 0 or 1.

Example: Multiply 1012 by 112

    101
   × 11
   -----
     101  (101 × 1)
    101   (101 × 1, shifted left)
   -----
    1111

Verification:
1012 = 510
112 = 310
5 × 3 = 15
11112 = 1510

Binary Division

Binary division follows the same long division method as decimal division.

Example: Divide 11012 by 102

      110.1
    --------
10 ) 1101.0
     -10
      --
       10
      -10
       --
        010
        -10
         --
          0

Verification:
11012 = 1310
102 = 210
13 ÷ 2 = 6.5
110.12 = 6.510