Binary Decimal Conversion


Binary Decimal Conversion



Compiled by David Barth, 26 August 2003



General computer configuration



A central processing unit (CPU) contains 16 general purpose registers numbered 0 though 15. Each register is 4 bytes long.

Decimal Instructions: operate on data in main storage.

Binary Instructions: operate on data in registers.

Operation field: Positions 10-14.

Mnemonic: Symbolic for a given operation.

Operands: Identify the data fields to be operated on.

Find decimal equivalent of a binary number



  1. Determine the positional value of each digit.
  2. Add the position values for all positions that contain a 1.
Example 1
  • Binary number: 11111 (base 2) = 32 (base 10)

  • 1x2 (4 power) + 1x2 (3 power) + 1x2 (2 power) + 1x2 (1 power) + 1x2 (0 power)

    • 1x2 (4 power) = 16

    • 1x2 (3 power) = 8

    • 1x2 (2 power) = 4

    • 1x2 (1 power) = 2

    • 1x2 (0 power) = 1
    • _______________________
    • 32

Example 2
  • Binary number: 11101 (base 2) = 29 (base 10)

  • 1x24 + 1x23 + 1x22 + 0x21 + 1x20


    • 1x2 (4 power) = 16

    • 1x2 (3 power) = 8

    • 1x2 (2 power) = 4

    • 0x2 (1 power) = 0

    • 1x2 (0 power) = 1
    • _______________________
    • 29

Remainder method for converting Decimal numbers to any other base



  1. Divide the number by the base.
  2. Indicate the remainder.
  3. Continue dividing into each quotient until the result is zero.
  4. The equivalent number is the base desired is the string of numeric remainders from the last division to the first.

Example

Convert 38 (base 10) to base 2:

  • 38/2 = 19 with a remainder of 0.

  • 19/2 = 9 with a remainder of 1.

  • 9/2 = 4 with a remainder of 1.

  • 4/2 = 2 with a remainder of 0.

  • 2/2 = 1 with a remainder of 0.

  • 1/2 = 0 with a remainder of 1.

The result is a string of remainders starting with the final division, working toward the first:

100110 (base 2)

Rules for adding binary numbers

  • 1 + 0 = 1
  • 0 + 0 = 0
  • 1 + 1 = 0, carry 1

Rules for subtracting binary numbers

  1. Complement the subtrahend by converting all 1s to 0s and all 0s to 1s.
  2. Proceed as in addition.
  3. Remove the high-order digit.
  4. Add a 1 to the total.

Example

11101 (Minuend)
11000 (Subtrahend)
_____
1. Complement the subtrahend.
11000 changed to 00111

2. Proceed as in addition:
11101
00111
_____
100100

3. Remove the high-order digit.

100100 = 00100

4. Add a 1 to the total.
00100 The result.
00001 Add 1 to the result.
_____
00101 (base 2) Final answer.

Rules for subtracting binary numbers when the subtrahend is larger than the minuend



  1. Complement the subtrahend by converting all 1s to 0s and all 0s to 1s.
  2. Proceed as in addition.
  3. Complement the result.
  4. Place a negative sign in front of the number.


Example

11000 Minuend
11101 Subtrahend
_____

1. Complement the subtrahend.
11101 changed to 00010

2. Proceed as in addition:
11000
00010
_____
11010

3. Complement the result.

11010 = 00101

4. Place a negative sign in front of the number.
-00101 (base 2) (or -101 (base 2))

Finding the binary equivalent of a negative number

  1. Represent the number as a positive value in binary form.
  2. Complement the number by replacing all 0s by 1s and all 1s by 0s.
  3. Add 1 to the result.

Example
1. Represent -10 in binary using 12 bits.

10/2 = 5 with a remainder of 0.

5/2 = 2 with a remainder of 1.

2/2 = 1 with a remainder of 0.

1/2 = 0 with a remainder of 1.

Result is 1010.

2. Show 12 bits:
1010 = 000000001010

3. Complement the number.
000000001010 = 111111110101

4. Add 1 to the result:
111111110101
000000000001
_____________
111111110110