12. Decimal to Binary Conversion

• Consider a decimal integer 105, which we would like to convert to a signed 8-bit quantity:

      position: 76543210
         value: ????????
  

   

1. 105 a positive, odd number.

    

2. Our task is to find values of bits a₇ through a₀ as follows:

   • 105 = a₇*2⁷ + a₆*2⁶ + a₅*2⁵ + a₄*2⁴ + a₃*2³ + a₂*2² + a₁*2¹ + a₀*2⁰

      

3. a₇ is zero, because 105 is positive:

       position: 76543210
          value: 0???????
   

    

4. a₀ is one, because 105 is odd:

       position: 76543210
          value: 0??????1
   

    

5. Subtracting one from 105 yields 104:

   • 104 = a₆*2⁶ + a₅*2⁵ + a₄*2⁴ + a₃*2³ + a₂*2² + a₁*2¹

      

6. Dividing both sides of the equation by 2 yields

   • 52 = a₆*2⁵ + a₅*2⁴ + a₄*2³ + a₃*2² + a₂*2¹ + a₁*2⁰

      

7. Since 52 is even, so a₁ equals to zero:

       position: 76543210
          value: 0?????01
   

    

8. Next, we can iterate back to

   • step 5, if the number is odd, and subtract again last digit, or

   • step 6, if the number is even, and divide both sides of the equation by two.

   • In our example dividing both sides by 2 yields

     • 26 = a₆*2⁴ + a₅*2³ + a₄*2² + a₃*2¹ + a₂*2⁰, therefore a₂ = 0:

           position: 76543210
              value: 0????001
       

      

9. Dividing both sides by 2 yields

   • 13 = a₆*2³ + a₅*2² + a₄*2¹ + a₃*2⁰, therefore a₃ = 1:

         position: 76543210
            value: 0???1001
     

      

10. Subtracting one from both sides yields

    • 12 = a₆*2³ + a₅*2² + a₄*2¹

       

11. Dividing both sides by 2 yields

    • 6 = a₆*2² + a₅*2¹ + a₄*2⁰, therefore a₄ = 0:

          position: 76543210
             value: 0??01001
      

       

12. Dividing both sides by 2 yields

    • 3 = a₆*2¹ + a₅*2⁰, therefore a₅ = 1:

          position: 76543210
             value: 0?101001
      

       

13. And, finally,

    • 1 = a₆*2⁰, therefore a₆ = 1:

          position: 76543210
             value: 01101001