4.3. Binary Conversions - Multiplication/Division Method

The biggest issue with the “table method” shown on the previous page is that it is hard to translate into an algorithm that a machine could run. The multiplication/division methods shown on this page are much easier to express in simple terms as they rely on mathematics more than things like “look at the table for _______”.

4.3.1. Decimal to Binary

The following algorithm converts a decimal number to a binary one:

Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done
        Otherwise, make the quotient your new number and go back to step 2

Here is an example of running the algorithm to convert \({11}_{10}\) to binary:

So the decimal number 11 in binary is \({1011}_{2}\).

4.3.2. Binary to Decimal

To convert a binary number to its decimal value, we can use almost the same trick, but in reverse:

Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done
        Otherwise, go back to step 2

If we follow this algorithm to convert \({1101}_{2}\) into to a decimal value, it would look like:

Step 1: number is 1101 and answer is 0
Step 2: answer is multiplied by 2 - it becomes 0
Step 3: Remove leftmost digit of number and add to answer
         number is now 101 and answer is 1
Step 4: number still has digits, go back to step 2
Step 2: answer is multiplied by 2 - it becomes 2
Step 3: Remove leftmost digit of number and add to answer
         number is now 01 and answer is 3
Step 4: number still has digits, go back to step 2
Step 2: answer is multiplied by 2 - it becomes 6
Step 3: Remove leftmost digit of number and add to answer
         number is now 1 and answer is 6
Step 4: number still has digits, go back to step 2
Step 2: answer is multiplied by 2 - it becomes 12
Step 3: Remove leftmost digit of number and add to answer
         number is now "" (empty) and answer is 13
Step 4: number is empty. We are done, the answer is 13.

As mentioned earlier, the advantage of this algorithm is that it very easily converts into relatively simple computer code. To demonstrate that, the algorithm is implemented in Python in the codelens below. You DO NOT need to worry about exactly how the Python language works. You will notice that turning the English algorithm above into code requires some changes, but the code shown follows the same process described above. See the “tips” box below the code lens for helping running the program.

Convert "1101" to decimal (Binary_Conversion)

Tip

How to use the “codelens”

  • You can click the “forward” button to watch the code run line by line.
  • The blue area that appears below the buttons will show you the current values of number and answer.
  • Use can use the large scroll bar to scroll the code to the right - at the end of each line of code is an explanation of what that line “says”

Self Check

    Q-8: If you use the division method to convert 49 to binary, how many times do you have to do step 2?
  • 5
  • You are dividing by 2 until you reach 0. 5 divisions won't get you there
  • 4
  • You are dividing by 2 until you reach 0. 4 divisions won't get you there
  • 6
  • 49
  • You are dividing by 2 until you reach 0. don't need that many divisions