5.3. 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 _______”.
5.3.1. Decimal to Binary¶
The following algorithm converts a decimal number to a binary one:
Step 1: Start with a blankanswer
and thenumber
your are converting Step 2: Divide yournumber
by 2 to make aquotient
and aremainder
Step 3: Place yourremainder
on the left side of youranswer
Step 4: If yourquotient
is 0, you are done Otherwise, make thequotient
your newnumber
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 \({1011}_{2}\).
5.3.2. 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 thenumber
your are converting and theanswer
of 0 Step 2: Multiply youranswer
by 2 Step 3: Remove the leftmost digit ofnumber
and add it to youranswer
Step 4: Ifnumber
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 andanswer
is 0 Step 2:answer
is multiplied by 2  it becomes 0 Step 3: Remove leftmost digit ofnumber
and add toanswer
number
is now 101 andanswer
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 ofnumber
and add toanswer
number
is now 01 andanswer
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 ofnumber
and add toanswer
number
is now 1 andanswer
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 ofnumber
and add toanswer
number
is now "" (empty) andanswer
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.
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
andanswer
.  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

Q8: 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