Divide and Conquer: A Four Step Process To Simplify Binary Division

First, we write out our division problem. We check to see whether our divisor has a radix point. In this example it hasn’t but if it did, we would keep moving the radix point one place to the right in both the divisor and dividend until the radix point had been eliminated from the divisor. This will allow us to write our division problem in terms of a divisor that is a whole-number.

Sub Problem 1

In step 1 of the process we perform division. We start by taking the first digit or our dividend (the 1 of 160.65) as our current working dividend and then look at whether our divisor (21) will fit into it. The answer in this case is no because 21 is greater than 1. We therefore put a 0 in the first column of our result (we write the result above the line as you can see from the illustration).

Following that we perform step 2 where we multiply our quotient by our divisor. In this case is 21 x 0 = 0 so we write 0 as the product.

In step 3, we subtract the product from our current working dividend to obtain a remainder. 1 – 0 = 1 so we put 1 as our remainder.

In the final step, we bring down the next digit of our original dividend (the first 6 of 160.65) to form a new working dividend of 16 and restart the process.

Sub Problem 2

Back at Step 1, we again look at whether our divisor can be divided into our working dividend. As 21 is greater than 16 the answer is no and we put a zero as the next number in our result.

Having calculated our quotient, we multiply it by our divisor to obtain a new product. 21 x 0 = 0 so we write 0 as the product.

The next step is to subtract our product from our current working dividend. 16 – 0 = 16 so we write 16 as the remainder.

The final step in this sub-problem is to bring down the next digit or our original quotient (in this case the 0 of the 160.65) to form a new working dividend of 160.

Sub Problem 3

Back at step 1 again, we attempt to divide our current working dividend by the divisor. This time we get an answer other than zero and with a bit of a guess calculate that 21 can fit into 160 around 7 times. We therefore put 7 as our the next digit in our result.

In step 2, we take the quotient we just calculated and multiply it by our divisor. In this case, 7 x 21 = 147 so we write that as the product.

The next step is to subtract the product from the current working dividend. 160 – 147 = 13 so we write 13 as our remainder.

The last step is to bring down the next number (the second 6 of 160.65) to create a new working dividend of 136.

Sub Problem 4

You should be getting the idea by now. We divide the current working dividend by the divisor. We guess that 21 fits into 136 around 6 times so we write that as the next digit in our result.

Taking the quotient from the last step we multiply it by the divisor. 6 x 21 = 126 which we write as the product.

Next up, we subtract the product from our current working dividend which in this case is 136 – 126 = 10 which we write as our remainder.

Finally we bring down the final digit or our original dividend (the 5 of 160.65) to form our new working dividend of 105.

Sub Problem 5

Nearly there and by now you should know the drill. Working dividend divided by the divisor gives us 105 / 21 = 5 which we take as our quotient and write as the next digit in our result.

Multiply our quotient by our divisor, 5 x 21 = 105 which we take as our product.

Subtract our product from the working dividend, 105 – 105 = 0 which we write as our remainder. As we have no more digits in our original dividend and have a remainder of zero we’re done. Looking above the line we therefore have our result. 160.65 / 21 = 7.65.

We start by writing out our problem. First, we check whether our divisor has a radix point which it has. To eliminate this, we move the radix point in both our dividend and divisor right by a single place and re-write our problem.

With the radix point eliminated in our divisor we are ready to go.

Sub Problems 1 – 3

I’ve collapsed the first three sub-problems into one as they are basically identical. In each, we follow the same four steps as we did with the decimal division example above. We take the first digit in our dividend and attempt to divide it by our divisor. In this case, it doesn’t fit so we write a 0 in the first column of our result, calculate the product (which is 0 and subtract it from our working dividend before bringing down the next digit of our original dividend to create a new working dividend. We repeat this process for the first three digits in our original dividend until eventually we reach the point where we have a working dividend of 1001.

Sub Problem 4

Back at step 1, we again attempt to divide our current working dividend (1001) by the divisor (111). This time, the divisor is less than the dividend and we write a 1 as the next digit in the result.

Moving on to step 2, we calculate the product 1 x 111 = 111 and then move to step 3.

Remember, step 3 is a subtraction operation. We learnt previously that we could perform subtraction by taking the two’s complement of the number we want to subtract and then add it instead.

Taking this approach, we calculate the two’s complement of 111 (remember the rules for two’s complement: flip all the bits and then add 1).

This gives us a two’s complement representation of 1001 which we then add to the current working dividend. Also remember how when we are adding a two’s complement number and we need to carry an additional bit into a new column we simply drop that bit. That is the case here. The result is a remainder of 100.

Step 4 sees us bring down the next bit in our original dividend to get a new working dividend of 100.

Sub Problem 5

Step 1 of sub-problem 5 sees us trying to divide our new working dividend of 100 by our divisor 111. As 111 is greater than 100 we put a zero as the next digit in our result. I’ll skip steps 2 and 3 (which basically have no effect) and then we bring down the next digit in the original dividend to get a new working dividend of 1000.

Sub Problem 6

In step 1 we divide our working dividend of 1000 by 111 which results in us writing a 1 as the next digit in our result. We then use the same two’s complement technique to subtract 111 from 1000. Again, we drop the carried bit and get a new remainder of 1. This is then combined with the next digit we bring down from the original dividend to create a new working dividend of 11.

Sub Problem 7 – 8

In sub-problem 7, we again go around the loop. This time, we attempt to divide our working dividend of 11 by our divisor 111. This results in 0 being added as the next digit or our result. We then perform the necessary multiplication and subtraction steps (which have no effect) and bring down the last and final digit in our original dividend to create a new working dividend of 111.

In our final sub problem, we follow the same process. This time we divide the working dividend of 111 by the divisor which is also 111. We write the quotient (1) as the next digit in the result and subtract the product of multiplying the quotient by our divisor (111) from our working dividend (111) to get a result of 0. With no remainder, and no more digits in our original dividend we have our final answer: 10010.001 / 11.1 = 101.01.