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Computer Science 306: Computer Architecture11 chapters | 105 lessons

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Lesson Transcript

Instructor:
*Sebastian Garces*

Sebastian has taught programming and computational thinking for University students and just got accepted at PURDUE for Master's degree

Binary numbers are the basis for all that is happening inside computers or electronic devices. In this lesson, learn how to multiply and divide binary numbers.

Binary division and multiplication are both pretty easy operations. Instead of dealing with a lot of numbers, you just need to make sure to set the 1 or 0 in the right place. For this reason, you need to make sure you are also familiar with binary addition and subtraction.

To perform a **binary multiplication** problem, we need to understand how addition works with binary numbers and follow the same process of multiplication and addition we would use with decimal numbers.

Let's say we want to multiply 4 by 3, which in binary will be 100 by 011 (or 11, it's the same):

We start with the first digit, in this case 1, and multiply 100 by 1, which gives the exact same number.

Then we move to the second digit and proceed to do the same. Since it's also 1, the number will remain intact, we just need to shift the number one digit to the left:

Later, since the last digit is a 0, we can skip it and start with the addition:

0 added to 0 is 0. And, 1 added to 0 is 1:

We can verify that the result is 12 by converting the binary product to decimal.

Now, let's try with a harder example. Let's multiply 12 by 15, which in binary will be 1100 by 1111.

First, you take the first digit from 1111 and multiply it by 1100, which gives the same number:

Now we're going to repeat the same process with each digit, shifting to the left with each of them, getting this descending series of 1100s:

Finally, we just need to add the numbers. This can be a little tricky when you add several 1's in a row.

Once we reach the column with the two consecutive 1's, remember that the result is 0 and we carry a 1 to the next column.

In the next column, we need to add the numbers by pairs. The first two 1's will add up to 0, carry a 1, then, the result 0 plus the 1 will add up to 1.

We keep repeating the same process until we finish all the addition:

We can check that 12 multiplied by 15 is 180, which in binary is 10110100.

To perform a **binary division**, we need to follow the same process as we do for dividing regular numbers but, in this case, we only need to decide if it's going to be a 1 or a 0.

To divide two numbers, which result is an exact division, we basically need to follow four steps: division, multiplication, subtraction, and next digit.

Let's say that we want to divide 18 by 3, which in binary will be 10010 divided by 00011 (or 11, it's the same).

First, we need to identify digit by digit. Is 11 less or equal than 1? No, so we get another digit. Is 11 less or equal than 10? No, so we repeat and get another digit. Is 11 less or equal than 100? Yes, so in this case we don't need to think how many times can 100 be divided by 11, instead, we just add a 1 to the product.

Now we need to multiply 11 by the product 1. Then, we need to subtract 11 from the digits we've considered so far, which in this case are three, 100:

Remember that 0 minus 1 requires us to carry a 1 from the next digit.

In this particular case, we can carry enough 1's to complete the subtraction, so we need to consider the subtraction as a whole:

The result of this subtraction is always 1. It doesn't matter if it's 10 - 1, 100 - 11, or 1000 - 111, it's always 1.

Now, we go ahead and get the next digit and repeat the four steps:

Is 11 less or equal than 11? Yes, so let's divide, which we know the result is 1:

Then, we multiply 11 by 1, which is 11:

Now, we need to subtract 11 by 11, which in this case will be 0:

Finally, we just need to get the next digit:

Since we got a new digit, we still need to go through the 4 steps again. In this case, 11 is greater than 0, which will add 0 to the product.

Next, 11 multiplied by 0 is 0. Then, 0 minus 0 is 0.

Given that we don't have any more digits to get down and the last residue is 0, we can conclude that the result of the division is 110, which converted to decimal is 6.

Let's briefly review. To do binary multiplication and division, we need to be familiar with the rules for binary addition and subtraction. For **binary multiplication**, we follow the same process as multiplying two decimal numbers where we multiply each digit of the second number by the first whole number, then we just need to add them, switching each resulting multiplication one digit to the left. For **binary division**, we need to follow the same four steps until we get to the end result: division, multiplication, subtraction, and next digit. Using these steps, we will get an accurate end result.

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Computer Science 306: Computer Architecture11 chapters | 105 lessons

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