Find the sum and the product of each of these pairs of numbers. Express your answers as a binary

step 1 Okay, so here we are adding and multiplying some binary numbers. And so for Part A, we will star

Find the sum and the product of each of these pairs of...

Key Concepts

Binary Number System

The binary number system is a base-2 numeral system that uses only two symbols, typically 0 and 1, to represent numbers. It is the foundation of computer science and digital electronics, where these binary digits (bits) are used to encode data and perform arithmetic operations.

Binary Arithmetic

Binary arithmetic involves performing mathematical operations, such as addition and multiplication, using binary numbers. The principles are similar to arithmetic in other bases but require careful attention to carrying and bit alignment, making it a crucial topic in computer architecture and programming.

Binary Addition

Binary addition is the process of adding two binary numbers together. The addition rules in binary are simpler than in the decimal system, with 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (which means writing down 0 and carrying over 1 to the next higher bit). Understanding this process is fundamental for various applications in computing.

Binary Multiplication

Binary multiplication is a method for multiplying two binary numbers. Similar to decimal multiplication, it involves multiplying each digit of one number by each digit of the other and then summing the appropriately shifted results. Since each multiplication step in binary is either 0 (if the digit is 0) or the multiplicand (if the digit is 1), the process is straightforward and essential for implementing algorithms in digital hardware and software.

Transcript

00:01 Okay, so here we are adding and multiplying some binary numbers.

00:05 And so for part a, we will start with, let's start with adding here.

00:10 So we'll have 1 -0 -0 -1 -1 -1, added to 111 -1.

00:26 1 -1.

00:28 And so when we add numbers, so if the operation is adding, make a little matrix of what's going to happen.

00:37 Well, 0 plus 0 is 0, 0 plus 1 is 1, but 1 plus 1 is 10.

00:44 Okay, so when we add 1 plus 1, we get a 0.

00:50 When we get a 10, right, so we have a 0, and we're going to put an additional 1 into sort of the next column.

00:57 Okay, well, here we have 1 plus 1 plus 1.

01:00 Okay, so you can kind of combine these two to be our 10, right? it's like because this a zero there and a one to the next column.

01:09 And then we'll take this zero, add it to this one, and that gives us one.

01:14 I'm going to get the exact same thing here.

01:16 So we can combine these to the top two ones into a 10 and move the one over.

01:23 We'll consider those as zero then, and the zero plus one becomes a one.

01:28 So now i have one plus zero, which is going to be one plus zero is one.

01:33 1 plus 0 is 1.

01:36 1 plus 0 is 1, and then we have a 10.

01:39 So added together, it should be 1 -1 -1 -1 -0.

01:47 So now we're going to multiply these same two together.

01:51 So that was, again, we had 1 -0 -1 -1.

01:57 We multiply that with 1 -1 -1 -0 -1 -1.

02:07 So we multiply, it starts with multiplication and then sort of ends in the process of addition.

02:17 So i didn't just have us do the multiplication, which gives us practice for both.

02:22 I don't know.

02:25 So we will start with, say, the first digit of this lower one.

02:34 And we're going to multiply this one by every element of the binary expression.

02:42 That is on the top.

02:44 When we multiply, so 1 times 0 is equal to 0, 1 times 1 is equal to 1, and 0 times 0 is equal to 0.

03:00 So here, we'll take this 1 and multiply it across, right? and so we'll get basically just the expression that is there.

03:16 Keep the columns i have.

03:19 Okay, so then we move on to this next one, which is a one.

03:23 And we'll do the same thing, except we're going to sort of place it one over.

03:30 So we're shifting it to the left.

03:41 We're going to be adding all these expressions that we're listing here.

03:45 Okay, so then we take the third element as we'll shift it over two places and do the same thing.

03:52 So since it's a one, you get one, one, one, zero, zero, and then in one.

04:01 Well, this next one is a zero, right? so we're still going to shift it over, but everything that we multiply by is going to make a zero.

04:17 Okay, then we have one again.

04:21 So we're going to shift again, and we'll do that two more times, shift again, and another one.

04:38 So we multiply the whole thing by one on top, but slide over.

04:47 And then we are adding all of these together.

04:53 Okay, so we want to start.

04:58 On the right here so that'll just give us a one here we have a one plus one which was 10 right so it's going to come down we'll get a zero then the remainder we need to add in to think a little remainder up there unless we have three ones so we're going to be combining the first two into a 10 so we'll end up adding a zero or adding a one to the next column and then we have a another 10 okay, so we'll put a zero here and we'll have to add another one at the top of the next column so here these two are going to become a 10 leaving a zero behind but putting a one in the next element and same with these two will combine with a 10.

06:03 So we're moving these two ones over by getting for this row another zero so we go here we're gonna have the same thing happen so these two are going to be a ten 10 and these two are going to be a 10.

06:20 So we get a 0, but we'll have two additional ones in the next column.

06:28 Then the same thing is going to happen because we have these two ones here and two ones here, so we get another 0, but we'll get two more ones in the next row.

06:42 And this time around, we have 110, 2 tens, and 3 of them, actually.

06:51 So we're going to stop with a 0, but in the next column we have added three ones here these two will combine to make ten sort of moving means that one goes to the next row and the zero stays behind these two are combined and these two will combine so essentially we get a whole row of zeros but then we add we move the ones over three separate times so it's another zero okay uh so then we look at this row and finally it looks like in our numbers so you can say that these two ones will combine to a 10, putting a 1 over here.

07:37 These two ones can combine to a 10, putting another one over here.

07:41 But we have one remaining, so we'll have a 1.

07:46 Okay.

07:48 When we go to the next row, we have the 2 ones that will combine to make a 10.

07:52 So we'll get a 0 here, but put the 1 over 1 row.

07:55 Those 2 ones will combine again, making a 10.

07:58 So we leave the 0, move the 1 over.

08:01 Same thing happens.

08:02 We get a 10, so we leave the 0 and move the 1 over, and again.

08:08 So when we multiply them, we get 1 ,000 ,000, 1 ,000 ,000 ,000, 0 ,000, 1.

08:21 Of course, you could also potentially convert them to decimal, multiply them, and then re -convert back to binary, if you so choose...