Convert each number from base 10 to the desired base. 45 45 into base 2 2

Welcome to the practice. For this problem, we'd like to convert 45 from base 10 into base 2. Our first step is to find the base and place values that we're converting to. We're told that we're converting to base 2. And the place values of base 2 are powers of 2. We have 1, then 2, then 2 squared, then 2 cubed, 24th, 25th, 26th, and so on and so forth, where we just increase the power on 2 by 1 as we move to the next place value. Now that we know what system we're working in, we're ready to identify the relevant place values. These are the place values that will actually be in our new world when we represent our quantity 45 in base two. Now, these are going to be all the powers of our base that are less than or equal to the number that we are converting, the number we started with. So we want to find all of the powers of 2 that are less than or equal to 45. Now, 1 is less or equal to 45, and so it's 22 squared is 4, which is less or equal to 45, and 2 cubed is 8, which is also less than or equal to 45. 24th is 6. 16, which is less than or equal to 45, and 25th is 32, which is also less than or equal to 45, but 26th is 64, and 64 is not less than or equal to 45, so 26th, as well as any higher power of 2, not going to be relevant here. We've identified the place values that we need. The highest place value is 25, followed by 24th, 2 cubed, 22, 2 and 1. And it is helpful to write these little lines or slots, because in step 3, we are going to be finding the digit of each place. one at a time. So it's helpful to have those little lines there that we can just fill in as we go. Now, I already mentioned step 3, which is where we're moving next. We want to find the digit in each of our place values, and we'll do that by starting with the highest place value, in this case, 25th, and dividing our base 10 number, this 45, by that place value. So we're going to take that 45. And we're going to do 45 divided by that highest place value 25th, and we said, of course, 25th is 32. Now, 32 does not go into 4, but it does go into 45 1 time, and we can subtract 32, we end up getting a remainder of 13. So now we have both our quotient and our remainder. Now, the quotient is the digit of that place value, so the one that number up top, the quotient is the digit in that place value, the place value for 25. The remainder becomes the new base 10 number that we'll be using when we do this same thing for the next highest place value 24. So, we're going to take the 13, that remainder, and divide it by 24, divided by 16. Now 16 goes into 130 times, and when we subtract the 0, we still get 13. So here our quotient, the number up top, is 0, and that will be the digit in the 24 place value. The remainder here is 13, so that's going to be the number that we're going to carry on when we do this step again for the next highest place value, 2 cubed. So we're going to take 13 and divide it by 2 cubed, which is 8. Now it does go into 13, it goes in one time, and when we subtract 8, we end up getting a remainder of 5. We're gonna take this number up top of the quotient, and that's the digit in the 2 cubed place values, so we'll fill in a 1. Then we're going to take our remainder, and we're going to use it when we do this again for the next place value. So let's come over here. Now we're going to take 5 that remainder, and divide it by 2 square, which is 44 goes into 51 time, when we subtract 4, we end up getting 1. So the number up top, the quotient, this 1, is going to be the digit in the 2 squared place value, so we'll slot in a 1 here. The remainder here was one, so that will be the number we carry on when we do this step again for the next highest place value. So we're gonna do 1 divided by that next place value of 2. Now 2 does not go into 1, it goes in 0 times, and when we subtract 0, we do get 1. So we're gonna take this number up top, this 0, and it's going to be the digit in the 2's place. Then we'll take our remainder of 1. And we'll use it with that final place value. So we'll take that remainder of 1 and divide it by 1, the final place value. Now 1 goes into 11 time, and then we're going to subtract 1 and we get 0. We're going to take that number up top, the quotient, and that will be the digit in the 1's place. Then we know that we finished either because we have filled in every place value, but also because we do end up getting a remainder of 0. So I'm just going to rewrite my number here. We get 101101, and of course we need the subscript for our base. So I'm gonna put a subscript of a 2 here, so we can say that 45 written in base two is 101101. Great work.