 This time we'll try some harder examples, so we have, start with this number in binary. Then I will just take one, multiply by two, give me two, add the second one, gives me three, multiply by two, gives me six, add the third bit, gives me six, multiply by two, gives me twelve, add the fourth bit, gives me with twelve, multiply by two, gives me twenty four, add the fifth bit, multiply by two, and then add the last bit. So this gives me fifty in base ten, and that is what I see over in my number line. If I started with the octal version of sixty two, I would do the same thing. I'll start with six, multiply by the base, six times eight is forty eight, then I'll add in the second digit, which will give me fifty as well. Hexadecimal would be the same way. I'll start with thirty two, take the three, multiply by my base. This time I'll get eighteen, and then one times three is three. That will give me forty eight, and then I'll add the second digit, and I'll get fifty in base ten again. If I take another large number like one hundred thousand, the binary version is... So I've picked the largest number in my number line, so I expect these two to still be equal. And I'll start with the one, and multiply by two. This gives me two, and I'll add the second bit. This gives me three, times two is six, add the third bit, times two is twelve, add the fourth bit. Now I might start getting lost with all these zeros, so I'm going to start marking these out as I use them. So you can see I've taken the first bit, the second bit, third bit, and there's the fourth bit. This leaves me with twelve, so I'll multiply by two and get twenty four. Then I'll add another zero, gives me twenty four again, multiply by two is forty eight. Add the next bit is forty eight, multiply by two will give me ninety six. Next bit is a one, ninety seven, and I'm going to come back up here. So I've got ninety seven times two, four, carry a one, eighteen nineteen, so I have one ninety four. And I'll add the next bit, this is one ninety five, multiply by two, zero carry a one, eighteen nineteen, two, three. And then I'll add the next bit, so I've got three ninety, multiply by two, zero, eight, seven. Add the next bit, so I have seven eighty one, multiply by two, two, sixteen, fourteen plus one is fifteen. Then I'll add the next bit, multiply this by two, so now I have four, twelve, eleven, and three. And then I'll add the next bit, thirty one, twenty five, and now I've got one, two, three, four, five more zeros. So I'm really not going to be doing any more adding, I'm just going to be doing a whole lot of multiplication. So I've got thirty one, twenty five, and I'll multiply by two, so that gives me four five, two, six. Now I'd add this zero, and then go back and multiply by two again, zero, carry a one, two, four, five, six times two is twelve. Add this zero, now I'll multiply by two, zero, zero, carry a one, five, and two. Add this zero, multiply by two, add this zero, multiply by two, and then finally add the last zero. And my result is, as you'd expect, one hundred thousand. If I start with the octal version of one hundred thousand, that's three, zero, three, two, four, zero. So when I'm done converting, I also expect to get a hundred thousand out again. And I'll start with my three this time, and again I'm multiplying by the base. Give me twenty four, then I'll add the second digit, let's me with twenty four, multiply by the base. So eight times four is thirty two, eight times two is sixteen, plus three is nineteen. Add the third digit, let's me one hundred ninety five, multiply by eight. Eight times five is forty, eight times nine is seventy two, plus four is seventy six, eight times one is eight, plus seven is fifteen. So I've used my three, the zero, and the second three. So the next thing is this two, which gives me fifteen sixty two, and I will multiply this by eight. So sixteen, forty eight plus one is forty nine, eight times five is forty plus four, and eight twelve. So now I'll add the next digit, which is a four, and this gives me twelve thousand five hundred. And I will multiply this by eight, zero, zero, forty, sixteen plus four is twenty, eight plus two is ten. And then lastly I would add the zero, and yet one hundred thousand in base ten. Finally I'll do the same thing for base sixteen, where I'll start with one, eight, six, a, zero. So I'll start with my one, and multiply by sixteen, give me sixteen, now I'll add my eight, this gives me twenty four, and I'll multiply this by sixteen, six times four is twenty four, six times two is twelve, plus two is fourteen, and then one times twenty four is twenty four. So when I add those together, I get three hundred eighty four, use my one and my eight, and now I'll use the six. So this gives me three hundred ninety, and I'll multiply this by sixteen, we add zero, six times nine is fifty four, six times three is eighteen, plus five is twenty three, and one times three ninety is three ninety. So now I want to add the a, except I don't have any a's in base ten, I'll need to convert the a into base ten. Fortunately a in base ten is just ten, so I'm going to add ten here, and this will give me sixty two fifty, which I will multiply by sixteen. So six times zero is zero, six times five is thirty, six times two is twelve, plus three is fifteen, six times six is thirty six, plus one is thirty seven, and one times sixty two fifty is sixty two fifty. So when I add these together, I get one hundred thousand in base ten, then I'll add my last zero, and I'll have a hundred thousand in base ten still.