 This time, we'll try some examples in arbitrary bases instead of well-known bases. So maybe I have a number in base five. And I want to subtract another number in base five. Again, the really hard thing here is just to remember which base you're in to know what the value of your tan is. So I can't subtract four from one, so I'll need to borrow something from that two. Two minus one is one. Now I have eleven. But I'm in base five, so this is really the same as six in decimal. Six minus four would obviously be two. Or I can say, well, four plus one would get me to ten, plus one more gets me to eleven. Either way, I'll write down a two. Now three is greater than one, so I can't do that. I'll have to go borrow something. So now I have eleven minus three. Well, eleven minus three isn't much different from eleven minus four. So that should be one more than whatever I got last time, or three. Now four is greater than two, so I'm going to need to borrow something. Now I've got twelve minus four. Well, four plus one is five, plus two more gets me to twelve. So I can say that's three. Or I could say that, well, really twelve in base five is the same as seven in base ten. So seven minus four would be three there too. And then three minus three is zero. If I try those same numbers in base six, I'll get a slightly different result. Because all of those borrows will work a little bit differently this time. So four is still greater than one, so I'll need to borrow something. Two becomes a one, my one becomes an eleven. And now I have eleven minus four. Before it was two, when I was in base five, this time my ten actually has a slightly greater value. So I'll get a slightly larger result. In this case I could say ten is six, so six plus one would give me seven. Seven minus four is three. Or I could say I've got ten minus four leaves me with two. Two plus one is three. Then three is less than one, so I'll need to borrow something. Now I have eleven minus three. Well, that's same as seven minus three, which is four. Or I could say I need ten minus three leaves me with three. Three plus one is four as well. And then two is less than four, so I'll borrow something from over here. Now I've got twelve minus four. So again this would be eight and decimal. Eight minus four is four. Or four plus two gives me ten, plus two more gives me twelve. And then three minus three is zero. So in this case I increased the size of my base by one, but kept the numbers the same. Which meant all of those borrows had a slightly greater value. Increased my base by one, so the value of those borrows went up by one as well. As a result, my number got one larger in each of those places. If I pick some different numbers and say base nine. Now one is less than two, so I'll need to borrow something. Now I have eleven minus two. And then base nine is the same as ten and decimal. Ten minus two would be eight. Or I can realize that again one is one less than two. So when I do the subtraction whatever I'm going to get is will be one less than my base. Which is again eight. Seven minus seven is zero. Two minus five. I can't do that, so I'll have to borrow something. Now I have twelve minus five. So that would be eleven if I was in decimal. And then eleven minus five would be six. Or I can do nine minus five is four. Four plus two is six. And then six minus six is zero. If I try something in say base twelve. Okay. So now I have A minus B. B is less than A. So I'll want to borrow something. Now I have one A minus B base twelve. So this should give me one less than my base. Which is again B. Or I can realize that ten minus B will leave me with one. A plus one is also B. Now I want to try five minus A. Which will require I borrow something from over here. Now I have fifteen minus A. Well ten minus A leaves me with two. Five plus two is seven. And then now I have one minus B. So I'll borrow something here. Four and eleven. So ten minus B leaves me with one. One plus one is two. And then four minus three is one. If I try something in base fifteen. So D is one less than E. So I'll need to borrow something. E minus one is D. Then I can have one D. So E is one less than our base of fifteen. So I could just realize that I'll add one to my D. And that'll give me E. Or I could try converting all of this into decimal. Where I'd say okay this is thirteen plus fifteen is twenty-eight. Twenty-eight minus fourteen is fourteen. And fourteen was E. Either way I'll get an E back out. Then D is two larger than B. So D minus B will leave me with two. C is also two larger than A. So I'll get two out. And then A is one larger than nine. So A minus nine is one. That's in base fifteen. So here I've got two numbers in base six. Four is less than five. So I'll borrow something from over here. Now I have fourteen minus five. This will leave me with five. Four minus three is one. Two minus three. I'll need to borrow something again. Four and a twelve. And twelve minus three is five. Four minus three is one.