Transcription

Many people's favorite duo is peanut butter and jelly, and for a YouTuber, their favorite duo is usually going to be like and subscribe. But my favorite duo has got to be chessboard and handful of dirty, slightly sticky pennies. Indeed, this classic pair makes a hell of a calculator. Today, I'm going to show you how to use this chessboard and a simple set of counters to do multiplication, division, and even taking square roots. The number doesn't even have to be a perfect square. We're going to do the square root of two as an example of that. This methodology was introduced by John Napier, who's best known for discovering logarithms. It was the final part of his 1617 treatise called Rabdology, and although it never caught on, John Napier seemed to be pretty fond of it. As this turns difficult arithmetic into just sliding counters around on a chessboard, he said it is perhaps better described as a lark rather than a labor. Lark is an old fangled term for something that's mischievous and daring done for fun. So if you're a real daredevil, you could try bringing a chessboard and some dirty pennies into your next math exam, but no doubt you will be promptly kicked out of the room, for this calculating device is simply too powerful. Let us promptly begin with a simple example of multiplication. Suppose we wanted to compute two times four. Then we're going to need two pennies. We'll put one penny down here on four, and one penny here on two. We then need only place a penny on the intersection of the row and column that we've just marked, thus we put a counter there, and then we can slide it diagonally down until we land in the first row. In this case, we just slide it down diagonally once, and we get the answer of eight. Now is a fine time to point out that both the columns and the rows are labeled with increasing powers of two, so you may be thinking about binary, which shouldn't be a big surprise, because since our only way of indicating a quantity is either putting a marker there or not putting a marker there, we basically just have a bunch of on-off switches, and the only way we're going to be able to express whatever integer we want with on-off switches is using base two. Though when this method was introduced in 1617, there was nothing all too common about binary. Indeed, John Napier described this as the one issue with this method, is that numbers must first be expressed in this strange form before then being reduced to a common form to get your answer. A slightly trickier multiplication is nine times six. To do nine times six, first we're going to need to express the two numbers in binary. To get nine, we'll use eight and one, and to get six, we will use four and two. So we place those pennies accordingly. Then just as before, we simply place pennies at the intersection of the rows and columns of the markers we've just placed. So we put a penny here, put a penny here, a penny there, and a penny there. The two key mathematical ideas underlying this process are one, the distributive property, where writing nine is eight plus one, and where writing six is four plus two, and then basically doing the distribution, four times eight, and two times eight, and so on. Then when we proceed to slide these pennies down diagonally, what we do is multiply by two, because we're moving to the left, where these powers of two increase, but we also divide by two, because we're moving down, where powers of two decrease, hence we're not actually changing the value that any penny represents when we slide it down diagonally. Let us then carry out that diagonal sliding. Just slide down diagonally until you get to that bottom row, and that is going to give us this. 32 plus 16 is 48, plus four is 52, plus two is 54, which is nine times six. Let's do one more example of multiplication, where something a little cute happens, before we move on to division. Let's do 24 times 12. So first, we'll express 24 down here in binary. That's just going to be 16 plus eight, and then for 12, we'll have eight plus four. Once again, we'll put pennies at the intersections of rows and columns, which is going to end up giving us this little square in the middle. We then proceed with that downwards diagonal sliding. Remember, each time we're going down this diagonal, we're multiplying and dividing by two, so the values are not being changed. You may notice the strange thing that happens here is these two pennies end up going down to the same spot on row one. This just means we have two copies of 64. If we wanted to then, we could take those out and exchange it for one copy of 128. Now we have two copies of 128, and if we had a bigger chessboard, we could exchange those for one copy of 256. Since we don't have a bigger chessboard, we'll just leave those two pennies there. Regardless, we add up the results here on row one to complete the multiplication. 24 times 12 is 128, plus 128, which is 256, plus 32, so 288. The objective of this highly advanced calculation device at all times is to change whatever the present operation is, be it multiplication, division, or square roots, and turn it into a routine addition problem. Let us now proceed to division. Let's try 24 divided by 4. So we'll first express 24 in binary, that's 16 and 8, and then 4, well that's just a 4 right over there. Now just as you saw how we would slide pennies down a diagonal to perform multiplication, division, being the opposite, it may be no surprise, we'll be sliding pennies diagonally up, and we're trying to form rectangles. So what we have to do for this division of 24 divided by 4 is take this first penny, let it enter onto the chessboard, and then slide it diagonally up until you encounter the row of this other penny. And then same thing for the penny on 8, enter it into the chessboard, and then slide it up diagonally until you encounter that row. You then see the result of the division is just the sum of the columns represented by these pennies. 4 plus 2 is 6, that's 24 divided by 4. You can see how if instead of doing the division we had done multiplication, we would just do what we just saw, but in reverse. This would be 6 times 4, and then we would just put markers up here at the intersections of the rows and columns, and then we would slide these pennies down diagonally until we got that product of 24. Let's try another division. Let's do 50 divided by 10. So we'll start by expressing 50 down here. That's going to be 32 plus 16 is 48, plus 2 is 50. And then we'll express 10, that is just 2 plus 8. This will obviously be a little bit trickier, but again we're just trying to create a rectangle by sliding these pennies up diagonally. The first penny should give us the upper left corner of our rectangle. So let's take this penny, enter it into the chessboard, and start sliding it up until we encounter the row of the topmost penny that's lined up vertically. So we would stop right there. Now immediately we have a little bit of an issue here, because if we move on to the next penny, bring it into the chessboard, and then slide it up diagonally, we see that we're getting a setup for a rectangle that we're just not going to be able to complete, and we can't have that. So we're going to have to make an exchange. This penny, which was on 16, we'll take that out and exchange it for two pennies on 8. Now we can form a rectangle as needed to complete the division. Slide a penny into the chessboard, and then we'll move it diagonally up until we get to this far top right corner of our rectangle. Then we can do the same thing with this next penny, except stop at the bottom left corner of the rectangle. Note that each time we stop with one of these pennies, we want to be on a row which is marked by one of these other pennies. So we're trying to line them up with the pennies that are representing the divisor, which in this case is 10. You can see with this last penny, we're able to bring it in, and then slide it up one on that diagonal, and thus we've completed our rectangle. No surprise then, we see that 50 divided by 10, just look at the columns, add them up, 4 plus 1, it is 5. As you've no doubt surmised at this point, this calculator is pretty OP, so it doesn't even matter if the division won't work out perfectly. Even with the remainder, this calculator will work just fine. Let's try 41 divided by 3 as our last division problem before we move on to square roots. For 41 divided by 3, we will first express 41 down here. So that's going to be 32 plus 8 is 40, plus 1 is 41. Then to express 3 on the vertical strip, we'll have 1 plus 2. Again, if we begin our highly advanced penny sliding technique here, we would bring that 32 penny onto the chessboard, slide it diagonally up until it's lined up with this penny, and that would be the top left corner of our rectangle. But we see that's not going to work. The rest of the pennies on the board are just not going to be able to finish this rectangle, which tells us we're going to need to do an exchange. We'll take away that 32 penny then, and exchange it for 2 pennies on 16. Now we can start to get somewhere. Bring one of these pennies into the chessboard, slide it up diagonally so it's lined up with that top penny vertically. You can see that there's nothing more we can do with this penny on 16. If we slide it in, there's just no place it belongs on this diagonal in order to finish forming some sort of rectangle. So we're going to take this penny on 16 and exchange it for 2 more on 8. So now we have a total of 3 pennies on 8. But now we can start to flesh out this rectangle. Bring a penny in, slide it up diagonally. We can also bring one more penny in, and we are continuing to build a rectangle. This next penny doesn't serve any purpose for us on 8, so we're going to exchange it for 2 pennies on 4, where we can continue to build our rectangle. It's tempting to slide one in and then up diagonally to try to finish building this big rectangle, but we're actually not going to be able to finish the rectangle if we do that, because then there's no good place for this penny except there, I guess. But that leaves a hole there. So in fact, what we want to do is just slide this penny into the chess board and leave it there. That way we still have this rectangular arrangement. And then this penny on 4 we'll exchange for 2 more pennies on 2. We can then bring one penny in, slide it up diagonally to the top of the rectangle, slide this last penny in that's on that 1, and then we have a rectangle, which is going to tell us our quotient. And this last penny that we weren't able to do anything with tells us our remainder. So what is 41 divided by 3? Well, let's look at the columns. It is 8 plus 4 plus 1, that's 13, with a remainder of 2. That's because 13 times 3 is 39, with 2 left over to get the 41. And boy howdy, this is a heck of a mischievous lark, isn't it? I mean, I haven't felt this devious since the last time I clicked Still Evaluating on my Wimrar program. Let's move on to taking square roots. Doing square roots on a chessboard is a heck of a trick. So how is this possible? Well, remember with the previous calculations, we were often trying to line up pennies from the first row with pennies that are here on this vertical strip. And we were doing things like this to get rectangular arrangements. Now the square root is what's called a unary operator. It operates on only a single number. So we're actually not going to have any number expressed here on the vertical strip. And now, instead of lining things up with pennies on that vertical strip, our key is this diagonal, which is representing perfect squares. This entry is 1 times 1, the square number 1. This entry is 2 times 2, the square number 4, and so on. So that's going to kind of be our target diagonal. And we're going to try to construct squares of pennies with that as our guide. This means instead of sliding a penny to line up with a particular row and having that be the top left corner of our rectangle, we might slide a penny up until we hit that diagonal and that would be the upper left corner of our square. The one case where we won't have to form a square is when we're taking the square root of a square number, which happens to be a power of 2. For example, what's the square root of 16? Well, I would put a penny on 16, bring it into the chessboard, and then we're going to slide it up diagonally until we encounter that diagonal. So I would go up right there and stop, and I see that this is 4. Note that just like sliding down the diagonal doesn't change a number because it's dividing and multiplying by 2, sliding up the diagonal doesn't change the number either for the same reason. So when this penny stops at this diagonal, instead of looking at it as 16, I can now see it in its square form of 4 times 4. So the square root of 16 is 4. Alright, let's ramp up the difficulty and try to find the square root of 9. I have no clue what that is, so it's a good thing I have this chessboard and these sticky pennies. Of course, we begin by expressing 9 on the first row, so 8 plus 1. Now it's easy to see when I slide this penny from 8 into the chessboard that I'm never going to encounter that diagonal of square numbers because I'm stuck on a black diagonal. So I immediately know that I have no interest in this 8 penny, I want to exchange it for 2 pennies on 4. From here, the process is pretty easy. Slide a penny into the chessboard and then move it up diagonally until you encounter the diagonal of squares. So I would stop there. Now I could try to do that again with this penny to keep building my square, but there's just no place for this penny to go that's going to continue building up the square. Note that to finish this square, I'm going to need an L-shape arrangement of coins here. That was what John Napier called a gnomon. So then I'm going to exchange this penny on 4 for 2 pennies on 2. And from here, it's easy to finish the square arrangement. Slide a penny in and then up diagonally, and then slide this penny in, and then slide this last penny in. Bam, we've got a square. So what's the square root of 9? Well, again, just look at the columns, or because it is a square, you could also look at the rows. Either way, we get 2 plus 1, which is 3, which is a relief to know. There is no limit to the size of numbers you could take square roots of with a chessboard as long as you were willing to make one of a necessarily large size. But let's push it up to 121. What is the square root of 121? First, we're going to have to form 121 with the powers of 2 down in row 1. So that's going to be 64 plus 32 is 96, plus 16 is 112, plus 8 is 120, plus 1 is 121. Now I'll bring this first penny into the chessboard, and I want to slide it up diagonally until I encounter that diagonal of square numbers. So we would slide it up and stop there. Now note the square arrangement we're trying to construct does not necessarily have to be all squished together and completely filled in like the last one we saw. They could be spaced out more, but necessarily we're going to have to have this corner penny and then some combination of L shapes filling out the rest of the square. Now it's tempting to take this next penny, move it onto the board, and then slide it up until we get there, starting to build some sort of shape out. But the thing is, because we're trying to construct a square, if we have a penny there, necessarily we're going to need one there as well. And with the remaining pennies, there's no way that's going to happen. Hence, this is not the right move. We're going to have to exchange this penny on 32 for two pennies on 16. Hence we would now have three pennies total on 16. We can then slide one penny up and diagonally until we get there, continuing to add to the top of the square. Now because of the symmetry of squares, that means we're going to need a penny there as well. But that can be accomplished. Just take the next penny, slide it up diagonally, and there we go. Then there's nothing for this next penny to do. If we were to slide it up diagonally to this place, well that's not good because we don't have a full L shape there. So we're going to want to exchange this guy on 16 for two more pennies on 8. So now we have three pennies on 8. We can then bring one penny in, slide it up diagonal to there, the corner of the square. If we have a penny there though, we're going to need one there as well. And that's okay. We can accomplish that with the very next penny. There is then no further purpose for this guy on 8 to serve. So we're going to switch it out for two pennies on 4. Then if we try to slide one of these pennies in and up diagonally to that far column, this is not going to work because then we'd say, well, let's slide this penny in there. But then we're not going to be able to fill in this L shape. We still need to get a penny there to complete this gnomon. So let's bring these guys back to 4. And what we'll do is use just one to fill in that space, thus finishing this gnomon. And then we're going to exchange this penny on 4 for two pennies on 2. And from here, it's easy to finish our square arrangement. You have to notice the symmetry here, which underlies this arrangement, viewed both from this angle and from this angle, the shape looks exactly the same. That's because it's representing 8 plus 2 plus 1, which is 11, times 8 plus 2 plus 1. We've re-expressed 121 as 11 times 11. Thus, we see that the square root of 121 is in fact 11. Now if we try to carry out this same procedure starting with 2 to find the square root of 2, we're not going to be able to get very far. And so in fact, what we're going to need to do is multiply 2 by 100, and then use this procedure on that. And that will allow us to compute the square root of 2 to the first decimal place, that is the tenths place. We could multiply by more copies of 100 if we wanted more precision, but we would of course need a larger board to deal with those bigger numbers. So we're going to multiply 2 by 100, thus we're going to express 200 here on the first row, and try taking the square root of that. That's going to give us root 2, accurate to the tenths place. So to get 200, we'll do 128 plus 64, that's 192. So then we'll just throw in an 8, and that gets us to 200. Of course, from the jump, I know that this penny on 128 is not going to get us anywhere, because it's on the blacks. So it's not going to encounter that diagonal square numbers. So let's swap it out for two more pennies on 64. Then this first penny from 64 will go onto the chessboard, and slide it up until it encounters that key diagonal of square numbers. There is then nothing more that these pennies on 64 can accomplish. They can't go anywhere that we would like them to. So we're going to exchange both of them for four pennies on 32. Then we can start to build out this square a little bit. Bring one penny in, slide it up here. If we have a penny there, we need to have one there, and that's no problem at all. We can slide another one right up there. To complete the gnomon, we'll also need to get a penny there. That's not going to be an issue as long as we just do an exchange. So we're going to swap these two pennies we have left on 32 for four pennies on 16. We can then bring one penny in and slide it up to complete that gnomon. The next penny we can bring into the chessboard and slide all the way up to there, continuing to add to our square. If we have a penny there, we're also going to need one there, and that is no problem at all. At this point, there's nothing left we can do with this last penny on 16, so we're going to swap it out for two more on 8. So now we have two pennies on 8. We can then bring one in and start to fill out this gnomon on the edge. Bring another one in and slide it there. Then we're going to need a penny in that corner, so we're going to swap this guy on 8 for two pennies on 4. We can bring one in and slide it up, and that's as far as we're going to be able to get. Since 200 isn't a perfect square, we're not going to be able to get rid of all pennies to form a perfect square. But what number have we expressed with this square? Well, it's 8 plus 4 plus 2, which is 14, or in our case, 1.4, which is the square root of 2 accurate to the tenths place. Irrational square roots on a chessboard. That's pretty incredible. I had math teachers as a kid that thought pretty highly of me, and I can only imagine how they would cringe at the level of villainy I've attained. Now that you've learned this centuries-old lark of a technique, there are two things for you to do. Check out Heidi Meyer's math channel, linked in the description. She had some videos that did a great job explaining this, much better than the books which I first read of this method in. And secondly, nobody tell Texas Instruments, alright? They kind of got a monopoly on this stuff in America, and if they knew what I just showed you, just hush hush, okay?