 Rounding is an important concept that appears to be very simple but is actually a very complex idea. To understand the concept of rounding, it helps to understand the concept of significant figures. First, in the expression of a number, any non-zero digit is significant. A zero might or might not be significant. Zeros will be significant if they are unnecessary, indicated with a bar over the top, or between significant figures. So let's try out a few examples. Let's find the number of significant figures in the number 100500. So the first thing we note is that any non-zero digit is significant. So that means in 100500 the significant figures are, first of all, the one and the five. Now we also have a couple of zeros here, so remember zeros are significant if they are unnecessary, indicated with a bar, or between significant figures. And we see that we do have a zero that's indicated with a bar, so it's significant. And finally, we know that the zeros between significant figures are also going to be significant. So the one and the five are significant figures, so the two zeros between them must also be significant. And finally, this last zero, the one after the zero with the bar, is not a significant figure, because if we omitted it, we would change the value of the number. And so that means this number has five significant figures, the one and the five, the zero with the bar over it, and the zeros between the one and the five. How about the number of significant figures in the number 100000? So again, the one is significant because it's a non-zero digit. The zeros after the decimal are actually unnecessary. If we omit them, we don't change the value of the number. And so the reasoning you might have here that will remind you why these are significant, I didn't need to write these zeros, and so the fact that I did is indicative that I think that they are important. And so the zeros after the decimal are significant because their omission would not change the value of the number. And then finally, because the one is significant and these zeros are significant, that also means these zeros must also be significant. And so the zeros before are significant since they are between one and a significant figure. And that means this number has six significant figures. Significant figures allow us to describe a number by the place value of the last significant figure. So for example, the last significant figure of 31500 is five, which is in the hundreds place. So 31500 is a hundred. On the other hand, if I put a bar over that next zero, then that becomes a significant figure, which is a ten. And so this number 31500 with a bar over the zero is a ten. And finally, if I write 31500.000, the last significant figure is the last zero, which is a thousandth. So this number 315.000 is a thousandth. So now we're ready to round numbers. When we round, we can round in at least three different ways. First, we can round up to the nearest number of a specific type. Next, we can round down to the nearest number of a specific type. And finally, we can simply round to the nearest of the number of a specific type. So let's round the number 4207 down to the nearest thousand. Let's round it up to the nearest thousand. And finally, we'll simply round it to the nearest thousand. Since we're trying to round to a thousand, we need to make sure that the thousand's place has the last significant digit. Well, the only way to make sure that happens is that all the digits after must be zeros. And so that means one of our possibilities is the number 420,000, where we've replaced all of the digits in the places after the thousands with zeros. Now this is one of the nearby thousands, but it's less. We should also find a nearby thousand that's more. And so we go to the next higher thousand, which will be 43,000. And so we've identified the nearby thousands, 42,000 and 43,000. So if we want to round down, we'll go to the lower number, 42,000. If we want to round up, we'll go to the higher number, 43,000. And then if we simply round, we want to go to the nearest number, and we see that the number 43,000 is a little bit closer than 42,000, so we'll round to 43,000. Let's try another example. We want to round the number 1053.135001, and we want to round this to the nearest tenth. So again, we'll want our tenth's place to have the last significant figure. Now because that's past the decimal point, if we write anything after that point, it will become significant. And so we need to make sure there is, in fact, nothing past the tenth's place. So one possibility is the number 1053.13 where we just drop all of the digits after the three. And this is a tenth that's less than our number. We want a tenth that's more, so we'll go to the next one, 1053.14. And so this gives us our choice that nearby tenths are going to be 1053.13 and 1053.14, and the one that's closer is 1053.14. What if we went around to three significant figures, the number 40358? Since we want a number with three significant figures, we'll want the first three digits of the number to be significant, and the others should be non-significant zeros. So a nearby number with three significant figures is going to be 40300, and this is nearby but less. A number that's nearby but more is going to be 40400. And so we have our nearby numbers with three significant figures, 40300 and 40400. And since the direction of rounding up or down is not specified, we'll round to the nearest of these numbers, 40400. We can also try and round a number to a specified number of significant figures. So let's round to three significant figures, the number 28970. So if we want our number to have three significant figures, it's important that the first three digits of the number be the significant figures, and all the rest have to be zeros. So our first three digits have to be the significant figures, so a nearby number with three significant figures is 28900, and this is a nearby number that's less. A nearby number that's more, where the first three digits are the significant figures, is 2900. However, we have to be careful here because as written, this number seems to only have two significant figures. So here's where we use that bar over the zero. We'll write this number as 2900 with a bar over that zero to indicate that it is significant. And so our two nearby numbers with three significant figures, 28900, 2900, and the closer one is 2900 with a bar over that zero. Every now and then, it'll also become important to round in other ways. So for example, suppose we may want to round to the nearest multiple of five, the number 46. As before, we want to have a choice between two possibilities, and here we want to find multiples of five that are close to 46. And we see that 45 is a multiple of five that's close by, but it's less than 46, and a multiple of five that's nearby but more than 46 is 50. And so the nearby multiples of five are 45 and 50, and the nearest one is 45.