 Let me ask you a question. How old are you? In seconds. Now, you were probably able to answer the question pretty quickly when I first asked it, until I added the extra stipulation of, in seconds. So let me make a guess. Or actually, let me make a guess presuming that you are one of my students. I'm going to estimate that you, my student, if I make an easy estimate, I'm going to say about 20 years old. That's an estimate to the closest decade. And if I do that, how do I convert that into seconds? Well, it's pretty straightforward with some numbers that we know. We know in 20 years there are 365 days in one year, and there are 24 hours for every one day. And there are 3,600 seconds for every one hour. Now, I could have split that up into minutes, 60 minutes per hour, and 60 seconds per minute. But when I take that, and I cancel out the appropriate units, so I end up with the final answer in seconds, I get a value here, and I'll pull out my precalculation instead of a calculator, and get 630,720,000 seconds. So how good is that estimate? Would that suffice to answer the question, how old are you in seconds? Well, if I made that guess, and then I really think about most of my students, most of my students tend to be in the last two years of high school, either juniors or seniors here in high school in the United States. So a better guess might actually be something like 17 years. Maybe instead of guessing to the nearest decade, I do a little better, and I make an approximation to the nearest year where they might be 16, 17, or 18, but 17 years is a better approximation. And when I do so, and follow the same multiplications, I get something that looks like this. 536,112,000 seconds. So is that a good estimate? Now, if I'm asking for it in seconds, why would I need it in seconds if years would have been good enough? Also, a lot of younger people I know, for example, my son, when he was 11, for the longest time he was 11, but as soon as he turns 11 and a half, he definitely made sure that we knew he was and a half. We could be a little more accurate than just that 17 years. Let's say some of my students are coming up on 18 and they like to think I'm very close to being 18 soon. So let's go ahead and assume that instead their age, they provide that extra and a half. How does that change the number of seconds they are old? Well, notice the actual number of seconds doesn't change, but my estimate for how many seconds they are does. Let's take a look here. 17 and a half years, 551,850,880,000 seconds. Okay. Can we do better? If I ask you to calculate how many seconds older you are, are you satisfied with that answer to the nearest half year or so? What if we got a little bit better? Because it's not too difficult to figure out, okay, what month is it now? What month is my birthday? Maybe instead of 17.5 years, my student is 17 years and four months. So that's going to be 17 times a 365, which we did, plus, let's see here, 17 times 365 plus four times. And how much is a month? Well, it varies, but let's estimate that at being about 30. And that would give us a number of days. And then we can translate that into 546,536 million, 480,000 seconds. All right. Notice my numbers are changing, but some of the numbers aren't changing so much. Here, I had six. Now I have five and I still have five and I still have five. So I seem to be getting a little bit more detail. I'm honing in. I'm getting to be what's called more precise. And my number over here, certain aspects of it are changing, but some of them aren't. Let's keep moving forward. What digits are changing as I get more precise with my age? Let's go to the nearest half month, 17 year, 4.5 months. So 17 times 365 plus 4.5 times 30 to estimate the number of days. And actually, I can put the number of days in here. For previously, this was 6,325 days. And then this becomes 6,340 days, because now I'm beginning to break this up, and then I do my final calculation of hours per day and seconds per hour. And in this case, I get 547,776,000 seconds. So let's continue and go a little bit more precise. 17 years, 4 months, and 10 days. Just probably a little less than half a month, but close. And if we do that, I get 17 times 365, but now that we're talking about days and we're getting a little bit closer here, maybe we have to think about the difference before. We assume there were 30 days per month. Maybe we have to start thinking about the difference that's happening in each month. So I'm going to say plus 30, plus 31, plus 31, plus 30 to represent the past few months that have happened. Challenge for you. Can you tell me which month we're doing this in? And then we'll add all those together. 6,337 days. And we get our value here of 547 million, I keep saying that, 516,800 seconds. So we started talking about different days, but some of you might have said, well, what about leap years? Aren't there extra days that have been counted in? Indeed, somebody who's been around for 17 years has lived through an extra day that's been added on. So interestingly, this was a problem that humans had for a while, settling on how long the calendar should actually be. And in fact, the Julien, it wasn't until 45 BC with the Julien calendar that we even started thinking about adding the leap years, that we had the day for leap year. And so let's see here. So we're going to go ahead and assume, instead of being 365 days a year, it's actually 365.25. So if we actually redo that same thing, 17 years, four months, 10 days, but now we're going to take into account leap year, 17 times 365.25, that'll add the quarter day in there. Now it doesn't exactly add the leap year's account exactly, but that's going to estimate and take into account adding a day for every four years. Plus R, let's see here. I believe this adds up to 122. And I forgot the plus 10 days, which wasn't, which was in there, but plus 10. So that's 132. The plus 10 was included in that 637. I just didn't write it. We end up with 6341.25 days, which if we again convert to seconds becomes 547,884, 547,884,000 seconds. All right. So you can understand here as we look, the more information, the more detail I have about my student or that whoever's calculating has about themselves, the more our numbers tend to not change as we add the detail, but we can have these numbers. The question is, is this good enough? Have we estimated the number of seconds closely enough for whatever your purposes are? Let's go ahead and try just a little bit closer. Okay, here it turns out that this 0.25, humans studied in history and figured out that even that wasn't quite close enough. In fact, it was after about a thousand years, I'm sorry, more than a thousand years, about 1,500 years in the year 1582, that they discovered that the Julian calendar was a little bit off. Even adding leap years was putting things so that the normal times considered different times a year were beginning to shift it. And in 1582, Thursday, October 4th, the next day after that was followed by Friday, October 15th. They had to add 11 days because the calendar had gotten so far off of the actual revolution of the earth that they needed to add an additional 11 days. And now we actually account for that by having some extra rules associated with leap year, things along the lines of we do observe leap year except, let's see if I can remember the details, as long as the, actually I'm not going to remember all the details there. Went on centuries, certain centuries, I believe we don't have a leap year except every 400th century, then we do have a leap year, something along those lines, but I'm going to leave that to you to go look at. The main point here is that this number 0.25 could be better. In fact, the number that we should be using if we wanted to get more precise, let's use the same four months, 10 days, but we'll do 17 times 365.2422. Now we've gotten a lot more precise with the amount of days per year. And when we translate that, we get 6,000, let's see here, 6341.1174 days and that ends up giving us 547,872,543.36 seconds. Now this is where I want to pay a little bit of attention because many of my students do this. We have now calculated this and we had lots of precision, lots of detailed information here. And when we multiplied this out, we ended up with lots of detailed information over here. The question is, is that detailed information any good? Suddenly we went from having lots of zeros over here or mostly zeros over here to having details down to the nearest 100th of a second. But we're only estimating to the nearest 10 days. And this is where we start leading to this idea of significant figures. Which numbers are important to keep? Which numbers are available to keep? Do we really need all this stuff or is it even meaningful at the end here when we're only estimating closer to the nearest day? Remember, a day is going to be something on the order of, let's see here, four times 3600 is 86,000 seconds. So are we worried about the nearest 10th of a second? Probably not. Maybe none of that at the end really has any meaning for us. Let's keep going. Let's get just a little bit more precise since we've got all this precision here. Maybe we can have a little bit more precision. We asked the student, okay, were you able to figure out four months, ten days from your birthday, yes, you know what time you were born. Let's add seven hours. So we add this whole thing 17 times 365.2422 plus 132. Plus since we're talking about days we'll say seven hours at a 24. So a fraction that we add in there. And then we got our number in days which is 6341.4091 days which we then get to be 547,897,746.24 seconds. When are you ever going to stop? Well, let's try one more. Because now we've gotten down to hours but can we get down to minutes? Seventeen years, four months, ten days, seven hours, four hours, 13 minutes. And we do the multiplication there. We have this same piece and this same piece and this same piece. And then we add 13 divided by 1440 which is the amount of minutes in a day. And we get 6341.4181 days and final value of 547,898,523.36 seconds. Now, you're probably asking yourself we've been watching this video for a number of minutes. So even if I took the time to figure out the number of seconds by the time I did the calculation the number of seconds would be obsolete. The question how old are you in seconds is changing every second. And do we even believe in this 0.36, this 0.23 there? It's changing as we move forward from every minute. Every minute we're changing here so some of that stuff is not important. The main reason for this exercise is to look and talk about how the number changed as we were beginning to hone in to what reality was. In this case, we think reality is somewhere close to this value. How close? Well, it probably depends on how close we are in minutes. If we are really to the nearest minute a minute is 60 seconds so somewhere around here. But let's look and see which of our various numbers how these things change as we move forward. Up here, I made a guess to the nearest decade. That too, I'm not quite sure about but I know it's pretty close. It's a reliable number within one or two values of it. Okay? And if we look here this number here, this 6 is different than the 5 down here but it's within one or two. So our reliable number here is this 6. Anything after it is kind of meaningless. I could have said 600 million seconds and been just as accurate. More or less. Then I got a little more careful. 17 years. So I knew the 1 was correct and I knew the 7 was close. Maybe it's 7, maybe it's 8, maybe it's 6. And when I did that you'll notice the 5 is the same all the way down. The 3 is the same. The 3 is now the one that's close. That's reliable but not perfect. This one is correct. That one is reliable but not perfect with a little underlining on it. This leads to a concept that I like to call the fuzzy digit. There is a number that we're not quite sure of and that we're going to it's going to be an estimate of what reality is. And in this case over here I was sure about this one. This one's a little fuzzy. My second digit is fuzzy. Over here my second digit is fuzzy. I get a little bit better. I'm certain about these two and this 5 is fuzzy. I do my calculation. I look over here. Well there's my 5 there. This 5 is not the same as the 4 but if we round them it's the same thing. So we're talking about something that's very close but also even better than close. It's close. It's the same if we round them. So this is the one that if I change this if I move this down a couple notches it would be 549 and then we'd be exactly the same. I know it's a little hand wavy but that's the idea here is that if I change this by a couple the other two would be pretty certain and this one is the fuzzy one. I get a little more precise. Let's look here. Now the 5 and the 4 are good. The 6 is still a little bit fuzzy but then I get a little closer. Now the 5, the 4 and the 7 are good and the one that's a little bit off is the 7 and as I continue down this line you can start seeing that I'm beginning to approach now the 8 is the same and this 8's a little bit off. The 8 is the same. Maybe that 7 is a little bit off. Now we have 8, 9, 7. The 9 is the same but that 7 is a little bit off. As we keep moving we have more faith more reliability in the numbers that we see. More and more digits are more and more reliable if we're trying to compare to what we consider to be the correct answer correct as we can get it. How reliable is this itself? Well who knows but if we're talking about being to the closest minutes we're probably pretty reliable if we're talking about 500 seconds, we're probably within 100 seconds of what we're talking about if we are close to the right minute. So I use this demonstration as the first piece of a discussion of significant figures. We'll talk a little bit more about some of this discussion in the second video.