CHAPTER 01.03 Review of Significant Digits

In this segment, we'll talk about reviewing significant digits which you have been exposed to in other classes as well. So, we have a number, like, for example, somebody says, hey, I got 257 dollars and 36 cents. So, in this case it makes perfect sense to use the decimal format or what we call as a fixed-point format, because we have so many digits, let's suppose three digits here for the integer part and two digits for the fractional part which stand for the cents and the integer part for the dollars. However, when we are talking about scientific and engineering calculations then things can become a little bit dicey. Let's look at, let's look at the thermal expansion quotient of steel for example it is given as 6.47 times, 6.47 micro inch per inch per degree Fahrenheit and if somebody said hey how many significant digits are here, we'll say six four, seven and yeah three significant digits. But if you were going to rewrite this number, let's suppose in the United States customary system of units and you are going to still use the decimal format, then what's going to happen is that you would write it like this; because 1 micro is 10 to the power minus 6, so, that's why you'll have five of these zeroes after the decimal point.

Imagine if somebody said hey, hey can you give me the value of alpha of steel within three significant, three decimal, within three decimal digits. So, you would say zero, zero, zero, zero here and that will be what will be the value of alpha of steel for three decimal digits. And you very well see that we cannot report our thermal expansion of steel as zero even when we're asking for three decimal digits. So, that's where the problem can take place. So, we've got to be very careful. So, one of the things which we can do here is that we can rewrite this number as 6.47 times 10 to the power minus 6 inch per inch per degree Fahrenheit and what's going to happen is that you'll have three significant digits being shown here, with the six being the first significant digit, four being the second significant, and seven being the third significant digit right there. And this particular format when you write it like this is called the scientific format or what we may also call this as the floating-point format. The reason why it's called a floating-point format is because this decimal point right here is floating from one place to another. It was here and now it is right there and that's why it's called the floating-point format. So, having said that, that is what is of concern when we talk about significant digits that we do understand what its significance is and what it means. So, what it means is that hey how many digits can we know reliably in a reported number, that's the bottom line about significant digits. But, how does it pose a problem for us is important to know.

Let's take another example, I live in Hillsborough county here in Florida, United States and if somebody, my friend asked me one day he said hey, hey how many people live in Hillsborough county in Florida and I said, I blurted out, I said that it is 1.5 million people and and you can very well see that hey, the number is not exactly this right! If I write it in the integer format, I'll get 1 5 followed by five zeros because my friend was not necessarily interested in knowing the exact number, but he wanted to just get a ballpark figure and in this case you can very well see that when I say 1.5 million, so, I can only maybe trust or say that hey I trust two significant digits right there, but I do not trust all of them. It's quite possible that hey, it s exactly 1.5 million but the actual population of my county here is 1471968 as per the 2019 data available to us. So, in that case now, it s not 1.5 million but as far as we know that all of these digits are significant and are correct so far as the knowing the population of Hillsborough county is concerned at that particular point. So, in this one we'll have seven significant digits, in this we have only two significant digits. But, if I want to show that hey this number has only two significant digits, I will write it as 1.5 times 10 to the power 6. If I want to show that all the digits are significant in this one, so then I will say 1.471968 times 10 to the power 6 and that's how I would represent the two numbers to show the correct number of significant digits.

Can you figure out how many significant digits are in your numbers? So, let's suppose we have 2.789 and this will also help us to figure out the rules of significant digits. So, if you look at 2.789, it has 4 significant digits guys. So, it has four significant digits and why is that so? Because all the non-zero digits are considered to be significant and it has all non-zero digits. Now, let s suppose somebody says 0.0439 and in this case, what we will have is any of the zeros which are to the left of the first non-zero number are not significant. So, this zero is not significant, this zero is not significant but this four, three and nine are significant. So, we have three significant digits here. So, you can see the rule there about that. Now I could have a number like 4.590 and now how many significant digits does it have? It has four significant digits here, we have four, five, nine and zero and someone can say hey why is zero the also a significant digit? Is because all the zeros which are to the right of the decimal point, which are being shown are considered to be significant. So, how about this number like 4008, 4008 has four significant digits, because all the, we have a non-zero number here and non-zero number here and we don't have to worry about the non-zero numbers which are in between. So, which brings us to the next one right here, so, that here in this case uh any non, any zeros which are between non-zero numbers they are significant. But, let's suppose if we have a number like 4208.07, what about in this case? In this case, we'll have six significant digits because all the zeros no matter where the decimal point is, they are between the two non-zero numbers here in this case 4 and 7. So, let's look at this number here 4000.0 and in this case what we'll have is we'll have five significant digits and why is the case as such? Is because all the zeros to the right of the decimal point are significant. So, which brings us again to another example like 4008.0, how many digits are correct? We'll have five significant digits here as well, because all the zeros to the right of the decimal point are considered to be significant.

So, having said or having talked about all these different rules here now the only one which is left over is a number like this; so somebody gives you 15000, you don't know exactly how many significant digits there are! So, in this case you can say there are two or maybe three or maybe four or maybe five significant digits. So, when somebody asks you how many significant digits are correct in this answer, you could, you should say two, three, four or five, not pick and choose based on what you think is correct.

But, this vagueness which we have now in these number of significant digits that can be addressed by using the scientific format or what we call as a floating point format, so both are the same just like fixed format is same as decimal format, scientific format is same as floating point format.

So, we can write it like this 1.5 times 10 to the power three, this will have two significant digits, one and five here. 1.50 times 10 to the power three that has three significant digits, the number is still fifteen thousand and then if you have 1.5000 let's suppose times 10 to the power three, that all the numbers which are in that 15000 are considered to be significant, you can write as 1.5 followed by three zeros and this will have five significant digits. And what all of this does is that it takes away all the vagueness of representing the number like this one, where it can have two, three, four or five significant digits based on this number right here. But if you want to signify exactly how many significant digits one can rely on, then you'll have to rewrite it in the scientific format.

So, all of the rules have been covered in the examples which I gave you, if you think that there is a number which you are unsure of put it in the comment statements, I will reply there and that's the end of this segment.