Scientific Notation and Significant Figures

Transcript

0:00Introduction

Hello everyone! Let's start with one small puzzle. You add three measurements: twelve point one one, eighteen point zero, and one point zero one two. Your calculator says thirty one point one two two. But the correct answer is thirty one point one. The last two digits are simply thrown away! Why? More digits means more accurate, right? Well, not always! Today you will see why those extra digits mean nothing. And remember, every numerical you solve in chemistry, JEE included, must be reported with the right number of digits. So, scientific notation and significant figures. Let's begin!

0:43The big idea: every measurement is uncertain

So, first the big idea. Every measured number carries some uncertainty, okay? Chemists handle this in two ways. First, scientific notation: we write the number as capital N, into ten to the power small n, where capital N is greater than or equal to one, and less than ten. Clear? For example, zero point zero zero four eight becomes four point eight into ten to the power minus three. And two lakh thirty four thousand becomes two point three four into ten to the power five. Just one non-zero digit before the decimal point, every time. Got it? Second, significant figures. We keep only the digits we are sure of, plus one uncertain digit. Those are the significant figures. That's it, that is the whole definition.

1:33Counting significant figures

Alright, now how do we count them? Rule one: every non-zero digit counts. Two hundred and eighty five has three significant figures. Easy. Rule two: zeros sitting between non-zero digits also count. So two point zero zero five has four, okay? Now tell me. Zero point zero zero three four. I can see five digits there, so five significant figures, right? Oh no, careful! Only two. Leading zeros, the zeros in front, never count. Only the three and the four are significant. See, this is where many students lose easy marks. Trailing zeros, the zeros at the end, are different. To the right of a decimal point, they count. So zero point two zero zero has three, and two point five zero also has three. But in a number without a decimal point, they are not counted. So one hundred has just one significant figure! Put a decimal point at the end, the little dot you see after one hundred, and now it has three. Even better, write one point zero zero into ten to the power two. Now everyone can see three significant figures. Crystal clear.

2:47Exact numbers and scientific notation

Now, two special cases. First, exact numbers: counted objects, defined values like one metre equals one hundred centimetres, and the coefficients in a formula. Now think: did anybody measure these? No! Nobody measures them, so there is no uncertainty at all. They have infinitely many significant figures, so they never limit your answer. Keep this in mind for the second example, okay? Second, in scientific notation only the digits of capital N count, never the power of ten. So six point zero two two into ten to the power twenty three has four significant figures. Simple, right?

3:28Significant figures in calculations

Alright, now calculations, the most important part. When you add or subtract, look at decimal places. The answer keeps as many decimal places as the number with the fewest decimal places, okay? But when you multiply or divide, look at significant figures instead. The answer keeps as many significant figures as the number with the fewest. Quick check. Two point five into one point two five. The calculator says three point one two five. How many figures do we keep? Multiplication, so count significant figures. One point two five has three, but two point five has only two, so the answer also gets only two: three point one. Look, students mix up these two rules all the time. Decimal places for adding and subtracting, significant figures for multiplying and dividing. Got it?

4:22Rounding off, and the round-once rule

Now, rounding off. Look at the first digit you drop. More than five? Raise the last kept digit by one. Less than five? Just leave it. But what if the digit you drop is exactly five? Raise, or leave? Raise the last kept digit only when it is odd. The arrow on screen means rounds to. So one point two three five: the kept digit is three, odd, so raise it, one point two four. But one point two four five: the kept digit is four, even, so leave it, again one point two four! Interesting, right? And one big JEE trap here. Don't round at every step; that makes your final answer drift. Carry one or two extra digits through the middle steps, and round only once, at the end. With me?

5:13Accuracy and precision

One last idea before the examples: accuracy and precision. They sound the same, but they are not. Precision is how close repeated measurements are to one another. Accuracy is how close they are to the true value. So tell me, can a balance be precise, but still not accurate? Yes, it can! If it is wrongly calibrated, its readings stay close to one another, so it is precise. But they are all away from the true value, so it is not accurate. Makes sense?

5:43Example 1: significant figures in an addition

Now, remember our puzzle from the start? Let's finish it. Evaluate twelve point one one, plus eighteen point zero, plus one point zero one two, to the correct number of significant figures. Pause the video, and try it first. Done? Okay, let's check. First, just add: thirty one point one two two. That is the calculator's answer. Now, this is addition, so which rule? Decimal places! So which number has the fewest decimal places? Eighteen point zero, yes! Just one decimal place. So the answer gets one decimal place too. The first dropped digit is two, less than five, so leave it: thirty one point one. See, puzzle solved! Eighteen point zero was known only to one decimal place, so the calculator's extra digits were never real.

6:36Example 2: molar mass of water to correct sig figs

Now a slightly harder one, JEE style. Find the molar mass of water, H two O, taking hydrogen as one point zero zero eight and oxygen as sixteen point zero zero, with the correct significant figures. Pause and try. Hint: first check which numbers are exact. Right, water has two hydrogen atoms, so two into one point zero zero eight. And this two has only one significant figure, so the answer can have only one significant figure, right? No, no, careful! This is the JEE trap. This two is an exact count. We counted two atoms; nobody measured them. And exact numbers have infinitely many significant figures, remember? So two into one point zero zero eight is two point zero one six, with all four figures kept. Now add the oxygen. Two point zero one six plus sixteen point zero zero is eighteen point zero one six. And notice, we did not round in the middle. Round only once, at the end, okay? Addition, so decimal places. Two point zero one six has three, but sixteen point zero zero has only two, so we keep two. The double arrow on screen means therefore. The dropped digit is six, more than five, so the one becomes two: eighteen point zero two gram per mole. See how every rule came together in one question?

8:08Quick revision: zeros, add and multiply rules, rounding

So, quick revision. Zeros: in between always count, leading zeros never, and trailing zeros only with a decimal point. Add or subtract: fewest decimal places. Multiply or divide: fewest significant figures. And exact numbers never limit you; round only once, at the end. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!