Significant Figures and the Rules for Counting Them

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Your friend measures a length as two point three zero eight centimetres. You write the same length in metres: zero point zero two three zero eight metres. Look, your number has more digits. So tell me, did you just become more precise, only by changing the unit? Hmm, that sounds too easy, right? Today we settle this with the rules for counting significant figures. And in JEE, measurement questions test these rules again and again. So, let's begin.

0:36What is a significant figure?

So, first, what is a significant figure? When you measure something, some digits you know for sure, and the last one is a small guess, okay? The significant figures are all the digits known reliably, plus the first uncertain digit. One guess, and then you stop. Let's try it. Take a metre scale, least count one millimetre. You read a length of twenty three point four centimetres. Now tell me, which digit is the guess? Yes, the last one, the four! The two and the three are sure, and the four is the estimated digit. So this reading has three significant figures. This double arrow on screen just means therefore, okay? And why do we care? More significant figures means a more precise measurement. So the count tells everyone how precise your reading is. Clear?

1:29Rules 1 to 3: non-zero digits and zeros

Now the rules. There are seven, but don't worry, each one is simple. Rule one: every non-zero digit is significant. So two point three four has three significant figures. This little arrow on screen points to the count, okay? Rule two: zeros sandwiched between non-zero digits are significant. In two point zero zero four, both zeros sit between the two and the four, so they count. Four significant figures. Alright, now a tricky one: zero point zero zero two seven zero. Six digits on paper. So six significant figures, right? No, no, careful! Rule three: leading zeros, the zeros before the first non-zero digit, are not significant. So we skip the three zeros in front. What remains is the two, the seven, and the last zero. Three significant figures. And why does that last zero count? That is our next rule.

2:31Rules 4 to 6: trailing zeros and powers of ten

Rule four: trailing zeros, the zeros at the end, are significant when they come after a decimal point. So four point five zero zero has four significant figures. And zero point three zero has two: the leading zero is out, the three and the end zero are in. Now tell me: four thousand five hundred, with no decimal point. How many significant figures? Two? Four? Nobody can tell! That is rule five. Without a decimal point, trailing zeros are ambiguous. Four thousand five hundred may have two, three or four significant figures. So how do we remove the doubt? Scientific notation. Four point five times ten to the power three: two significant figures. Four point five zero times ten to the power three: three. Four point five zero zero times ten to the power three: four. Now there is no doubt at all. Nice and clear. And rule six: the power of ten does not affect the count. Six point zero two times ten to the power twenty three has three significant figures: six, zero, two. The power of ten only sets the size, okay?

3:45Rule 7: exact numbers, and a topper's trick

Rule seven is different. Quick question: in a dozen eggs, did anyone measure that twelve? No, you just counted it! Some numbers are not measured at all; they are exact. And exact numbers have infinitely many significant figures. For example, objects you count, like the twelve eggs in a dozen. Defined conversions, like one metre equals one hundred centimetres. Or the two in two pi r. Nobody measured that two, right? So exact numbers never limit the precision of a result. Now let me share a trick that toppers use. Rules one to four, packed into one shortcut: if the number has a decimal point, start at the first non-zero digit, and count every digit till the end. Try it on zero point zero zero two seven zero. Start counting at the two, so we count two, seven and zero. Three significant figures. That's it, done in one second. Got it?

4:45JEE trap: changing the unit

Now let's go back to our puzzle. Two point three zero eight centimetres is twenty three point zero eight millimetres, or zero point zero two three zero eight metres. The metres version has more digits. So, is it more precise? No! Look at those zeros in front. They are leading zeros, so they don't count. Start at the two: two, three, zero, eight. Four significant figures, same as before. And in micrometres? This symbol, mu, means micro. It is two point three zero eight times ten to the power four micrometres. Again, four. Every time, four significant figures. Changing units never changes the number of significant figures. So no, you cannot become more precise just by changing the unit. Puzzle solved! But here is the JEE trap. Students write twenty three thousand and eighty micrometres. But wait, that number has no decimal point, so its last zero is ambiguous, rule five! Always write two point three zero eight times ten to the power four. With me?

5:55Example 1: count significant figures in six numbers

Okay, Example one. Count the significant figures in these six numbers. Zero point zero zero one zero. One point zero zero one zero. One hundred point zero. Three point zero zero times ten to the power eight. Five times ten to the power minus three. And four point five zero zero times ten to the power three. Pause the video, and try it first. Done? Okay, let's check. Zero point zero zero one zero: start at the one. One, zero: two significant figures. One point zero zero one zero: the zeros between the ones count, and the end zero is after the decimal point. Five. Now one hundred point zero. Only one non-zero digit, so one significant figure, right? Hmm, careful. There is a decimal point, so use the trick: start at the one and count till the end. One, zero, zero, zero: four significant figures, absolutely. That little point zero at the end changes everything. Last three, in scientific notation. Remember, the power of ten does not count. Three point zero zero: three. Five: just one. Four point five zero zero: four. See, once you know the rules, it is just counting.

7:17Example 2: exact speed of light vs measured G

Now a slightly harder one, JEE level. Since twenty nineteen, the speed of light, c, is fixed as exactly two hundred ninety nine million, seven hundred ninety two thousand, four hundred fifty eight metres per second. How many significant figures does it have? And the measured value of the gravitational constant, capital G, is six point six seven four times ten to the power minus eleven, in newton metre squared per kilogram squared. How many here? Pause and try. Done? Okay, let's check. Start with the speed of light. Nine digits, and not a single zero. So nine significant figures, right? No, no, careful! Read the question again. Look at the word exactly. This value is fixed by definition, not measured. So, just like the two in two pi r, it is an exact number, with unlimited significant figures. It never limits the precision of your answer. Now capital G, which is measured, so our rules apply. The power of ten does not count, so look only at six point six seven four. Six, six, seven, four: four significant figures. And notice, the minus eleven in the power has nothing to do with it. Got it?

8:32Quick revision: zeros, powers of ten and exact numbers

So, quick revision. Leading zeros never count, and trailing zeros after a decimal point always do. For a number like four thousand five hundred, use scientific notation, and remember, the power of ten never counts. And changing units never changes the count, while exact numbers have unlimited significant figures. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!