Rounding Off

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. Round two point seven four five to three significant figures. Easy, right? The dropped digit is five, and five always rounds up, so two point seven five. Now round two point seven three five. Two point seven four. But look, by the rules we use for measurements, the answer for both of them is two point seven four! Wait, how? Why did the first five not go up? Think about it. Today we will learn the three rules of rounding off, and that last rule is exactly where students lose marks. In JEE, whenever you write an answer to the right number of significant figures, you round it off. So, let's begin.

0:47Look at the first digit you drop

So, how does rounding off work? First, decide how many digits you need to keep. Then look at the first digit you are dropping, okay? Just that one digit, not the whole number, decides what happens to the last digit you keep. Clear? Let's try one. Two point seven four three, to three significant figures. We keep two point seven four, and the first dropped digit is three. So what happens to the four? Yes, nothing at all! Three is less than five. So, rule one. If the first dropped digit is less than five, leave the last kept digit unchanged. Two point seven four three becomes two point seven four. That is what the little arrow on screen means: becomes. Got it? Rule two. If the first dropped digit is more than five, increase the last kept digit by one. Two point seven four six: the dropped digit is six, more than five, so the four goes up to five. Two point seven five. Really simple.

1:45Five, with something after it

Alright, now a slightly tricky one. Two point seven four five one, to three significant figures. We keep two point seven four, and the first dropped digit is five. Is it less than five? No. More than five? Also no. So what do we do? Now look at what comes after that five. There is a one. So rule two has a second part. If the dropped digit is five, followed by any non-zero digits, we increase the last kept digit by one, okay? Five followed by one is a little more than five, right? So two point seven four five one becomes two point seven five. Remember this: if anything non-zero hides behind that five, round up. So far so good? Now comes the real twist.

2:31When the dropped digit is exactly five

Now, our puzzle. Two point seven four five. We keep two point seven four, and we drop a five, with nothing after it. So round up, two point seven five, right? No, no, careful! This is the JEE trap. Rule three. When the dropped digit is exactly five, with nothing but zeros after it, make the last kept digit even, okay? Even means zero, two, four, six or eight. In two point seven four five, the last kept digit is four. Four is already even. So just keep it: two point seven four. And two point seven three five? The last kept digit is three, and three is odd. To make it even, round up: three becomes four. Two point seven four. See? Puzzle solved. Both numbers give two point seven four. Absolutely correct! But why such a strange rule? Think about it. If an exact five always went up, our rounded values would always lean a little upwards. With the even rule, half the time we round up, and half the time we round down. So there is no bias. Got it?

3:36Topper's trick: three questions, in order

Now let me share how toppers do this in the exam, with zero confusion. They ask three questions, in order. One: is the first dropped digit less than five? Then keep the last digit as it is. The double arrow means, then do this. Two: is it more than five, or a five with some non-zero digit after it? Then round up. So always read the whole tail after the five, not just the five. And three: is it exactly five? Then remember one short line: odd goes up, even stays. Three questions, and rounding becomes automatic. With me?

4:14Example 1: round five numbers to three significant figures

Now, let's practise. Round each of these to three significant figures. Four point three six five seven. Zero point zero four two three eight five. Two point eight four five. Two point eight five five. And three point one two five zero one. Pause the video, and try it first. Done? Okay, let's check. First, four point three six five seven. We keep four point three six, and the first dropped digit is five. Six is even, so keep it: four point three six, right? No, no! Read the whole tail. After the five there is a seven, a non-zero digit. So this is not an exact five. Round up: four point three seven. Next, zero point zero four two three eight five. Remember, leading zeros are not significant, so the three significant figures are four, two and three. The first dropped digit is eight, more than five. So round up: zero point zero four two four. Now two point eight four five. Exactly five, with nothing after it. The last kept digit is four, which is even, so keep it: two point eight four. And two point eight five five. Again exactly five. But now the last kept digit is five, which is odd. So round up: two point eight six. See how the even rule works both ways? Last one, three point one two five zero one. The five is followed by zero one. Is that an exact five? No! There is a one hiding in there, a non-zero digit. So round up: three point one three.

5:54Example 2: round 0.0004567 and 1.995 (it gives 2.00)

Now a slightly harder one, JEE level. Round zero point zero zero zero four five six seven to two significant figures. And round one point nine nine five to three significant figures. Pause the video and try it. Hint: watch the zeros carefully. Done? Okay, let's check. First one. The leading zeros are not significant, so the first two significant figures are four and five, okay? The first dropped digit is six. The sign on screen means greater than: six is more than five. So the five goes up to six. Zero point zero zero zero four six. And notice, we write all the zeros before the four exactly as they were. Now one point nine nine five, to three significant figures. We keep one point nine nine, and the dropped digit is exactly five. The last kept digit is nine, and nine is odd. So, odd goes up! Now the last nine goes up by one and becomes ten, so we carry. One point nine nine becomes two. So the answer is just two, right? No, no, careful! How many significant figures did the question ask for? Three, exactly! So we must write two point zero zero. Those two zeros after the decimal point are significant, and they show the three significant figures. And notice, this time the five did make us round up, because nine is odd. So five does not always go up, and it does not always stay. The even rule decides.

7:29Quick revision: the three rounding rules and the even rule

So, quick revision. If the first dropped digit is less than five, keep. If it is more than five, or a five with non-zero digits after it, round up. If it is exactly five, make the last kept digit even: odd goes up, even stays. And keep the zeros your answer needs: one point nine nine five becomes two point zero zero, not two. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!