Scientific Notation and Reporting a Result

Transcript

0:00Introduction

Hello everyone! Okay, one small question before we start. The mass of the Earth is about five, nine, seven, and then zero, zero, zero, zero, a long, long line of zeros, in kilograms. Imagine writing that in your exam paper. How many zeros? Did you miss one? And a deeper question: which of those zeros did anyone actually measure? Difficult to say, right? Today we fix both problems with one simple form, scientific notation. And then we learn how a physicist reports a result with its error. Now, JEE tests this directly, and it is also where students throw away easy marks. So, let's begin.

0:43Scientific notation

So, here is the form. Any number, call it capital N, is written as a times ten to the power b, okay? Notice the two parts: a number in front, and a power of ten behind it. The power of ten carries the size, so that long line of zeros simply goes away. Clear? Alright, now a question. Take four thousand five hundred. I write it as forty five times ten squared. The value is correct, right? Forty five hundreds make four thousand five hundred. So, is this proper scientific notation? Many students say yes, the value matches, done. But no! There is one condition on the front number. It must be at least one, and less than ten: one is less than or equal to a, and a is less than ten, okay? Forty five is more than ten, so it is not allowed. And zero point four five times ten to the four? Also not allowed, because now the front number is less than one. The proper way: four point five times ten cubed. Exactly one digit before the decimal point, and never zero. Always. Now the real magic of this form. The digits of a are exactly the significant figures. And the power of ten does not count at all. Six point zero two times ten to the twenty three, on screen, has three significant figures: six, zero, two. The ten to the twenty three adds nothing to the count. Got it? Remember four thousand five hundred from our lesson on counting significant figures? How many does it have? Two, three, or four. Nobody can tell, right? Trailing zeros in a number without a decimal point are ambiguous. Scientific notation removes the doubt completely. Four point five times ten cubed: two significant figures. Four point five zero times ten cubed: three. Four point five zero zero times ten cubed: four. Same size every time, but the front number tells you exactly how much was measured. With me?

2:52Reporting a result with its error

Now, the second half: reporting a result. When we quote a value with its error, we use this plus or minus sign. Suppose a student measures a time and writes: two point six two four, plus or minus zero point one zero eight, seconds. Looks very precise, right? So many digits. Is this a good report? Many students would happily write exactly this. But no, it is not. Look. The error is about one tenth of a second. So the value is already uncertain in the first decimal place. Then what is the four, sitting in the third decimal place, telling us? Nothing! It is lost inside the error. Those extra digits are not precision; they are just calculator noise. So, rule one. The error itself is quoted to one significant figure, at most two. Here we keep two. Zero point one zero eight: the first digit we drop is eight, more than five, so we round up. Zero point one one. Clear? Rule two. The value goes to the same decimal place as the error. Our error, zero point one one, ends at the second decimal place. So the value also stops at the second decimal place. Two point six two four: the digit we drop is four, less than five, so it stays two point six two. And here is a topper's habit: always round the error first. The error decides where the value stops, never the other way round. So the proper report is: two point six two, plus or minus zero point one one, seconds. Fewer digits, but every digit is honest. Nice and clean. And if an option in the exam shows all the calculator digits, don't get tempted, okay?

4:37Example 1: mass of the Earth in scientific notation

So, remember the mass of the Earth from the start? Let's finish it. The mass of the Earth is about five, nine, seven, followed by that long line of zeros, in kilograms, known to three significant figures. Write it properly. Pause the video, and try it first. Done? Okay, let's check. Three significant figures, so the front number gets exactly three digits: five, nine, seven. Put the point after the first digit: five point nine seven. Is it between one and ten? Yes. Perfect. Now the power of ten. Let's count the zeros on screen. After five, nine, seven, there are twenty two zeros. So, ten to the twenty two, right? No, no, careful! This is the trap. The decimal point sits at the very end of the number. To land just after the five, it must jump over all twenty two zeros, and also over the nine and the seven. Twenty two plus two: twenty four places. So b is twenty four, okay? So the mass of the Earth is five point nine seven times ten to the twenty four kilograms. See, mystery solved. Those twenty two zeros were never measured; they only showed the size, and now the ten to the twenty four does that job. Three digits in front, exactly the three significant figures. And five point nine seven zero? That would claim four significant figures, so it is wrong here.

6:05Example 2: Earth's radius to four and to two sig figs

Now a slightly harder one. The Earth's radius is six thousand three hundred and seventy one kilometres. Write it in metre, first with four significant figures, and then with two. Pause and try. Hint: change to metre first. Done? Okay, let's check. One kilometre is one thousand metres, that is, ten cubed metres. So the radius is six three seven one times ten cubed metres. Is this the final answer? Hold on, no. Look at the front number: six thousand three hundred and seventy one, way bigger than ten. Not allowed, right? So move the point three places to the left: six point three seven one. The front number became a thousand times smaller, so the power goes up by three: ten cubed becomes ten to the six. Six point three seven one times ten to the six metres. Four digits in front, four significant figures. Done. Now with two significant figures. Keep two digits, six point three, and just cut off the rest? No, no, careful! Cutting is not rounding. Remember the rounding rule: look at the first digit you drop. It is seven, more than five. So the last kept digit goes up by one: three becomes four. Six point four. Six point four times ten to the six metres. Notice, the power of ten did not change; only the front number lost digits. Now imagine writing it in long form: six million four hundred thousand metres. Could anyone tell it has only two significant figures? Not at all. That is exactly why we use scientific notation.

7:50Quick revision: scientific notation and reporting the error

So, quick revision. Write any number as a times ten to the power b, with the front number at least one and less than ten. The digits of that front number are exactly the significant figures. And when you report a result, round the error to one, at most two, significant figures, and stop the value at the same decimal place as the error. Now go and try the practice problems on this page. Only three, and you will finish them in no time. All the best!