Order of Magnitude and Estimation

Transcript

0:00Introduction

Hello everyone! So, one small question before we start. Put your hand on your chest. Feel that beat? Your heart beats about seventy two times every minute. Now tell me: in a life of seventy years, how many times will it beat? Thousands? Lakhs? Crores? Look, you don't need a calculator to get close. Physicists answer such questions with one smart number: the order of magnitude. By the end of this video, you will answer this one yourself. And remember, in JEE, estimation questions are quick marks, but only if you know the exact rule. So, let's begin.

0:41Order of magnitude: the nearest power of ten

So, step one. Write the number N as small a into ten to the power b, okay? This is just the scientific notation you already know. Here a, the number in front, goes from one up to ten: it can be one, but never ten. That is what the two signs on screen say: one is less than or equal to a, and a is less than ten. Clear? Now, the order of magnitude. Is it simply the power b? Hmm, not always! It depends on the number in front. If small a is five or less, nothing changes. The order of magnitude is just b, okay? But if small a is more than five, we round up, to the next power of ten. The order of magnitude becomes b plus one. So five is the boundary. Write this down, it is important. So, in simple words, the order of magnitude is the power of ten nearest to the number. Not the exact value, just its size, in powers of ten. Really useful when you only need a rough answer.

1:44Five or 3.16? Two conventions

Now, one small warning. The rule you just learnt is the NCERT rule, with the boundary at five. But some books round on the logarithmic scale instead. They use root ten, about three point one six, as the boundary, in place of five. Root ten is the square root sign you see on screen, okay? So, do these two rules always disagree? Think about it. No! Only when small a lies between three point one six and five. Above five, both rules round up. Up to three point one six, both keep b as it is. Only that middle band is different. And in JEE questions, the NCERT rule is used, boundary at five. Just remember that, and you are safe. Got it?

2:32A feel for the scales: lengths

Alright, now let's build a feel for sizes. Toppers remember a few anchor values. Look at the lengths: a nucleus is about ten to the power minus fifteen metre, an atom ten to the power minus ten metre, and a human, ten to the power zero, that is, one metre. Going up: the Earth's radius is ten to the power seven metre, Earth to Sun is ten to the power eleven metre, and the observable universe, ten to the power twenty six metre. What a range, right? Quick question for you. How many orders of magnitude larger is an atom than its nucleus? Use the first line. Divide them: ten to the power minus ten, over ten to the power minus fifteen, gives ten to the power five. So yes, the atom is five orders of magnitude larger! And careful: five orders means a factor of ten to the power five, not five times, okay?

3:29A feel for the scales: masses and times

Now, masses. An electron is about ten to the power minus thirty kilogram, a proton ten to the power minus twenty seven kilogram, and a human, ten to the power two kilogram, that is, of the order of a hundred kilogram. And the giants: the Earth, ten to the power twenty five kilogram, and the Sun, ten to the power thirty kilogram. Quick one: how many orders of magnitude heavier is the Sun than the Earth? Thirty minus twenty five: five orders, exactly! Same trick as the atom and the nucleus, just divide the powers of ten. Now times. Light crossing a nucleus takes about ten to the power minus twenty three second. One period of visible light is about ten to the power minus fifteen second. And a day is about ten to the power five second, a human lifetime about ten to the power nine second, and the age of the universe, ten to the power seventeen second. Here is the topper's habit: after any estimate, compare your answer with these anchors. If it is way off, check your working. With me?

4:35Example 1: order of magnitude of c, electron mass and more

Now, example one. Find the order of magnitude of four famous numbers. The speed of light, three into ten to the power eight metre per second. The electron's mass, nine point one into ten to the power minus thirty one kilogram. The Earth's radius, six point four into ten to the power six metre. And Avogadro's number, six point zero two into ten to the power twenty three. Pause the video, and try it first. Done? Okay, let's check. Speed of light: small a is three. Three is not more than five, so we keep b. Order of magnitude: eight. Now the electron. Nine point one into ten to the power minus thirty one. So the order is minus thirty one, right? No, no, careful! This is the JEE trap. Look at the number in front. Nine point one is more than five, so we add one. And here is the second trap: minus thirty one plus one is minus thirty, not minus thirty two! So the order is minus thirty. And look, our table also said the electron is ten to the power minus thirty kilogram. Perfect match! Earth's radius: six point four is more than five, so six plus one, order seven. Again it matches the table, ten to the power seven metre. And Avogadro's number: six point zero two is more than five, so twenty three plus one, order twenty four. Four numbers, four answers: eight, minus thirty, seven and twenty four. That's it!

6:13Example 2: heartbeats in a 70-year lifetime

Now, our puzzle from the start, JEE pattern. Estimate the number of heartbeats in a lifetime of seventy years, at seventy two beats per minute, and give its order of magnitude. Pause and try. Hint: go step by step, from minutes to hours, to days, to years. Done? Okay, let's check. Seventy two beats every minute, and sixty minutes in an hour, so multiply by sixty. Twenty four hours in a day: times twenty four. Wow, that is already about one lakh beats every single day! Then times three hundred sixty five days, and times seventy years. Work it out, and you get about two point six five into ten to the power nine beats. That is about two hundred sixty five crore heartbeats! Now the order of magnitude. Which boundary should we use, five or three point one six? In JEE, five, the NCERT rule. And here, small a is two point six five, below three point one six and below five, so both rules agree anyway. Order of magnitude: nine. Now the topper's habit: check against the anchors. From our table, a human lifetime is about ten to the power nine seconds, and the heart beats a little more than once every second. So order nine makes perfect sense. Your heart beats about ten to the power nine times in a lifetime. Puzzle solved!

7:43Quick revision: the NCERT rule and the negative-power trap

So, quick revision. Write the number as small a into ten to the power b, with small a from one up to ten. If small a is five or less, the order is just b, okay? If it is more than five, the order is b plus one: the NCERT rule, the one JEE uses. And careful with negative powers: nine point one into ten to the power minus thirty one has order minus thirty, not minus thirty two. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!