Comparing Large Numbers and Approximations

Transcript

0:00Introduction

Hello everyone! Okay, a quick contest before we start. On one side, ninety nine to the power fifty, plus one hundred to the power fifty. On the other side, just one number: one hundred and one to the power fifty. Which side is bigger? Two huge numbers against one, so the left side wins, right? Hmm, don't be so sure. These numbers have about a hundred digits each, so an ordinary calculator cannot settle it. The binomial theorem will. Questions like this do come up in JEE, so let's begin.

0:37Idea 1: a round number plus a little

So, let's warm up with a smaller contest. Which is larger: two to the power fifty, or ten to the power fifteen? The small raised number is the power, okay? Take a moment and decide. Most students say ten to the power fifteen, because ten is so much bigger than two. But don't guess; look for a round number. Two to the power ten is one thousand and twenty four: one thousand, plus twenty four. Very close to a round number, right? Now, fifty is ten times five. So two to the power fifty is two to the power ten, in brackets, to the power five. And that is one thousand plus twenty four, to the power five. With me? Now expand with the binomial theorem. First term: one thousand to the power five. Then five C one, times one thousand to the power four, times twenty four. The capital C with five at the top left and one at the bottom right is read five C one, and the dot means times. The three dots mean more terms. Now, are these other terms positive or negative? All positive, yes, exactly! So the sum is greater than its first term; that is the greater than sign. Greater than one thousand to the power five, which is ten to the power fifteen. So two to the power fifty is larger! Remember: when the extra terms are all positive, drop them, and you get an inequality. Clear?

2:05Idea 2: different powers? Divide.

Now, the second idea, and first, one fact we proved in Chapter one of this unit. Two is less than or equal to, one plus one over n, in brackets, to the power n, which is less than three. It holds for every n belonging to the double struck capital N, the set of natural numbers. Okay? Now look at this contest. One thousand to the power one thousand, against one thousand and one to the power nine hundred and ninety nine. One has the bigger base, the other has one extra power. Which one wins? Here the powers are different, so divide one by the other. Write one thousand and one to the power nine hundred and ninety nine as one thousand and one to the power one thousand, over one thousand and one. Then the ratio becomes one plus one over one thousand, to the power one thousand, all over one thousand and one. See our fact, with n equal to one thousand? Right? The top is less than three, by our fact. So the ratio is less than three over one thousand and one, which is less than one. Got it? Ratio less than one means the number at the bottom is bigger. So one thousand to the power one thousand is larger! If you picked the bigger base, that is the trap. That one extra power of one thousand is worth far more. Clear?

3:29Idea 3: approximations, term by term

Now the third idea: approximations. Here is the expansion, with n a positive integer. One plus x, to the power n, equals one, plus n C one x, plus n C two x squared, plus n C three x cubed, and so on. Now, if x is a tiny decimal, what happens to x squared and x cubed? They get tinier and tinier, yes! So the later terms soon stop mattering. The rule: keep adding terms, one by one, until the next term is too small to change the decimal places you need. But how many terms is enough? Let's see it in Example one. With me?

4:10Example 1: (1.002)^12 correct to four decimal places

Now, Example one. Find one point zero zero two, to the power twelve, correct to four decimal places. Hint: write it as one plus a small number. Pause the video and try it. Done? Okay, let's check. One point zero zero two is one plus zero point zero zero two. So we need one plus zero point zero zero two, to the power twelve. Simple, right? Now expand. Twelve C one is twelve, twelve C two is sixty six, and twelve C three is two hundred and twenty. So: one, plus twelve times zero point zero zero two, plus sixty six times zero point zero zero two squared, plus two hundred and twenty times its cube, and so on. Each bracket means times. Okay? Now, twelve times zero point zero zero two is zero point zero two four. So the answer is one point zero two four, that is, one point zero two four zero to four places, right? Oh no, careful! You stopped too early. Sixty six times zero point zero zero two squared is zero point zero zero zero two six four, right in the fourth decimal place! The next term is about zero point zero zero zero zero zero one eight; the wavy equals sign means approximately. It only touches the sixth decimal place. Clear? Now add them up: one point zero two four two six five eight. Later terms are far too small to matter. So, correct to four decimal places, the answer is one point zero two four three, not one point zero two four zero. One small term made the difference. Got it?

5:56Example 2: which is larger, 99^50 + 100^50 or 101^50

Now Example two, and this is our puzzle from the start. Which is larger: ninety nine to the power fifty plus one hundred to the power fifty, or one hundred and one to the power fifty? Hint: find the round number, and look at a difference. Pause the video and try it. Done? Okay, let's check. Ninety nine and one hundred and one: which round number sits right between them? One hundred, exactly! So let's look at one hundred and one to the power fifty, minus ninety nine to the power fifty. That is one hundred plus one, to the power fifty, minus, one hundred minus one, to the power fifty. Okay? Now expand both. The two expansions have the same terms, except that in the second one, the second, fourth and sixth terms, and so on, carry a minus sign. So when we subtract, which terms cancel? The first, third and fifth terms cancel, and the others double! So the difference is two times, in square brackets, fifty C one times one hundred to the power forty nine, plus fifty C three times one hundred to the power forty seven, and so on. All positive. Clear? Keep only the first term in the bracket and drop the rest, just like our warm up. The difference is greater than two times fifty times one hundred to the power forty nine. Two times fifty is one hundred, so that is one hundred to the power fifty. Beautiful, right? So one hundred and one to the power fifty, minus ninety nine to the power fifty, is more than one hundred to the power fifty. Move ninety nine to the power fifty across: one hundred and one to the power fifty is larger! One number beats the sum of two. With me?

7:47Quick revision: expand, divide, approximate term by term

So, quick revision. One: write the number as a round number, plus or minus a little, expand, and drop positive terms to get an inequality. Two: when the powers are different, divide, and use one plus one over n, to the power n, less than three. Three: for approximations, add terms until the next one is too small to matter. Never stop too early. Now go and try the practice problems on this page. All the best!