Rounding to a nearest neighbour, and choosing which one

0:00Here is a number. Six crore, seventy two lakh, eighty five thousand, one hundred and eighty three. Now I ask you for the nearest round number. And exactly as I asked it, that question has no answer. It depends entirely on how round you want it to be. The nearest thousand is one answer. The nearest ten thousand is another. Then lakh, then ten lakh, then crore. Five questions, and five separate answers.

0:31So a number does not have a nearest neighbour. It has five of them, one for every place. And here is the part that catches people out. Those five do not line up behind each other. Two of them, only one place apart, sit on opposite sides of the number. Let's do the easiest one first, and do it properly. The nearest thousand. Our number sits between two multiples of a thousand.

1:01Below it, eighty five thousand. Above it, eighty six thousand. That is the whole question. Which of the two is closer? It is one hundred and eighty three above the lower one. It is eight hundred and seventeen below the upper one. So the lower one wins, and the nearest thousand ends in eighty five thousand. Notice what we did not do. We did not chop the last three digits off.

1:30Chopping always hands you the one below. Sometimes the one above is closer, and chopping is simply wrong. Same number, next place up. The nearest ten thousand. Now the two candidates are eighty thousand and ninety thousand. Our number is five thousand one hundred and eighty three above the first. And four thousand eight hundred and seventeen below the second. So this time the upper one wins. The nearest ten thousand goes up.

2:03One place further out, and the direction has already changed. Keep going. The nearest lakh, where the candidates are seventy two lakh and seventy three lakh. We are eighty five thousand past the lower one, and under fifteen thousand short of the upper. So the nearest lakh goes up as well. Six crore seventy three lakh. Now the nearest ten lakh, and this is where it gets interesting. The candidates are six crore seventy lakh, and six crore eighty lakh.

2:38Our number is two lakh eighty five thousand above the lower one. And seven lakh fifteen thousand below the upper one. So the nearest ten lakh goes down. Six crore seventy lakh. Now read that against the line just before it. The nearest lakh sat above our number. The nearest ten lakh sits below it. One place apart, and they are on opposite sides of the thing they are describing.

3:06So these are not rungs on a ladder you climb. Every one of them is its own question. Let's finish the set. The nearest crore. Six crore is seventy two lakh below us. Seven crore is twenty seven lakh above us. So the nearest crore is seven crore, and it goes up. Here are all five together, with an arrow beside each one. Down. Up. Up. Down. Up. Three of them sit above our number and two sit below it.

3:38If rounding were just chopping digits off, every arrow would point down. They do not. Rounding is a comparison, not a deletion. And the list itself is not increasing, not decreasing, not sorted at all. So here is a shortcut that looks obvious and is not. You already have the nearest lakh. Why not round that again to get the nearest ten lakh? Try it on a fresh number. Six crore seventy four lakh ninety thousand.

4:12Asked directly there is no doubt. It sits four lakh ninety thousand above six crore seventy lakh, and five lakh ten thousand below six crore eighty. The lower mark is nearer, and that is the answer. Now the lazy way. Round to the nearest lakh and you land on six crore seventy five lakh, exactly halfway between those two marks. Halfway is the one position that cannot tell you which way to go.

4:43And that is not bad luck with this number. Whenever this shortcut fails, it fails in exactly this way. Rounding to a place lands you on a multiple of that place, and the halfway mark of the next place out is one of those multiples. So the shortcut can always reach the spot where nothing decides. Go back to the original, every time. Here is a question worth sitting with for a moment.

5:14Could a number have all five of its neighbours the same? Nearest thousand, ten thousand, lakh, ten lakh and crore, every one of them five crore. Yes. And not just one such number. Which of the five conditions is tightest? The nearest thousand. For five crore to be your nearest thousand, you have to be within five hundred of it. And once you are that close, the other four come for free. You are deep inside every wider window.

5:45So the answer is a run of one thousand consecutive whole numbers, sitting either side of five crore. Which thousand exactly, nobody can say. Five hundred away is a dead heat, and you have just seen that a dead heat decides nothing. Now let's put all of that to work, and stop calculating altogether. Add these two. Four lakh sixty three thousand one hundred and twenty eight, plus four lakh nineteen thousand six hundred and eighty two.

6:20Do not add them. Two people look at it and each say something. Ana says it is near eight lakh, and above it. Ben says it is near nine lakh, and below it. That sounds like a disagreement, and it is not one. Near eight lakh and above. Near nine lakh and below. Both can be true of the same number at once. And both of them are true here.

6:45So an estimate is not a guess. It is a claim about where the answer lives, and claims can be checked. If both are right, is there nothing to choose between them? There certainly is. The exact sum is eight lakh eighty two thousand eight hundred and ten. Ana said eight lakh. She is eighty two thousand eight hundred and ten away. Ben said nine lakh. He is seventeen thousand one hundred and ninety away.

7:16Ben is nearly five times closer than Ana is. Both correct, one far more useful. Closeness is a completely different question from correctness. And here is the useful part. You can pin that sum down without ever adding it. Is it above eight lakh fifty thousand? It is, and here is the one line reason. The first is past four lakh fifty thousand, the second past four lakh. Those floors add to eight lakh fifty thousand, so the total has to clear it.

7:53Now squeeze it from the other side. This is the prettiest move in the topic. That second number, four lakh nineteen thousand six hundred and eighty two, is just short of four lakh twenty thousand. Short by three hundred and eighteen. So swap it out. Add four lakh twenty thousand instead, and the total is eight lakh eighty three thousand one hundred and twenty eight. And because we replaced one number with something bigger, the real sum has to be below that.

8:26Below it by exactly three hundred and eighteen, in fact. The same amount we handed it. So the sum is trapped between eight lakh fifty thousand and that ceiling. A window about thirty three thousand wide, from a gap of a whole lakh. And we still have not done the addition. The same move works on a subtraction, and the two of them swap places. Fourteen lakh sixty three thousand one hundred and twenty eight, take away four lakh ninety thousand and twenty.

9:01Ana says near ten lakh and below it. Ben says near nine lakh and above it. Both of them right again. The answer is nine lakh seventy three thousand one hundred and eight. But this time it is Ana who is closer. Twenty six thousand out, against Ben's seventy three thousand. So being the better estimator is not a property of a person. It is a property of the question.

9:29And the tightening works here too. Take away five lakh instead, and you would have exactly nine lakh sixty three thousand one hundred and twenty eight left. We take away less than five lakh, so the true answer sits above it. Pinned from both sides, still nothing subtracted. Here is one to try. Take any two six digit numbers, trap their sum between two round numbers without adding, then add them and see how close you got.