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Rounding. To begin, we will discuss a couple ideas that may initially not seem to be related to the idea of rounding, but they will be important in that discussion. We need to consider the nature of decimals on the number line. So the first question, how many numbers are between 3 and 4? Think about this for a minute, on the number line, how many numbers are between 3 and 4?
Of course, there are no integers between 3 and 4, but that’s not the question. I’m not asking about integers, I’m asking about numbers in general. There are decimals between 3 and 4, how many of them? Well the answer of course is infinity! There’s an infinite number of decimals between any two integers. So that’s mind blowing fact number one.
Second question. How many numbers are between 3.99 and 4? Well again the answer is infinity. And then the next question, now we’re getting a bit ridiculous here. How many numbers are between three and a whole bunch of nines after it, looks like there are 20 nines after it, between that number and four?
How many decimals are in that tiny, infinitesimal little gap. Well, turns out, there’s still infinity, this is a really deep idea. Between any two numbers on the number line, no matter how close those two numbers are, there lies an infinity of points between them. That is a very deep idea about the nature of decimals and the nature of the number line.
This fact, this purple fact will not be tested on the GRE, but knowing this will give you important perspective in the discussion of rounding. Rounding, so now we know about rounding, let’s start very easy. Suppose we know rounding to the nearest integer and suppose there is only one digit to the right of the decimal point. The rule is very easy.
If the number in the tenth’s place is 0 to 4, then we round down. If the number in the tenth’s place is 5 through 9, then we round up, very straightforward. So here, we’ll select those, first of all, that are in the first category the 0 through 4 category, those all get rounded down. Now the other ones are in the five through nine category, they all have digits in the tenth’s place that are five through nine and of course those ones get rounded up.
So far, so good. Now, we add a twist. We are still rounding to the nearest integer, but now we could have any number of decimals to the right of the decimal point. Here’s the tricky part. We look only at the number at the tenth’s place and no other number.
If the number in the tenth’s place is zero to four, we round down. If the number in the tenth’s place is five to nine, we round up. The subsequent decimals, anything beyond the tenth’s place is absolutely irrelevant. Well, here’s what I mean by that. So of course most people are pretty clear on the fact, that, numbers like 7.12 or 7.384, those get rounded down and of course, 7.96, 7.532, those get rounded up to 8.
Most people get that, but where it gets tricky is when we have a 4 in the tenth’s place with more digits behind it. So for example, 7.45, that has a 4 in the tenth’s place, so that gets rounded down to 7, 7.49, gets rounded down to 7. 7.49999 gets rounded down to 7, now this starts to confuse people, because people look at that and they say, well jeez, 7.49999, that’s essentially 7.5 but we have to be very careful here.
How many decimals separate 7.49999 from 7.5? Well as we just talked about, there’s an infinity of decimals separating them and because there’s an infinity of difference, one gets rounded down but 7.5 would get rounded up. As you might infer, a very common mistake and here’s the mistake I’m trying to target.
The mistake of double rounding. People look at something like 6.48 and we have to round to the nearest integer. And so they do this they say, oh well, first of all let’s look at that 8, that 8 gets rounded up, so the 4 gets rounded up to 5 and now the 5 has to round up the 6 to 7. And so what we’ve done is we’ve done rounding in two stages here, this is 100% incorrect t his is not what you are supposed to do.
Instead, you have to look at that four, the eight is irrelevant, look at the four, the four gets rounded down, and so 6.48 gets rounded down to six. Very important to avoid double rounding, which is a very common mistake. Now we are ready to generalize. The GRE could ask you to round to any decimal place. The nearest hundreds, the nearest thousandths, etc.
When you are asked to round to a place, look only at the digit in the next place to the right, the next smallest place. The same rule as above, 0 to 4, round it down, 5 through 9, round it up. For example, 65,536. If we’re gonna round this to the nearer thousand, well, we look at the number in the next place, the hundreds place.
That number is a 5, so we’re gonna round up and it’s, that means it gets rounded up to 66,000, that’s the nearest thousand. So here’s some practice ones and I’d suggest pause the video here and practice these on your own. And here are the answers.
In this video, we discussed the rules of double-rounding and covered the mistake of double rounding
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