# Rounding to the Nearest: The Basics

We frequently get questions about how to round; so many different issues arise that I won’t try to fit them all into one post. Children have trouble learning how to do it, and sometimes their parents are surprised to find that they are being taught a different way than they learned. There are several common misconceptions and errors. And those are just the basics. Some other time I will get into issues of why we round at all, how we decide how to round (up, down, nearest), and even how to round negative numbers. So let’s get started.

## Why look at the next digit?

The first thing to tell a student is that rounding to the nearest whole number (or nearest tenth, or nearest hundred, or whatever) means just what it says: making a round (approximate) number that is the nearest of its type to the given number. From that definition, we move on to procedures for finding that number efficiently, which is a separate issue. So many students seem to learn only the rules, and not the meaning, so that when they mix up the rule, they can end up with a nonsensical number — one that is not at all near the target, or that is not the type of number they are aiming for, for example.

Here is a question from Keith in 2006 where I got into the main issues:

Different Techniques for Rounding Off Decimal Numbers

I am a little split on the different ideas of rounding decimals.  I understand that in basic math they tell you to take the decimal place just to the right of what you are going to round to (ex: 1.45 to the tenth is 1.5).

However what I was taught in advanced mathematics is when you are taking a number that has multiple numbers beyond the decimal place that you should take the complete number in account when rounding.  (ex: 1.4457 would round to 1.5 or 2.0).  I don't know if it is just the way different teachers teach or maybe it is a different form of math that involves this kind of rounding.  Could you help clarify?

There were several different things he might be confused about, so I chose to give him an overview of what rounding is, starting with the definition:

First, the essential idea in rounding is stated in the words we use when we state a problem fully:

Round 1.45 to the nearest tenth.

This means

Find the multiple of 0.1 that is closest to the number 1.45.

If we just do exactly what this says (which does, indeed, take the entire number into account), then we can't help getting the right answer--if there really is one!  The two nearest multiples of 0.1 are the tenth above and the tenth below, namely 1.4 and 1.5.  If our number had been, say, 1.445, we could find how far it is from 1.4 and from 1.5, and choose 1.4 because that is, indeed, closer.  In the case we are considering, however, we find that both numbers are exactly the same distance from 1.45, so there really is NO correct answer.

So if we want to have one "correct" answer, we have to arbitrarily choose one of the two.  This is where trouble comes in: we have a choice, and we might make different choices depending on our concerns.

What we see here is that, first, using the definition directly is a little laborious, so we need a simple procedure; but, second, there are cases (when a number is exactly halfway between two nearest numbers) when the definition doesn’t give us an answer, and we have to make an arbitrary choice (unless we chose to say the number couldn’t be rounded!). This means that the simple procedure we invent may not be the same as someone else might prefer; it depends on our criteria for the arbitrary choice.

In teaching children, and in using rounding for simple purposes such as estimation, we want the simplest possible method.  A reasonable way to do it is this: Any number between 1.4 and 1.45 will round down; any number between 1.45 and 1.5 will round up.  The first group all have the NEXT digit less than 5 (0, 1, 2, 3, or 4); the second group all have their next digit five or more (5, 6, 7, 8, or 9).  If we arbitrarily choose to round 1.45 up, then the rule becomes very simple: ANY number with a 5 in the next digit will round up.  This rule gives the correct answer whenever there is one correct answer, and gives a valid answer when there are two.

This, in my experience, is the usual elementary method taught in America, and probably in many other places. We have often been asked why we “always round up on 5”, and the reason is just that this produces the simplest possible rule to teach — not that it is the only correct thing to do. Note that the definition “takes the complete number into account”, in Keith’s words; but this procedure only needs to look at the next digit in order to accomplish the same result.

## Is that the only way?

There are times when we have reason to do something other than what is simplest:

In some settings, we care about the statistical properties of our rounding; we don't want to skew averages by always rounding up in this odd situation, so we'd like to round up half the time and round down half the time.  One common solution is to always round such "exactly between" numbers so that the last digit is EVEN.  Then our 1.45 will round to 1.4 rather than to 1.5.  Note that this rule is identical to the other in all cases except exact halfway numbers like 1.45; both methods give the same answer when there is only one valid answer.  It makes a different choice in the special case.  And it "looks at the entire number" only in the sense that it checks whether there are any other digits following a 5.  If there are, then we follow the basic rule; if not, we round to an even number.  But there is nothing else beyond the next digit that matters.

Even this method can lead to biases (for example, it will lead to too many even answers!); so if that matters, you might need to just randomly decide whether to round halfway numbers up or down.

So this method is just a trick that can help in some special statistical situations; it is not a cure-all. And it is not, as some people tell us, the only right way because it is what they were taught.

Here is an example, from 2001, where Debbie asked about the “round-to-even” rule, and turned out to have missed the most important detail:

Rounding Decimals: Even/Odd Issues

This is a strange question... I remember learning a way to round certain numbers, but now I think I am going crazy!  Please help - here is what I remember:

When an even decimal (or any even number) is followed by a 5, you round down. When an odd decimal is followed by a 5, you round up.  For example: 75.45 = 75.4, but 75.55=75.6. Or rounding to the nearest 10: 145 = 140, but 155 = 160.

Can you let me know if I learned the "rule" incorrectly?  It seemed from the info on your page, I did learn it incorrectly; but it has persisted in my memory!  As you've said on your page, statistically we should "round down the first 5 (1-4) and round up the last 5 (5-9)."

She had seen what we had written to students who asked about the simple method (whom we hadn’t told that there was any other way). I explained the round-to-nearest-even rule, as I did above, and also referred to a previous answer we had given in 1999 to a similar question to hers (Rounding Up or Down on a 5).

Debbie wrote back:

Thanks very much! Now I know I've not gone completely nuts! But the caveat you suggest is one I do overlook, and should be considered ("for example, you wouldn't round 75.451 down to 75.4, because there is another digit beyond the 5, and 75.451 is closer to 75.5 than to 75.4.").

And thanks for replying so quickly.  I've bookmarked your page - even though I teach college students, sometimes they need to be reminded of stuff they already learned!

The fact that this detail is easily overlooked is one reason we usually teach only the simple method to kids: there’s less to pay attention to. (And note that teachers need to be reminded, too. That includes me.)

## Why not look beyond the next digit?

Now let’s look at a different issue involving what digits we should look at:

Rounding 3.445 to the Tenths Place

In my daughter's 6th grade math class, they are told to address the digit to the right of the place being rounded. For example, 3.445 rounded to the tenths place, would be 3.4, since the number to the right of the tenth place is less than 5.

However, doesn't the presence of the 5 in the one-thousandths place round the 4 hundredths to 5 hundredths, which in turn would round the 4 tenths to 5 tenths?

Doctor Rick and I gave supplementary answers to this. He explained the reason for the method (in a little more detail than I did above, if you want to see it done right), showing why there is no need to look beyond the next digit. I went one step further, and emphasized it is actually wrong to look beyond it:

The problem here is that you have to round all at once, not one digit at a time. Rounding twice, to different digits, doesn't do what you would think it would.

Here's what happens: Since 3.445 is closer to 3.4 than to 3.5, it must round to 3.4; the border between 3.4 and 3.5 is at 3.45, and 3.445 is below that. But if you first round it to the nearest hundredth, it becomes 3.45, moving it from "below the border" to "right on the border" and allowing a second rounding to move it "over the border" to 3.5.

It's as if the border patrol were to decree that anyone within ten feet of the boundary fence should be considered to be on the fence; and then said that anyone on the fence should be arrested for illegal entry. That wouldn't be right, since people ten feet outside of the country would be treated as if they were inside!

In case it isn’t clear what I am saying, here is the picture I had in mind:

                    3.445 :
\/ :
+----+----+----+----+----+----+----+----+----+----+
3.4  3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.5

I have marked 3.445, which clearly is to the left of the border between 3.4 territory and 3.5 territory. The nearest tenth is clearly 3.4. But if we first rounded it to the nearest hundredth, 3.45, well, that is the boundary for the nearest tenth, and by our standard rule, when we now round to the nearest tenth, anything on the boundary rounds up to 3.5.

## What if the result makes no sense?

Let’s look at one more issue. Another teacher, Susan, in 2000, asked this:

Rounding Down to Nothing

In my third grade class, we are working on rounding numbers. A question came up about rounding the numbers 0, 1, 2, 3, and 4 to the nearest 10. The nearest 10 would be zero, but that seems to say that you would be "rounding" down to nothing. That seems inaccurate, since as you do have "some." Would you round down to 1? Would you consider negative numbers here? We can't seem to find information on this subject.

It just felt wrong to them: How could rounding change something to nothing? Well, this is where a central feature of math, making definitions and sticking to them, comes to the rescue. Even if it feels wrong, we have to follow the rules. But it helps when we know why the rule makes sense:

The problem with rounding small numbers down to zero is not that the rounding itself is wrong, but that one would not ordinarily want to do it. As you say, rounding loses a lot of accuracy - in fact, it loses all the information you have. For precisely that reason, we would rarely round a number that way.

For example, suppose you measured the height of everyone in your class, and got numbers like, say, 1.234 meters. If I asked you to round them to the nearest ten meters, you'd probably question my choice, since they would all round to zero. Even if we round to the nearest meter, we'll lose all our information, since all the numbers will be 1. Instead, we would probably choose to round to the nearest centimeter, in order to avoid losing data.

But if you were measuring the heights of mountains, with some numbers in the kilometers and others (say, in Delaware) only a few meters, then rounding to the nearest ten meters would make sense, even if some "mountains" (sand dunes?) rounded to 0. You would still have useful information; the zero would tell you a lot about the height compared to real mountains.

... If the result is zero, it's not the answer that's wrong, but the question.

Context is everything. You choose to round in a certain way for a reason; if you round to the nearest ten meters, then it must be because not everything you are measuring will round to zero. But if this means your height rounds to zero, you do it, because any other answer would not really be the nearest ten meters! In math, we don’t lie just to avoid hurting a number’s feelings; truth matters.

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