When the decimals don't matter: our rounding to the nearest integer calculator comes in handy. With our simple tool, you will find:

  • How to round to the nearest integer: the simple rules;

  • The calculations to round to the nearest integer in the case of half-integer numbers; and

  • Neat and exhaustive examples.

What to do with that 0.50.5 won't be a mystery anymore: you will learn how to manage those always confusing numbers, or, at least, discover a tool that will do that for you!

What does it mean to round to the nearest integer?

Rounding to the nearest integer is a simple mathematical operation we apply to decimal numbers. Ok, what's a decimal number? A decimal number is a number where we can identify an integer part (11, 22, 33, etc.) and an excess (decimal part). We separate these two quantities with the decimal separator:

1.37 \textcolor{red}{1}\textcolor{black}{.}\textcolor{blue}{37}

In the example above, the integer part is red; the fractional excess is blue. We often need to get rid of the decimal part (say that we don't need it, or as we know from the significant figures calculator, that precision would be too high). The result of the calculations to round to the nearest integer is an integer number, as you can understand from the name. But what are the rules we use to do so?

How do I round to the nearest integer?

The following rules apply to every interval between two pairs of contiguous integers (e.g. 11-22, 1010-1111, 9-9-8-8, etc.). In any such interval, you can find infinite decimal numbers. For most of them, the calculations to round to the nearest integer are pretty straightforward. Consider the generic integer number ii. Here are the possible situations you can meet when considering the decimal number nn:

  • For n=i.0n = i.0, we round to ii;

  • For n > i.0n\ \text{\textgreater}\ i.0 and n < i.5n\ \text{\textless}\ i.5, we round to ii (you can simply erase the decimal part);

  • For n=i.5n=i.5, check the next section;

  • For n > i.5n\ \text{\textgreater}\ i.5 and n < i+1.0n\ \text{\textless}\ i+1.0 (where i+1i+1 is the nearest higher integer), we round to i+1i+1; and

  • For n=i+1.0n=i+1.0, we round to i+1i+1.

You can see a symmetry, with similar behaviors separated by the value i.5i.5. Before checking what to do in the case of half-integer numbers, we can see some examples of the situations we listed above.

How to round on a line
In this image you can see how to round to the nearest integer. We will talk about the red .5.5 in the next section.

Take 4.84.8. Since it falls in the second half of the integral, we round it to 4+1=54+1 = 5. What about 5.05.0, then? Simply cut the .0.0: 5.055.0\rightarrow 5.

Let's consider the other half of the interval: in the case of 1.31.3, we round to 11. What about longer decimal parts? Let's consider pi, 3.1415926535893.141592653589. Yup, as you can see, the decimal part falls in the first half, from 3.03.0 and 3.53.5: rounding pi to the nearest integer returns 33, as many engineers will tell you!

For negative numbers, follow the same rules!

How to round to the nearest integer when the decimal part is 0.5

What happens in the case of half-integers? Every decimal number ending with 0.50.5 has the same "distance" from the two integers defining the interval. How do we round to the nearest integer, then? We use a convention called half up rounding, where 0.50.5 goes up:

i.05i+1i.05\rightarrow i+1

Notice that this operation returns different results for negative and positive numbers:

8.59 8.5\rightarrow9

While:

8.58-8.5\rightarrow-8

Half up rounding is not the only way to round half-integer numbers to the nearest integer.

  • Rounding half down returns the smallest integer between the extremes: 4.544.5\rightarrow4, and 4.55-4.5\rightarrow-5.

  • Rounding half even returns alternating results:

i.5={i+1 if i is oddi if i is eveni.5=\begin{cases} i+1&\ \mathrm{if\ }i\mathrm{\ is\ odd}\\ i&\ \mathrm{if\ }i\mathrm{\ is\ even} \end{cases}

Notice that this way of calculating the rounding to the nearest integer always returns an even number.

Notice that you can express the half up policy using the ceiling function. The "twin" function, floor, gives us another rounding policy. Let's see them!

Floor and ceiling are a pair of particular mathematical functions that return, respectively, the closest smallest integer number and the closest largest integer number:

floor(x)=x=max(n is integer,nx)\begin{split} &\mathrm{floor}(x) = \left\lfloor x\right\rfloor\\ &= \mathrm{max}(n\ \mathrm{is\ integer}, n\leq x) \end{split}

And:

ceil(x)=x=min(n is integer,nx)\begin{split} &\mathrm{ceil}(x) = \left\lceil x\right\rceil\\ &= \mathrm{min}(n\ \mathrm{is\ integer}, n\geq x) \end{split}

Using these functions, we define:

  • The half ceil policy, which returns the ceiling of i.05i.05: 4.554.5\rightarrow5, and 4.54-4.5\rightarrow -4.

  • The half floor policy, where you apply the floor function, always obtaining the smaller integer: 4.544.5\rightarrow4, and 4.55-4.5\rightarrow-5.

One last policy is the so-called away from 00. In this case, we round half-integers to the integer being on the opposite side of 00 on the number line: 4.554.5\rightarrow5, and 4.55-4.5\rightarrow-5.

How to use our rounding to the nearest integer calculator

Using our tool is straightforward: input the number, and we will round it to the nearest integer according to the rules outlined in this article.

If you want to access more advanced options, click on advanced mode. You will find other rounding techniques (for example, the desired power of ten) or policies for half-integer numbers.

More rounding tools!

Rounding to the nearest integer is just one of many ways to round a number. Omni took care of it: try our other rounding tools:

  • The rounding calculator (for a general tool to cover all your needs);
  • The round to the nearest ten;
  • The round to the nearest tenth;
  • The round to the nearest hundred;
  • The round to the nearest hundredth;
  • The round to the nearest thousand;
  • The round to the nearest thousandth;
  • The round to the nearest dollar;
  • The round to the nearest penny; and
  • The round to the nearest cent.

FAQ

How do I round a number to the nearest integer?

To round a decimal number to the nearest integer, take a look at its fractional part:

  • If the decimal part is between .0 (included) and 0.5 (excluded), round to the integer part.

  • If the decimal part is between .5 (excluded) and .0 (excluded), round the number to the integer part +1.

  • If the decimal part is .5 (half-integer), always round to the greater closest integer.

How do I round -4.5?

To round -4.5, follow these easy steps:

  1. Choose the rounding policy: the standard one is the so-called "half up".

  2. Add 0.5 to the number:

    4.5 + 0.5 = 5.0

  3. Round according to the standard procedures: the result is 5.

Should I always round half-integer numbers to the higher integer?

No! Before rounding a half-integer number, be sure about the policy in use. Here are the most commonly used:

  • Half up (also known as half ceiling): you always round half-integers to the highest closest integer;

  • Half even: you always round to the closest even number;

  • Half down: you always round to the number closer to 0;

  • Half floor: the result is always the smallest neighboring integer number; and

  • Away from 0: round to the number lying away from 0 on the number line.

How do I round negative numbers to the nearest integer?

Negative numbers follow the same rules defined for positive numbers when rounding them to the nearest integer apart from half-integer numbers.

In the case of half-integer numbers, you must choose a convention to round to the closest number. In the half up convention, treat .5 as a "greater" fractional part and round the number to the integer to the right on the number line.

Davide Borchia
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For non-standard rounding modes, check out the advanced mode.
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