Introduction
When we get a decimal quantity from a measurement or from a calculation, rounding allows us to express that quantity to a desired level of accuracy. It is most useful as a way to communicate only the amount of information that we think is necessary.
Round decimal to a specific place value
If you ask someone their age, they will usually reply with a whole number of years: "I am 14." But unless it happens to be their birthday, we know that they are probably a little bit older than exactly 14 years, perhaps 14.38276 years. By rounding their age to a whole number, what they mean is something like "I have lived for at least 14 years", or maybe "I have ridden the Earth for 14 whole loops around the Sun". The number of whole years is what is most important.
Now imagine you want to purchase some ham from the butcher using your bank card. Say the butcher is selling ham for \$13.95\text{ per kg} and you want 0.25\text{ kg}. Then the cost would be given by 13.95\times0.25=\$3.4875. But we cannot transfer amounts of money less than 1 cent, so the cost will need to be rounded to two decimal places. This means you would ultimately pay \$3.49 for the 0.25\text{ kg} of ham.
What can we notice when comparing a rounded number with the original number? The examples above show that rounding a number makes it less precise. But the benefit of this precision loss is that usually we are not interested in the exact value of a quantity, so a rounded number is a simpler number.
Let's look at what it means to round a decimal to the nearest whole number. If we choose any decimal number, it will either be exactly a whole number, or it will be somewhere in between two whole numbers. The rounded value is the whole number it is closest to. Use the applet below to see how this works.
Exploration
Move the point on the number line to see how rounding off a decimal works.
Notice that the rounded whole number depends only on the value of the digit in the tenths place.
If the digit in the tenths place is 0,1,2,3, or 4 then the number rounds down, and if the digit in the tenths place is 5,6,7,8, or 9 then the number rounds up.
We use this same approach when rounding to any place value. For example, if we want to round a decimal to the nearest tenth, we would look at the digits in the tenths and hundredths places. Similarly, if we want to round a decimal to the nearest hundredth, then we look at the digits in the hundredths and thousandths places.
Examples
Example 1
Round 328.864 to the nearest hundredth.
If we are rounding to a nearest place value, check the place value next to it whether it is lower, higher, or exactly 5.
If it is lower than 5, we need to round down.
If it is higher or equal to 5, we need to round up.
Round to a number of decimal places
Instead of specifying that a number be rounded to the nearest whole number, or the nearest tenth or hundredth, we can simply state how many digits we want to keep in the decimal.
For example, rounding 5.3175 to the nearest hundredth gave 5.32, and 5.32 has only two digits after the decimal point, so this rounding is to two decimal places. We can write this like so:
| \displaystyle 5.3175 | \displaystyle = | \displaystyle 5.32 | (to 2 d.p.) |
where "d.p." stands for "decimal place(s)". Let's look at some other questions.
Examples
Example 2
Round 79.38 to one decimal place.
Example 3
Round 1.599\,64 to three decimal places.
Instead of specifying that a number be rounded to the nearest whole number, or the nearest tenth or hundredth, we can simply state how many digits we want to keep in the decimal.