Often we use rounding to make calculations simpler. A simple example of this is when you measure something using a ruler. You might say it is 4 centimeters long when really you know it is slightly longer than 4 centimeters; for example, 4 centimeters and a bit, perhaps 4 and a quarter centimeters.
For many things, being close to accurate is good enough. We lose accuracy when we round, but the resulting number is typically easier to use.
Other concrete examples are shoe sizes. Shoe sizes are often in whole numbers: for example, 6, 7, 8, or half sizes: 6.5, 7.5, 8.5. It is unlikely that you are exactly a whole or half size, but the shoe fit is good enough. Or, when someone asks how far from campus you live, you are unlikely to say 3.125 miles and more likely to say 3 miles. More accurate is to say you live 3.125 miles away (though saying that might make people think you are slightly weird), and so we would typically say 3 miles. We lose accuracy, but things are easier.
When we report numbers in APA style, we round numbers. In other words, in this class we use APA style, and so when we write numbers we round to two decimal places. The two exceptions to this rule are p values, which are always rounded to three decimal places, and percentages, which are rounded to one decimal place. Given this, the examples here focus on two decimal places.
Another place we use rounding is with the calculation tools we use—whether a calculator, a phone, or a computer.
Here are two examples to illustrate this point.
The first considers the relatively simple example of what happens when we divide 1 by 3, a third. While the fraction 1/3 is clean and easy to understand, if we write this value in decimal form we get 0.3333333 and the 3s keep on going. For calculation processes, we therefore round to 0.3333 or some variation, for example, 0.33. So, although the real value is 0.3333333333, we round to 0.33.
The second example involves π—not that pie, but the mathematical pi.
We do not need to know what π is for the course, but you likely remember it from a math class. It has to do with circles and you can look it up on the Internet if you want to know more.
π is an irrational number, which means its decimal representation never ends— 3.14159265358979323846264338327950288419716939937510 and so on... π has been calculated to trillions of digits and no pattern emerges.
For practical purposes, we use a rounded value of π in calculations such as 3.14159, or 3.14—Pi Day on the 14th of March.
Before we talk about how to round, we should do a quick clarification of what it means to have two decimal places.
A number with two decimal places only has two digits after the decimal point. For example, 8.41, 9.89, 26.76, .54, or 515.90. (Note that we still write the zero even if the last digit is zero, as in the example 515.90. In other words, it is not written as 515.9, but rather 515.90.)
Having discussed the need for rounding, we can now consider how to round. As mentioned, we will focus on rounding to two decimal places.
When we round to two decimal places, we look at the third decimal place to determine what happens to the number in the second decimal place. So, first let us make sure we are comfortable with knowing where the second and third decimal places are.
Here are two examples to show where the second and third decimal place are:
The rounding rule is:
The rounding rule for numbers ending in 0, 1, 2, 3, or 4: For these numbers, we round down.
So, for 3.923, we know we look at the 3, the third decimal place, to determine how to round. 3 falls under the rule of 4 or smaller, and so we do nothing to the prior digit and therefore, the answer is 3.92.
Another example is 5.921. Here we look at 1 the third decimal place to determine what we do to the 2 in the second decimal place. 1 falls under the rule of 4 or smaller, and so we do nothing to the prior digit and therefore, the answer is 5.92.
Here are two additional examples:
These examples are known as “rounding down” because the rounded number is smaller than the original number.
The rounding rule for numbers ending in 5, 6, 7, 8, 9: For these numbers, we round up.
So, for 1.687, we look at 7, the third decimal place, to determine how to round. 7 falls under the rule of 5 or bigger, and so we increase the second decimal place by one number; the 8 changes to a 9 and so the answer is 1.69.
2.778: Here we look at 8, the third decimal place, to determine what we do to the 7 in the second decimal place. 8 falls under the rule of 5 or bigger, and so we increase the second decimal place by one number; the 7 changes to a 8 and so the answer is 2.78.
Here are two additional examples:
These examples are known as “rounding up” because the rounded number is bigger than the original number.
Sometimes we might have a number such as 3.898, and here the third decimal place is 8, which falls under the rule of 5 or bigger. So we increase the second decimal place by one number—but the second digit is 9 so we end up with the answer being 3.90.
We saw earlier that 3.923 rounds to 3.92. Suppose the number was 3.9238? This still rounds to 3.92, as we ignore the 8 in the fourth decimal place when we round to two decimal places.
It does not matter to us what happens in the fourth decimal place onwards, we only care about the third decimal place and how that impacts the second decimal place.
If we were rounding to three decimal places, we look at the fourth decimal place; if rounding to four decimal places, we look at the fifth decimal place, and so on. But once the concept of rounding is understood, it is easy to generalize to round to a different number of decimal places.
Similarly, if we were rounding to one decimal place, we look at the second decimal place.