Rounding Numbers
There are three basic ways to round numbers. The one that most people know is rounding to the nearest whole number, or to the nearest ten, but there are two other ways to round which are essentially very similar to this method. They are rounding to two (or three) deciml places and rounding to three (or four) significant figures.
Why do we need to round numbers?
You may see it reported that a TV program had 23 million viewers. This is not actually true because some of those viewers fell asleep half way through the program and some people lied about watching the program. The true number of viewers was somewhere between 22½ million and 23½ million and the published figure was rounded to the nearest million. Similarly if you used your calculator to find the square root of 1000, you would get something like:
√1000 = 31.622776601683793319988935444327
The most important digits are the 3, 1 and 6 at the beginning. The least important digits are the 3, 2 and 7 at the end. It is bad practice to write down all the digits that the calculator shows so we choose only to write down a few of the most important ones.
Rounding to the nearest ten and the nearest whole number
Looking at the number above, it should be seen that, to the nearest ten, the square root of 1000 is 30. The above number is between 30 and 40 and it is nearer to 30. This is an example of rounding down.
To the nearest whole number, the square root of 1000 is actually 32 because the number given above is closer to 32 than to 31. This is an example of rounding up.
Round up or round down?
To decide whether to round up or down we look at the digit next to those that we plan to write down. When we rounded to the nearest ten, this digit was the 1 in the units place. When we rounded to the nearest whole number, this digit was the first digit after the decimal point, the 6. If this digit is less than 5, we round down, if it is equal to or more than 5, we round up. This rule is used whether we are rounding to the nearest million, rounding to 2 decimal places or rounding to 3 significant figures.
Decimal Places (dp)
Rounding to 1 decimal place is the same as rounding to the nearest tenth. We just write write one number after the decimal point and we look at the second number after the decimal point to decide whether to round up or down. The second number after the decimal point is 2, so we round down to 31.6. An acceptable abbreviation for decimal places is dp, so we could write this down as:
√1000 = 31.6 (1 dp)
Similarly:
√1000 = 31.623 (3 dp) Because the 4th decimal place is 7, we round up.
Try changing this number and the number of decimal places it should be rounded to. Is the result as you would expect?
Significant Figures (sf)
Every digit in any number is a significant figure irrespective of whether it comes before or after the decimal point, except for zero (zero sometimes is significant, sometimes not). So the number 31.6 has three significant figures, the 3, the 1 and the 6. The number 31.623 has five significant figures. The rounding rules are the same - look at the next digit, if it's 5 or more round up, otherwise round down. The abbreviation for significant figures is sf, so we can say that:
√1000 = 31.6 (3 sf)
√1000 = 31.623 (5 sf)
Zero causes a couple of small problems with significant figures. In the first example below, zero is a significant figure, in the second example it is not:
√1000 = 31.62277660 (10 sf)
√1000 = 30 (1 sf)
Normally common sense will tell you whether to count zero as a significant figure.
How accurately should I round?
If a test question asks you to round your answers to 1 decimal place (1 dp), then this is what you should do. If the instructions do not specify haw accurately answers should be given, then 3 significant figures (3 sf) is usually considered acceptable. It is good practice to write (3 sf) after such answers. Answers should never be given to 10 significant figures (the above example was used just to demonstrate the point about zeros).