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How to Round Numbers

There are several different methods for rounding decimal values to a target precision. The rounding method we will use in physics can be expressed simply as "round 0…4 down and 5…9 up."

Rounding is commonly done to properly express the precision of your answer. The target precision is usually based on the precision of the given quantities (e.g., measurements) and referred to as the number of significant figures (or significant digits) in the result. The following detailed procedure places rounding into this important context:

1. Identify Last Significant Digit - Decide which decimal place in your answer should be the last (rightmost) significant digit. For example, when rounding 49.783 to three significant figures, the last significant digit is the tenths (10^-1) digit (currently 7). When rounding 45802.7 to three significant figures, the last significant digit is the hundreds (10²) digit (currently 8).
2. Evaluate First Insignificant Digit - The next digit to the right will be the first digit to be discarded (i.e., set to zero). For example, when rounding 49.783 to three significant figures, the first insignificant digit is the hundredths (10^-2) digit (currently 8). When rounding 45802.7 to three significant figures, the first insignificant digit is the tens (10¹) digit (currently 0).
3. Round 0…4 Down and 5…9 Up - If the first insignificant digit is 0…4, remove it and all digits to its right from your answer (set removed digits left of decimal point to 0.) If the first insignificant digit is 5…9, remove it and all digits to its right, but increase the first significant digit by 1 (i.e., round it up).
4. Express Significance of Result - The simplest way to express which digits in your answer are significant is to always convert your answer to scientific notation. That said, it is often more readable to use the closest SI prefix (e.g., 23.4mm instead of 2.34 x 10^-3 meters). However, this can be awkward when the rightmost significant digit becomes zero, but is still left of the decimal point. Normally, the zeros immediately left of the decimal point are not considered significant, so you must draw a horizontal bar over them to make it clear that they are significant. I usually go with Scientific Notation instead.

Note: Negative numbers are treated in the same manner, so technically "rounding down" means "rounding toward zero", and "rounding up" means "rounding away from zero".

Rounding Examples

Here are some rounding examples to make sure you understand the "edge cases":

An Even Better Method

Dr. Kromm incorrectly describes an even better rounding method, called the "Round-to-even" method. When properly applied, that method differs from our method only when the first insignificant digit is 5 and there are no more insignificant digits to be removed. See the Rounding page on Wikipedia for a description of this method and a rationale for why it's better, but we're not going to bother with this because our results won't be collected into large data sets for statistical analysis.