0.1 Significant digits

Significant digits

Introduction

Though easy to comprehend, significant digits play an important role in engineering calculations. In this module, the rules are presented that govern how to determine which digits present in a number are significant. In addition, several applications are used to illustrate how significant digits can be used to express the results of engineering calculations.

Basic rules

A number can be thought of as a string of digits. Significant digits represent those digits present in a number that carry significance or importance to the precision of the number.

Rule 1: All digits other than 0 are significant. For example, the number 46 has two significant digits and the number 25.8 has three significant digits.

Rule 2: Zeros appearing anywhere between two non-zero digits are significant. Let us consider the number 506.72. The 0 that occurs between the 5 and 6 is significant according to Rule 2. Thus the number 506.72 has five significant digits.

Rule 3: Leading zeros are not significant. For example, 0.00489 has threee significant digits.

Rule 4: Trailing zeros in a number containing a decimal point are significant. For example, the number 36.500 has five significant digits.

Rule 5: Zeroes at the end of a number are significant only if they are behind a decimal point. Let us consider 4,600 as the number. It is not clear whether the zeros at the end of the number are significant. As a result, there could be two, three or four significant digits present. To avoid ambiguity, one may express the number by means of scientific notation. If the number is written as 4.6 ×10 ³ , then it has two significant digits. If the number is written as 4.60 ×10 ³ , then it has three significant digits. Lastly, if the number is written as 4.600 ×10 ³ , then it has four significant digits.

Rounding

The concept of significant digits is often used in connection with rounding. Rounding to n significant digits is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way.

Let us consider the population of a town. The population of the town might be known to the nearest thousand, say 12,000. Now let us consider the population of a state. The might be known only to the nearest million and might be stated as 12,000,000. The former number might be in error by hundreds while the latter number might be in error by hundreds of thousands of individuals. Despite this, the two numbers have the same significant digits. They are 5 and 2. This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.

The rules for rounding a number to n significant digits are:

Start with the leftmost non-zero digit (e.g. the "7" in 7400, or the "4" in 0.0456).

• Keep n digits. Replace the rest with zeros.
• Round up by one if appropriate. For example, if rounding 0.89 to 1 significant digit, the result would be 0.9.
• Or, round down by one if appropriate. For example, if rounding 0.042 to one significant digit, the result would be 0.04