Introduction

Given the approximation 83.53273 to the correct value 83.52923, rather than calculating the relative error, it is quite easy to see that the three most significant digits are the same, and therefore we would like to say that the approximation is correct to three significant digits. This topic gives a rigorous definition to the intuitive idea of counting the number of significant digits.

One word of warning: the concept of significant digits may not always be intuitive, and it is a very course approximation of the accuracy of an approximation. In general, it is safer to use the actual relative error reserving significant digits as a brief summary.

Theory

Looking at an approximation 2.75303 to an actual value of 2.75194, we note that the three most significant digits are equal, and therefore one may state that the approximation has three significant digits of accuracy. One problem with simply looking at the digits is given by the following two examples:

1. 1.9 as an approximation to 1.1 may appear to have one significant digit, but with a relative error of 0.73, this seems unreasonable.
2. 1.9999 as an approximation to 2.0001 may appear to have no significant digits, but the relative error is 0.00010 which is almost the same relative error as the approximation 1.9239 is to 1.9237 .

Thus, we need a more mathematical definition of the number of significant digits, and we will therefore generalize the following observation:

Given the exact value of 0.999⋅⋅⋅, we note that the relative error of the approximation 0.95 is 0.04999⋅⋅⋅, and we would like to state that 0.95 is correct to 1 significant digit. Similarly, the relative error of the approximation 0.995 is 0.004999⋅⋅⋅, and we would like to state that 0.995 is correct to 2 significant digits. Using this as our model, we come up with the following definition:

An approximation a approximates a correct value x to n significant digits if n is the largest integer for which the inequality

continues to hold true. If the relative error is greater than 0.5, then we will simply state that the approximation has zero significant digits.

Solving the previous equation for n, we have that the number of significant digits is equal to

where ⌊x⌋ is the floor function, also known as the greatest integer less than function.

Calculating Significant Digits

Given a relative error Erel, find the largest integer n such that Erel < 0.5 10-n. If the relative error is greater than 0.5, state that the approximation does not have any significant digits.

In general, the number of significant digits between a number and its approximation are equal to the number of leading digits which are equal, though this is only a rule of thumb, and if the most significant digit is 1 or 2, it is most useful to ignore it when counting the number of significant digits.

Examples

1. What is the number of significant digits of the approximation 3.14 to the value π?

Erel = |3.14 - π|/|π| ≈ 0.00051 ≤ 0.005 = 0.5 ⋅ 10-2, and therefore it is correct to two significant digits.

This example demonstrates a weakness in the concept of significant digits: in this example, it would be almost better to say that 3.14 approximates π to almost or approximately three significant digits.

2. What is the number of significant digits of the label 240 Ω when the correct value is 243.32753 Ω?

Erel = |240 - 243.32753|/|243.32753| ≈ 0.014 ≤ 0.05 = 0.5 ⋅ 10-1, and therefore it is correct to one significant digit.

3. To how many significant digits is the approximation 1.998532 when the actual value is 2.001959?

Erel = |1.998532 - 2.001959|/|2.001959| ≈ 0.0017 ≤ 0.005 = 0.5 ⋅ 10-2 and therefore it is correct to two digits.

Questions

1. To how many significant digits does 22/7 approximate π? (3)

2. To how many significant digits does 355/113 approximate pi π? (6)

3. A capacitor is labeled as 100 mF whereas it is actually 108.2532 mF. To how many significant digits does the label approximate the actual capacitance? (0)

Matlab

Given the relative error 0.00036, we can easily calculate the number of significant digits (3) using the formula:

>> floor( -log10( 0.00036 / 0.5 ) )

Maple

Given the relative error 0.00036, we can easily calculate the number of significant digits (3) using the formula:

floor( -log[10]( 0.00036 / 0.5 ) );