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1.2: Significant Digits and Resolution

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    A key element of any measurement or derived value is the resulting resolution. Resolution refers to the finest change or variation that can be discerned by a measurement system. For digital measurement systems, this is typically the last or lowest level digit displayed. For example, a bathroom scale may show weights in whole pounds. Thus, one pound would be the resolution of the measurement. Even if the scale was otherwise perfectly accurate, we could not be assured of a person's weight to within better than one pound using this scale as there is no way of indicating fractions of a pound.

    Related to resolution is a value's number of significant digits. Significant digits can be thought of as representing potential percentage accuracy in measurement or computation. Continuing with the bathroom scale example, consider what happens when weighing a 156 pound adult versus a small child of 23 pounds. As the scale only resolves to one pound, that presents us with an uncertainty of one pound out of 156 for the adult, but a much larger uncertainty of one pound out of 23 for the child. The 156 pound measurement has three significant digits (i.e., units, tens and hundreds) while the 23 pound measurement has but two significant digits (units and tens).

    In general, leading and trailing zeroes are not considered significant. For example, the value 173.58 has five significant digits while the value 0.00143 has only three significant digits (the “143” portion) as does 0.000000143. Similarly, if we compute the value 63/3.0, we arrive at 21, with two significant digits. If your calculator shows 21.0 or 21.00, those extra trailing zeroes do not increase accuracy and are not considered significant. An exception to this rule is when measuring values in the laboratory. If a high resolution voltmeter indicates a value of, say, 120.0 volts, those last two zeroes are considered significant in that they reflect the resolution of the measurement (i.e, the meter is capable of reading down to tenths of volts).

    When performing calculations, the results will generally be no more accurate than the accuracy of the initial measurements. Consequently, it is senseless to divide two measured values obtained with three significant digits and report the result with ten significant digits, even if that's what shows up on the calculator. The same would be true for multiplication. For these sorts of calculations, we can't expect the result to be any better than the “weakest link” in terms of resolution and resulting significant digits. For example, consider the value 3.5 divided by 2.3. Both of these values have two significant digits. Using a standard calculator, we find an answer of 1.52173913. The result has nine significant digits implying much greater accuracy and resolution than we started with, and thus is misleading. To two significant digits, the answer would be rounded to 1.5. On a long chain of calculations it may be advisable to round the intermediate results to a further digit and then round the final answer as indicated previously. This will help to mitigate accumulated errors.

    When it comes to addition and subtraction, the larger value will tend to indicate the number of significant digits available, particularly in the laboratory. This is due to the resolution of the measurement (that is, its finest digit). For example, suppose we have to add two distances. We drive a car from a parking lot to its exit, a distance we record with a tape measure as being 51.17 feet. We then drive from the exit to the next city, which the car's odometer records as being 60.0 miles. Is it fair to say the total distance is 60 miles plus 51.17 feet? No, it is not. Why? Because we cannot expect the car's odometer to be accurate to within 0.01 feet, like the tape measure. In fact, the odometer is reading out a value with one-tenth mile resolution. One-tenth of a mile is 528 feet, or more than ten times the entire measurement given by the tape. We would simply ignore the tape measurement because it is smaller than the resolution of the odometer. The proper result is 60.0 miles. Now in contrast, if the tape measure had indicated 552.7 feet, or slightly more than 0.1 miles, we could safely say the total distance was 60.1 miles, rounding the result to finest resolution digit of the more coarsely measured value. Of course, this example is a bit contrived but it is designed to show the limitations of measurement devices. It is also complicated by the fact that we are using using USA customary unit (miles, feet), complete with their odd conversion factors. Fortunately, virtually all current work in science and technology uses the much simpler metric system, which we will be examining shortly.

     

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    Example 1.2.1

    Determine the number of significant digits in the following values.

    A. 12.6

    B. 0.0034

    C. 43.001

    D. 5400

    Answers:

    A. All three digits are significant.

    B. Two. Only the 3 and 4 count because leading zeroes are not significant.

    C. Five. Embedded zeroes are significant.

    D. Two. Trailing zeroes are not generally considered significant.

     

    Example 1.2.2

    Perform the following computations, leaving the answer with the appropriate number of significant digits.

    A. \(55 \cdot 10.1\)

    B. 2312.5 / 16.2

    C. 1756.2 + 345.1

    D. 750.2 − 0.004

    Answers:

    A. 555.5 which rounds to 560 (55 has only two significant digits).

    B. 142.7469136... which rounds to 143 (16.2 has three significant digits).

    C. 2101.3 (both values have resolution to tenths place).

    D. 750.2 (do not extend the answer beyond the more coarse resolution of tenths place).

     

    Summary

    • Leading zeros are not significant
    • Trailing zeros are usually not significant, but can be when higher resolution measuring equipment provides trailing zeros in the measurement 
    • All non-zero digits are significant and all zero digits between non-zero digits are significant
    • When multiplying and dividing, the value with the least number of significant digits indicates the number of significant digits in the result
    • When adding and subtracting, the value with the least resolution (i.e., the most coarse) will set the limit of resolution in the result

     

    This page titled 1.2: Significant Digits and Resolution is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James M. Fiore via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.