Significant Figures in Physics

Significant figures refer to the digits in a number that accurately convey the value of a measured quantity. They include all certain digits and the first uncertain digit.

These figures essentially indicate the level of precision and accuracy with which a measurement is taken.

When measuring a physical quantity—such as the volume of a liquid, temperature, or the length of an object—you use an instrument with its own inherent accuracy.

Significant figures include all the digits that a measuring tool can reliably detect, based on its level of sensitivity.

In other words, the number of significant figures reflects the tool’s precision: the more sensitive the instrument, the more significant figures it can record.

An Example

Imagine measuring the length of a pencil using a ruler with 0.1 cm increments.

You record a measurement of 15.2 cm.

measurement example

In this instance, the digits "1" and "5" (the whole number) are certain, while the digit "2" in the decimal place carries some uncertainty, depending on the precision of the instrument, defined by its smallest division.

In this example, the measurement 15.2 cm has three significant figures: the first two are certain digits (or exact digits), while the third is uncertain but still significant.

Among these significant figures, the certain digits are those that remain stable within the measurement's uncertainty range. Essentially, certain digits are the ones we can rely on most confidently, as they don’t fluctuate with the measurement error. In this case, the pencil measures 15.2 cm using a millimeter-marked ruler, with an uncertainty of $ \pm 0.5 \ mm $, or $ \pm 0.05 \ cm $: $$ 15.2 \, \text{cm} \pm 0.05 \, \text{cm} $$ So, the actual length of the pencil lies somewhere between 15.15 cm and 15.25 cm. The digits that don’t change within this range are 1 and 5, making these the certain digits in the measurement.

Why Are Significant Figures Important?

Significant figures not only indicate the numerical value of a measurement but also the precision with which it has been determined.

It's crucial to account for them when performing calculations with measured values because each additional digit provides a different level of precision in interpreting the measurement.

For instance, if you write 15.20 cm instead of 15.2 cm, you are indicating that the measurement was taken with a precision of 0.01 cm, and thus, the "0" after the "2" is also significant. In this case, the number has four significant figures. This is why writing 15.20 cm is different from 15.2 cm; they don’t provide the same level of information. The first measurement is much more precise.

How to Count Significant Figures

Here are some general rules:

Non-zero digits are always significant.
For example, in 1234, all four digits are significant.
Zeros between significant digits (captive zeros) are also significant.
For instance, in 102, all three digits are significant.
Leading zeros are not significant as they only act as placeholders for the decimal point.
For example, in 0.0056, only the digits 5 and 6 are significant.
Trailing zeros in a decimal number are significant.
For instance, 45.00 has four significant figures because the trailing zeros indicate the measurement's precision.
Trailing zeros in a whole number without a decimal point may or may not be significant.
For example, in 1500, the number of significant figures depends on the context or notation used.

Practical Examples

To clarify how to count significant figures, consider these examples:

0.0045 has two significant figures: the initial zeros are not significant as they merely position the "4" after the decimal point.
8.300 has four significant figures: the trailing zero is significant because it reflects the measurement’s precision.
0.0560 has three significant figures: the initial zeros are not counted, but the trailing zero is, as it indicates the measurement’s precision.

In some cases, however, trailing zeros in an integer can cause ambiguity.

For instance, a value like 1400 m could have two, three, or four significant figures, depending on whether the zeros are viewed as part of the measurement’s precision or merely as placeholders.

To eliminate ambiguity, scientific notation is used.

This method allows you to clearly specify the number of significant figures in a number. For example:

Writing 1.4 × 10³ m indicates that the value has two significant figures, with an uncertainty margin of 0.1 m.
Writing 1.40 × 10³ m indicates that the value has three significant figures, with an uncertainty margin of 0.01 m.
Writing 1.400 × 10³ m indicates that the value has four significant figures, with an uncertainty margin of 0.001 m.

As shown, the number remains the same, but the level of uncertainty varies significantly based on the precision.

Working with Significant Figures

When performing mathematical operations on measurements of physical quantities, it’s essential to follow the rules for significant figures. These rules ensure that the precision of the result doesn’t exceed that of the quantities involved.

Rounding a Number
Rounding a number follows a straightforward guideline:
- If the first digit to drop is less than 5, the preceding digit stays the same.
  For example, rounding $4.372$ from 4 significant figures to 3 gives $4.37$. Since the last significant digit ($4.37\color{red}2$) is less than 5, the previous digit ($4.3\color{red}72$) remains unchanged ($4.3\color{red}7$).
- If the first digit to drop is 5 or more, the preceding digit increases by one.
  For example, rounding $4.376$ from 4 significant figures to 3 results in $4.38$. Here, the last significant digit ($4.37\color{red}6$) is 5 or more, so the previous digit ($4.3\color{red}76$) is rounded up, making the rounded number $4.3\color{red}8$.
Multiplying or Dividing Two Quantities
For multiplication or division, the result should have the same number of significant figures as the quantity with the fewest significant figures. This means the precision of the result is constrained by the least precise measurement.
For instance, consider the following multiplication: $$ (4.376 \, m) \cdot (1.4 \, m) = 6.1264 \, m^2 $$ Since the less precise quantity has 2 significant figures (1.4 m), the result should be rounded to 2 significant figures as well: $$ (4.376 \, m) \cdot (1.4 \, m) = 6.1264 \, m^2 \approx 6.1 \, m^2 $$
Multiplying or Dividing a Quantity by a Number
When a physical quantity is multiplied or divided by a pure number, the result retains the same number of significant figures as the original quantity. This is because the precision of a pure number does not affect the precision of the outcome; only the physical quantity sets the number of significant figures in the result.
For example, multiply $5.678 \, m$ by $3$: $$ (5.678 \, m) \cdot 3 = 17.034 \, m $$ Since $5.678$ has 4 significant figures, the result (17.034 m) should also be rounded to 4 significant figures, yielding $17.03 \, m$. $$ (5.678 \, m) \cdot 3 = 17.034 \, m \approx 17.03 \, m $$
Adding or Subtracting Two Quantities
For addition or subtraction, the result should match the number of decimal places of the quantity with the fewest decimal places. Round each term to align with the least precise quantity in terms of decimal places. In an alternative approach, you can add the original terms and then round the result.
For example, add $12.3 \, s$ and $4.678 \, s$: $$ 12.3 \, s + 4.678 \, s $$ Here, $12.3$ s has only one decimal place, so you must round the other term to one decimal place as well ($4.678 \, s \rightarrow 4.7 \, s$). $$ 12.3 \, s + 4.678 \, s = 12.3 \, s + 4.7 \, s $$ With all terms adjusted to the same decimal precision, proceed with the addition. The result here is $17.0 \, s$. $$ 12.3 \, s + 4.678 \, s = 12.3 \, s + 4.7 \, s = 17.0 \, s $$

Following these simple guidelines helps you avoid assigning unrealistic precision to your results, maintaining the scientific integrity essential in physical measurements. This approach makes data interpretation clear and reliable.

In conclusion, significant figures are not just a technical or stylistic detail; they are essential for accurately expressing measurement results in scientific contexts.

Each additional or omitted digit can alter the interpretation of a measurement, thus affecting calculations and decisions based on that information.

Significant figures are crucial in physics, chemistry, and other scientific fields to communicate measurement results accurately, avoid ambiguity, and reflect the precision of the measuring instrument used.