Significant Figures

What Are Significant Figures?

In science, numbers are based on measurements. Since every measurement has some uncertainty, we must report our numbers in a way that reflects the measurement’s precision. The standard convention is to write all of the certain digits in a measurement along with the first uncertain digit. These digits are collectively called significant figures.

Quick Reference

The Significant Figures reference page is a crib sheet for all significant figures rules.

The rules for handling significant figures are not arbitrary. They are a universal language that scientists use to communicate the honesty of their data. Before we learn the abstract rules for counting which digits are “significant,” it is essential to understand why they exist. The concept of significance is born directly from the act of measurement itself.

Measurement, Uncertainty, and Significant Figures

Reading Measurements: Certainty and Estimation

In the world of chemistry, a measurement is more than just a number; it is a statement about our knowledge and its limits. Every measurement made with an analog instrument, like a ruler or a graduated cylinder, has some degree of uncertainty. The practice of using significant figures is how we honestly communicate both what we know for sure and where we have estimated. The number of significant figures in any measurement is the sum of the certain digits plus the one, and only one, estimated (or uncertain) digit. This practice ensures that the measurement we record is an honest representation of the instrument’s precision.

Consider the graduated cylinder shown below. Our goal is to determine the volume of the liquid as accurately as possible.

To read the volume correctly, we follow a systematic process:

  1. Identify the Certain Digits. Look at the numbered markings. We can see the liquid level is above the “40” mL mark. The smaller markings represent 1 mL increments. The bottom of the curved meniscus is clearly above the 42 mL mark but below the 43 mL mark. Therefore, we are certain that the volume is “42-point-something” milliliters. The digits ‘4’ and ‘2’ are our certain digits.

  2. Estimate the Uncertain Digit. Now, we must estimate the liquid’s position between the 42 mL and 43 mL lines. It is not exactly halfway (which would be 42.5 mL), nor is it right at the top (42.9 mL). It appears to be about eight-tenths of the way up from the 42 mL line. We will therefore estimate this last digit as an “8”.

  3. Combine to Form the Full Measurement. By combining our certain digits with our single uncertain digit, we report the volume as 42.8 mL.

This measurement contains three significant figures.

The curved surface of a liquid in a narrow container like a graduated cylinder is called a meniscus. This curve is the result of a competition between two types of intermolecular forces:

  • Cohesive Forces: The attraction between the liquid’s own molecules.
  • Adhesive Forces: The attraction between the liquid’s molecules and the walls of the container.

For water (and most water-based solutions) in a glass cylinder, the adhesive forces are stronger than the cohesive forces. The water molecules are more attracted to the glass than to each other, causing the liquid to “climb” the walls of the container. This creates the familiar concave (U-shaped) meniscus.

Why do we read the bottom?

For a concave meniscus, the bottom of the curve is the lowest point of the bulk liquid and is the most distinct and reproducible point to measure. To ensure that every scientist records the same volume from the same measurement, it is the universal convention to read the volume from the bottom of a concave meniscus.

(A small number of liquids, like mercury, have stronger cohesive forces than adhesive forces and form a convex (domed) meniscus. In those rare cases, the measurement is read from the top of the dome.)

Now, let’s apply the same principle to measuring the length of an object with a ruler. This ruler is marked in centimeters (cm), and the smaller tick marks represent millimeters (mm), or tenths of a centimeter.

A common point of confusion arises when a measurement appears to land on a marked line. How do we report the significant figures in this case?

  1. Identify the Certain Digits. The tip of the pencil aligns with the “8” on the ruler. Because the ruler has markings for every millimeter (every 0.1 cm), we can see that the tip is not at 7.9 cm or 8.1 cm. It is exactly on the 8.0 cm line. Therefore, we are certain about the digit in the ones place (the ‘8’) and the digit in the tenths place (a ‘0’). Our certain digits are ‘8’ and ‘0’.

  2. Estimate the Uncertain Digit. The rule of measurement is to always estimate one digit beyond the smallest graduation mark. The smallest marks on this ruler are millimeters (0.1 cm). This means we must estimate to the hundredths of a centimeter (0.01 cm). Since the pencil’s tip falls exactly on the 8.0 cm line, our best estimate for the hundredths place is ‘0’. This final ‘0’ is our uncertain digit, but it is a necessary one.

  3. Combine to Form the Full Measurement. Combining our certain digits (8.0) with our uncertain digit (0) gives a final measurement of 8.00 cm.

Reporting the measurement as “8 cm” would be incorrect as it would imply that we are uncertain about the tenths place. Reporting “8.0 cm” would also be incomplete, as it implies uncertainty in the hundredths place when the ruler’s precision allows us to estimate that digit. The value 8.00 cm correctly communicates that we are certain the length is 8.0-something and that our estimate for the next digit is zero.

This measurement contains three significant figures. The trailing zeros after the decimal point are significant because they reflect the precision of the instrument used.

When using a digital instrument like an analytical balance, reading a measurement is different from using an analog tool. The instrument performs the measurement and provides a direct numerical readout, so you do not need to estimate a digit.

However, a common misconception is that this makes the reading perfectly certain. In reality, every measurement has uncertainty. For any digital instrument, the rule is: the last digit on the display is the uncertain digit.

Let’s break this down with the example:

  1. Read the Full Display: The balance shows a mass of 12.105 g.

  2. Understand the Precision vs. Uncertainty:

    • The manufacturer’s precision, d = 0.001 g (where ‘d’ stands for scale division), indicates the smallest increment the balance can resolve. In this case, it is the thousandths place (the third decimal).
    • This means the digits 12.10 are considered certain, assuming the balance is properly calibrated and leveled.
    • The final digit, the 5, is the uncertain digit. It represents the limit of the instrument’s resolution, and the true value could be slightly different (e.g., 12.104 g or 12.106 g).

  3. Record the Full, Significant Measurement: A valid measurement includes all certain digits plus the one uncertain digit. Therefore, we record the full value displayed. All digits shown on the screen are significant.

    • The measurement is 12.105 g.
    • This reading has five significant figures.

This is also why trailing zeros after a decimal point on a digital display are always significant. If the balance read 12.100 g, it would also have five significant figures. The final ‘0’ would be the uncertain digit, representing the instrument’s best estimate at that decimal place.

Identifying Significant Figures

The rules for identifying which digits in a number are significant can be organized into three main categories: non-zero digits, the handling of zeros, and special cases for exact numbers.

This is the simplest rule: All non-zero digits are always significant.

  • 123 has 3 significant figures.
  • 9.87 has 3 significant figures.

A zero’s significance depends entirely on its location in the number.

TipThe Pacific-Atlantic Mnemonic

A popular way to remember the zero rules:

  • If a decimal point is Present, start counting from the first non-zero digit on the Pacific (left) side and count all digits to the end.
  • If a decimal point is Absent, start counting from the first non-zero digit on the Atlantic (right) side and count all digits to the beginning.

A zero in a measurement has one of two jobs: it is either a placeholder to locate the decimal point, or it is a significant digit that indicates a specific level of precision. The following rules are designed to help you determine which job a zero is doing based on its location.

A. Captive Zeros

Zeros that are “captured” between non-zero digits are always significant.

  • 101.3 has 4 significant figures.
  • 5007 has 4 significant figures.

B. Leading Zeros

Zeros that come before all non-zero digits are never significant. They are placeholders that indicate the scale of the number.

  • 0.54 has 2 significant figures.
  • 0.0032 has 2 significant figures.

C. Trailing Zeros

The significance of zeros at the end of a number is the most common point of confusion. Their meaning depends on the presence of a decimal point.

  • Case 1: Decimal Point is Present Trailing zeros are significant because they indicate a specific level of precision.

    • 50.0 has 3 sf (the decimal point makes the final zero significant)
    • 0.0300 has 3 sf (the trailing zeros are significant)

  • Case 2: No Decimal Point (The Ambiguity Rule)

    Trailing zeros in a number without an explicit decimal point are ambiguous. For the number 1200, it is unclear if the measurement is precise to the hundreds, tens, or ones place.

    WarningClassroom Convention for Ambiguous Cases

    Unless specified otherwise, assume trailing zeros in a number without a decimal point are not significant. Therefore, treat 1200 as having 2 significant figures.

    To remove this ambiguity, either place a decimal point at the end of the number or use scientific notation.

Exact Numbers

Numbers obtained from counting discrete objects or from definitions are considered exact. They have an infinite number of significant figures and never limit the precision of a calculation.

  • Counting: 35 students, 12 beakers
  • Definitions: 1 inch = 2.54 cm (by definition), 1 minute = 60 seconds

Physical Constants

Some physical constants are defined values (exact), while others are measured values (inexact).

  • Defined (Exact): The speed of light is defined as c = 299,792,458 m/s. This has infinite significant figures.
  • Rounded (Inexact): An exact constant rounded to reduced precision such as the speed of light to 3 decimal places, c = 2.998 × 108 m/s, is inexact.
  • Measured (Inexact): The molar gas constant, R = 8.31446… J mol⁻¹ K⁻¹, is based on measurement and has an associated uncertainty.
ImportantA Note on Treatment of Physical Constants

Sometimes instructors will have you treat all given physical constants (even rounded) as exact. This book will treat all provided physical constants in a manner that follows the actual rules of significant figures.


Summary Rules for Identification

Table 1: Rules for Identifying Significant Figures
TipWhat About Other Kinds of Numbers?

The real world is full of numbers that aren’t direct measurements: prices, wages, dosages, recipes, and even conversion factors given in a problem. How do we handle their significant figures?

This is a crucial and often confusing topic. To master these special cases, read our detailed guide: A Guide to Exact, Counted, and Stipulated Values

Significant Figures in Calculations

Addition and Subtraction

The Rule: When adding or subtracting, the precision of the result is limited by the least precise number in the calculation. The final answer must be rounded to the same place value as the last significant digit of the least precise input.

Think of it as a chain: its strength is determined by its weakest link. In addition or subtraction, the “weakest link” is the number whose final significant digit holds the largest place value (e.g., a number precise to the hundreds place is less precise than a number precise to the tenths place).

Example: Numbers with Decimals

Let’s add 12.11, 18.0, and 1.013.

  12.11   (hundredths place)
  18.0    (tenths place)   ← Least Precise Place
+  1.013  (thousandths place)
  31.123

The raw sum is 31.123. The least precise number, 18.0, is known to the tenths place. Therefore, the last significant digit in the answer is in the tenths place.

  • Unrounded answer showing last significant digit: 31.123
  • Final rounded answer: 31.1

Example: Numbers without Decimals

Let’s add 10,500 and 235. Assume the trailing zeros in 10,500 are not significant.

  10,500   (hundreds place)   ← Least Precise Place
+    235   (ones place)
  10,735

The raw sum is 10,735. The least precise number is known to the hundreds place. Therefore, the last significant digit in the answer is in the hundreds place.

  • Unrounded answer showing last significant digit: 10,735
  • Final rounded answer: 10,700

Multiplication and Division

The Rule: The result is limited by the measurement with the fewest number of significant figures.

Logarithms and Antilogarithms

The rules for significant figures with logarithms are unique because the result contains two distinct pieces of information. To handle this, we separate the result of a logarithm into two named parts:

  • The Characteristic: The integer part of the logarithm (the number before the decimal point). It represents the order of magnitude, specifically the integer power of the logarithm’s base.
  • The Mantissa: The decimal part of the logarithm (the numbers after the decimal point). It contains the precision of the original number, and its number of digits is related to the significant figures.

Example:

log(x) = 4.090

  • The characteristic is 4.
  • The mantissa is .090.

Think of this as being similar to scientific notation, where the exponent gives the magnitude and the coefficient gives the significant figures. Since most calculations in general chemistry use the base-10 logarithm (log₁₀, often written as log), the characteristic conveniently corresponds to the power of 10.

Logarithms (log, ln)

The number of digits in the mantissa of the result of a log is equal to the number of significant figures in the original number.

Example:

log(1.23 × 104)

  • The original number has 3 significant figures.
  • Therefore, the mantissa of the result must have 3 digits.

Result: 4.0899… → Rounded result: 4.090

Antilogarithms (10x, ex)

The number of significant figures in the result of the antilogarithm is equal to the number of digits in the mantissa of the original exponent.

Example:

100.52

  • The exponent’s mantissa is .52, which has 2 digits.
  • Therefore, the result must be rounded to 2 significant figures.

Result: 3.311… → Rounded result: 3.3

Example:

102.45

  • The exponent’s mantissa is .45, which has 2 digits.
  • Therefore, the result must be rounded to 2 significant figures.

Result: 281.838… → Rounded result: 2.8 × 102

Summary Rules for Operations

Table 2: Rules for Operations with Significant Figures

Rounding Rules

Because the final significant digit in a calculated answer is an estimate, we must follow a consistent set of rules for how to round that number based on the digits that follow it. There are two common methods.

Textbook Rounding (round half up)

This method is commonly used in introductory courses.

  • If the digit next beyond the one to be retained is 5 or greater, round up the last kept digit.
  • If the digit next beyond the one to be retained is 4 or less, keep the last digit the same.

Examples (rounding to 3 sig figs):

  • 1.236… → 1.24 (round up)
  • 1.235… → 1.24 (round up)
  • 1.234… → 1.23 (keep same)

Banker’s Rounding (round half to even)

This method is preferred by IUPAC and NIST1 and is statistically less biased.

  • If the digit next beyond the one to be retained is exactly 5, round the last kept digit to the nearest even number.
  • If the digit next beyond the one to be retained is greater than 5, round up the last kept digit.
  • If the digit next beyond the one to be retained is less than 5, keep the last digit the same.
  • If two or more figures are to the right of the last figure to be retained, consider them as a group in rounding decisions. Thus, in 2.4(501), the group (501) is greater than 5 while for 2.5(499), (499) is less than 5

Examples (rounding to 3 sig figs):

  • 1.235 → 1.24 (exactly 5, round to even 4)
  • 1.225 → 1.22 (exactly 5, round to even 2)
  • 1.236 → 1.24 (> 5, round up)
  • 1.234 → 1.23 (< 5, keep same)
  • 1.2351 → 1.24 (> 5, round up)
  • 1.2349 → 1.23 (< 5, keep same)
NoteWhich Rounding Rule Do I Follow?

Always follow the rounding rule specified by your instructor or textbook. This book will use Banker’s Rounding (round half to even).

TipBanker’s Rounding is Better

Want to know why Banker’s Rounding is better or other rounding rules that exist depending on the application? Read Beyond the 5: Why Scientists Use Banker’s Rounding for more!

Multi-Step Calculations

The Rule: Do not round intermediate results. Carry extra digits through all steps and only round the final answer, applying the order of operations and tracking significance at each step.

Example: (2.54 × 0.0028) / (1.12 + 0.235)

  1. Numerator Parentheses (Multiplication): 2.54 × 0.0028 = 0.007112. The result is limited by 0.0028, which has 2 sig figs. Our intermediate is 0.007112.

  2. Denominator Parentheses (Addition): 1.12 + 0.235 = 1.355. The result’s last significant digit is in the hundredths place. So, our intermediate value is 1.355 (3 sig figs).

  3. Final Division: 0.007112 / 1.355. We are dividing a number with 2 significant figures by a number with 3 significant figures. The final answer must be limited to 2 significant figures.

    • Calculator shows: 0.0052487…
    • The last significant digit is the second non-zero digit: 0.0052487…

  4. Round the Final Answer: The final answer, rounded to 2 significant figures, is 0.0052.

ImportantScaling vs Summing

8.0 + 8.0 = 16.0 but 2 x 8.0 = 16. When should you add and when should you multiply? A common and confusing paradox arises when an operation can be viewed in two ways and the choice matters. Read the Scaling vs. Summing article to find out!

References

(1)
National Institute of Standards and Technology. GLP 9: Rounding Expanded Uncertainties and Calibration Values; Good Laboratory Practice; National Institute of Standards; Technology, 2019. https://www.nist.gov/system/files/documents/2019/05/14/glp-9-rounding-20190506.pdf.