KAHIBARO
Table of Contents
Why Significant Figures Matter in Physics
In physics, every measured number has limits in how precise it is. Significant figures are a way to show that limitation directly in the number itself. When you write $3.2$ meters instead of $3.200$ meters, you are not just writing fewer digits, you are saying something about how well the length was known.
Significant figures, often called “sig figs,” are digits in a number that carry information about its precision. They help you avoid pretending your results are more accurate than your measurements allow.
Key idea: Significant figures show the precision of a measured or calculated quantity. You must not report more significant figures than your measurements justify.
In this chapter, you will learn how to count significant figures in different types of numbers and how to handle them when doing calculations. We will not yet discuss measurement techniques or error analysis in detail, because these belong to later chapters.
The scale suggests $3.0$ meters, not $3.000$ meters, because its markings limit the precision.
What Counts as a Significant Figure
A significant figure is any digit that contributes to the precision of the number. This includes all nonzero digits, some zeros, and excludes zeros that only locate the decimal point.
General rules for counting significant figures
You can summarize the rules in a compact form:
Rules for counting significant figures
- All nonzero digits are significant.
- Zeros between nonzero digits are significant.
- Leading zeros (to the left of the first nonzero digit) are not significant.
- Trailing zeros (to the right of the last nonzero digit) are significant if there is a decimal point shown.
- Trailing zeros in a whole number with no decimal point are ambiguous and are usually not treated as significant unless specified.
The following table illustrates these rules.
| Number | How to interpret | Significant figures |
|---|---|---|
| $3$ | One nonzero digit | 1 |
| $27$ | Two nonzero digits | 2 |
| $2.7$ | Two nonzero digits | 2 |
| $0.003$ | Leading zeros only position the decimal | 1 |
| $0.0030$ | One nonzero digit plus one trailing zero (decimal) | 2 |
| $100$ | Trailing zeros, no decimal, ambiguous | Often taken as 1 |
| $100.$ | Decimal shown, zeros are significant | 3 |
| $1.00$ | All digits, including trailing zeros, are precise | 3 |
| $20.0$ | Decimal point implies trailing zero is significant | 3 |
| $2000$ | Ambiguous trailing zeros | Often taken as 1 |
| $2.000\times 10^3$ | All shown digits are significant | 4 |
Nonzero digits and zeros between digits
All nonzero digits are always significant. For example, $2.56$ has three significant figures. If zeros appear between nonzero digits, they are significant, as in $2003$ which has four significant figures, or $1.08$ which has three.
Leading zeros
Leading zeros are the zeros before the first nonzero digit. They only show where the decimal point is placed and do not increase precision. For example, $0.00045$ has two significant figures, the digits $4$ and $5$.
In scientific notation, these leading zeros disappear. The number $0.00045$ is written as $4.5\times 10^{-4}$, which makes the two significant figures clearer.
Trailing zeros with a decimal point
If a number has a decimal point, zeros at the end are significant, because writing them states that the quantity was measured to that place.
For example, $3.0$ has two significant figures. It suggests that the value is known to the nearest tenth. The number $3.00$ has three significant figures. It is known to the nearest hundredth.
Similarly, $0.0500$ has three significant figures: $5$, $0$, and $0$ after the $5$.
Trailing zeros without a decimal point
A number like $1500$ is tricky. Does it have two significant figures ($1.5\times 10^3$) or four ($1.500\times 10^3$)? Written as $1500$ with no decimal point, it is ambiguous. In many introductory physics contexts, if nothing more is said, it is often treated as having two significant figures, but conventions can vary.
To avoid confusion, scientists and engineers prefer scientific notation. Then $1.5\times 10^3$ clearly has two significant figures, while $1.500\times 10^3$ clearly has four.
Significant Figures and Scientific Notation
Scientific notation is very useful when working with significant figures, especially for very large or very small numbers. It makes the number of significant figures explicit.
In scientific notation, a number is written as
$$
N = a\times 10^n,
$$
where the coefficient $a$ is between $1$ and $10$ in absolute value, and $n$ is an integer.
Only the digits written in $a$ are counted as significant. For example:
| Scientific notation | Significant figures |
|---|---|
| $3\times 10^5$ | 1 |
| $3.0\times 10^5$ | 2 |
| $3.00\times 10^5$ | 3 |
| $3.000\times 10^5$ | 4 |
| $4.50\times 10^{-3}$ | 3 |
| $4.500\times 10^{-3}$ | 4 |
Using scientific notation removes the ambiguity you saw with $1000$ or $0.00030$ written in ordinary decimal form.
Rounding to the Correct Number of Significant Figures
When you perform calculations, you often obtain a long string of digits from a calculator. You then need to round the result to the appropriate number of significant figures, matching the precision of the data that went into the calculation.
Basic rounding rules
To round a number to a given number of significant figures:
- Identify the last digit you want to keep.
- Look at the next digit to the right.
- If that next digit is less than $5$, keep the last digit the same and drop all remaining digits.
- If the next digit is $5$ or greater, increase the last kept digit by $1$ and drop all remaining digits.
For example:
| Original value | Rounded to 3 sig figs |
|---|---|
| $3.14159$ | $3.14$ (next digit $1<5$) |
| $3.14659$ | $3.15$ (next digit $6\ge 5$) |
| $0.0009876$ | $0.000988$ |
| $12749$ | $1.27\times 10^4$ |
When you round, you must preserve the position of the decimal by sometimes switching to scientific notation. Rounding $12749$ to three significant figures gives $1.27\times 10^4$, not $127$.
Rounding and intermediate steps
In physics calculations, it is usually best to keep extra digits during intermediate steps, then round only at the end. If you round too early, small rounding changes can build up and noticeably affect your final answer.
A common practice is to keep at least one more significant figure during intermediate steps than you expect to keep in your final result.
Significant Figures in Calculations
There are two main types of operations in basic physics problems: addition and subtraction, and multiplication and division. Significant figures behave differently in these two cases.
Addition and subtraction
For addition and subtraction, the key idea is to look at decimal places, not the total number of significant figures.
Rule for addition and subtraction:
The result should be rounded to the same decimal place as the least precise term in the sum or difference.
This means that if one of the numbers is only known to the nearest tenth, the result must also be given to at most the nearest tenth.
For example:
| Calculation | Raw result | Properly reported |
|---|---|---|
| $12.3 + 0.45$ | $12.75$ | $12.8$ (nearest tenth) |
| $5.67 - 2.0$ | $3.67$ | $3.7$ (nearest tenth) |
| $100.0 + 0.003$ | $100.003$ | $100.0$ (nearest tenth) |
In the last example, $100.0$ is known only to the nearest tenth, while $0.003$ is known to the nearest thousandth. The sum cannot be more precise than $100.0$, so you must round the final answer to $100.0$.
A useful way to think about this is to line up the decimal points and see where the least certain digit appears.
Multiplication and division
For multiplication and division, the rule is different.
Rule for multiplication and division:
The result should have the same number of significant figures as the factor with the fewest significant figures.
Here you ignore decimal places and count only total significant figures in each factor.
For example:
| Calculation | Factors’ sig figs | Raw result | Properly reported |
|---|---|---|---|
| $2.5\times 3.42$ | $2$ and $3$ | $8.55$ | $8.6$ (2 sig figs) |
| $4.00\times 2.0$ | $3$ and $2$ | $8.0$ | $8.0$ (2 sig figs) |
| $10.5 / 3.2$ | $3$ and $2$ | $3.28125$ | $3.3$ (2 sig figs) |
| $0.00340\times 20.0$ | $3$ and $3$ | $0.0680$ | $0.0680$ (3 sig figs) |
In the last example, both factors have three significant figures, so the product is properly written with three significant figures.
Combined operations
Many physics problems involve a mixture of operations, such as multiplying two numbers and then adding a third. The correct handling of significant figures in these mixed cases is to treat each step according to its rule, but in practice you often keep extra digits and apply the rules carefully at the end.
A reliable strategy is:
- Do the calculation in full with your calculator.
- Decide how many significant figures the final answer should have, by examining the measurement data and the type of operations.
- Round the final answer once, at the end, following the appropriate rule that dominates the overall precision.
In more advanced physics, strict significant figure rules are often replaced by a more detailed treatment of uncertainties, but the basic ideas remain similar.
Exact Numbers and Counting
Not every number in a physics problem is subject to significant figure limits. Some numbers are exact, which means they do not introduce any uncertainty and can be treated as having infinitely many significant figures.
Examples of exact numbers include:
- Pure counting: If you have $12$ identical resistors, the number $12$ is exact.
- Defined constants: Sometimes a unit definition makes a quantity exact. For example, there are exactly $100$ centimeters in $1$ meter.
- Exact integers in formulas: Factors like $2$ in $2\pi r$ or $1/2$ in $\frac{1}{2}mv^2$ are exact, they are part of the mathematical definition.
Rule for exact numbers:
Exact numbers do not limit the number of significant figures in a result. Only measured or experimentally determined quantities do.
For example, if you measure a length as $2.34$ m and multiply by an exact factor of $2$, your result is $4.68$ m, still with three significant figures.
Using Significant Figures in Physics Problems
When you solve physics problems, you usually proceed in three stages. You identify the known quantities, use a physical law to relate them, and compute the unknown. Significant figures enter mainly at the start and the end.
At the start, you identify how many significant figures each measured quantity has. At the end, you adjust your final numerical answer so that it is consistent with the least precise of the input data, using the rules for the type of calculation you have performed.
For example, suppose a car travels $35.2$ km in $0.75$ h and you want its average speed. You compute
$$
v = \frac{35.2\ \text{km}}{0.75\ \text{h}} \approx 46.933\ldots\ \text{km/h}.
$$
The distance has three significant figures, the time has two. Since you divided, the result should have two significant figures. You report the speed as $47\ \text{km/h}$, not as $46.933\ \text{km/h}$.
Significant figures do not replace careful thinking about measurements and uncertainties, but they provide a simple first rule for reporting reasonable answers in introductory physics.