Two students measure the same wire. One writes 8.4 cm; the other copies 8.400000 cm from a calculator. The second answer looks more impressive, but the calculator has not inspected the ruler. We need a way to show what a measurement actually supports.
Significant figures are a compact way of reporting measured information. They are not a test of how many digits a calculator can display. This chapter starts with the meaning of a reading, builds the counting and rounding rules, and then shows how to carry that information through a calculation. You need decimal place value and ordinary arithmetic; calculus is not required.
What you will learn to do
- Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal.
- Count meaningful digits without counting place-holding zeros or powers of ten.
- Choose the appropriate reporting rule for addition, multiplication and mixed calculations.
- Distinguish reporting precision from accuracy, exact constants and a stated uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Does writing 2.50 m instead of 2.5 m change the length?
Compare your reasoning
The numerical length is equal. In a measurement report, the extra zero communicates a finer reported precision. This chapter separates the quantity from the information carried by its notation.
Revisit the explanation →01 / Significant figures
Start with the measurement, not the rule
A measurement compares a physical quantity with a unit. The number 4.7 cm tells us both a size and a chosen reporting place: tenths of a centimetre. Writing 4.70 cm retains a hundredths place. The extra zero says something about the report even though adding a trailing zero does not change the central value.
Think of the last reported digit as the edge of the information being communicated. In the usual introductory convention, significant figures include the reliably established digits and the final estimated or uncertain digit. That final digit is useful; it is not a licence to append another six guessed digits.
A real instrument also has calibration errors, drift, alignment and reading uncertainty. Therefore the last printed digit alone cannot establish a guaranteed interval for the true length. We will first study how to report numbers, then separate that convention from a complete uncertainty statement.
TRY THE IDEA
One length, three reporting resolutions
Keep the model length fixed. Change the reporting step before moving the strip. Predict which readout changes first when the strip becomes slightly longer.
Reported length4.4 cm
Rounding interval only4.35 to 4.45 cm
Nearest-step rounding; exact ties go to an even final digit. The labelled interval shows the two halfway boundaries, not a measured confidence interval. Try 4.35 cm at a 0.1 cm step.
STOP & THINK
Two reports are 4.7 cm and 4.70 cm. What follows from the notation alone?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures / Start with the measurement, not the rule Problem: Two reports are 4.7 cm and 4.70 cm. What follows from the notation alone? Choices: 1. The second object is longer 2. The second report retains a finer decimal place 3. The second instrument is certainly better calibrated Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: Start with the measurement, not the rule Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
02 / Significant figures
Decide what each zero is doing
Begin at the first nonzero digit. Zeros to its left only locate the decimal point; they do not record extra precision. In 0.00380 m, the early zeros say that the length is much smaller than a metre. The digits 3, 8 and the final 0 form the reported significant part.
A zero between meaningful digits is different. In 3.008 m, deleting either interior zero changes the value, so both count. A zero after the decimal at the end of a measured number also counts: 3.80 records a finer reporting place than 3.8.
An integer such as 4800 m is ambiguous unless a convention or uncertainty is supplied. It might mean a value reported to hundreds, tens or units. Some books adopt a house rule for these zeros; do not mistake that rule for information present in the bare number. Scientific notation removes the ambiguity.
| Report | Significant figures | Reason |
|---|---|---|
| 0.00380 m | 3 | Leading zeros locate the decimal; the final zero is retained. |
| 3.008 m | 4 | Both zeros lie inside the reported significant part. |
| 3.800 × 10³ m | 4 | Only the coefficient determines the digit count. |
| 4800 m | Not uniquely specified | Trailing integer zeros need context. |
| 4.80 × 10³ m | 3 | The coefficient explicitly retains its final zero. |
WORKED EXAMPLE
A unit conversion must preserve the information
A length is reported as 0.00620 m. Express it in millimetres without changing its significant-figure information.
Convert the physical unit exactly, then preserve the three meaningful digits.
The first meaningful digit is 6; the digits 6, 2 and the final zero give three significant figures.
The metre-to-millimetre conversion is an exact definition, so it adds no measurement limitation.
Retain the last zero. Writing 6.2 mm would discard reporting information; writing 6.200 mm would add an unsupported place.
What this gives usThe equivalent report is 6.20 mm, with three significant figures.
Check the result another way
The last place was 0.00001 m and is now 0.01 mm. These are the same physical length increment.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: A unit conversion must preserve the information Problem: A length is reported as 0.00620 m. Express it in millimetres without changing its significant-figure information. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
STOP & THINK
How many significant figures are in the measured value 0.04050 kg?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures / Decide what each zero is doing Problem: How many significant figures are in the measured value 0.04050 kg? Requested answer unit: figures Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: Decide what each zero is doing Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Significant figures
Separate measured values from exact information
Suppose you measure the total mass of five identical washers and divide by five. The count of five washers is exact if none is missing. It does not restrict the answer to one significant figure. A measured mass of 5 g would be a different kind of information.
Defined conversion factors are exact too: 100 cm equals 1 m by definition. The factor 2 in a diameter-to-radius relation is mathematical, not an imprecise measurement. Such values are not the limiting inputs in the usual significant-figure rule.
The mathematical constant π is exact, but a decimal approximation such as 3.14 is not identical to π. Likewise, 22/7 is an exact rational number, but is only an approximation to π. Keep a calculator value of π until the final rounding unless a question explicitly instructs you to use a specified approximation.
WORKED EXAMPLE
An exact count does not limit the answer
Five washers have a measured total mass of 18.65 g. What average mass should be reported?
The mass has four significant figures. The count five is exact.
Divide the measured total by the exact number of washers.
The numerical quotient is 3.73, but retaining four significant figures requires the final zero.
This is an average mass. The calculation alone does not prove all five individual masses are identical.
What this gives usReport the average as 3.730 g.
Check the result another way
Multiplying the unrounded average by the exact count recovers 18.65 g.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: An exact count does not limit the answer Problem: Five washers have a measured total mass of 18.65 g. What average mass should be reported? Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: Separate measured values from exact information Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Significant figures
Round once, and state what happens at a tie
To round to a chosen place, keep every digit through that place and inspect the entire discarded part. Less than halfway leaves the retained value unchanged. More than halfway increases the last retained digit by one. A carry can cross the decimal point or change the power of ten.
Exactly halfway requires a convention. In this chapter, an exact tie is rounded to the nearest value whose final retained digit is even. Thus 2.450 rounded to two significant figures is 2.4, while 2.550 becomes 2.6. If the discarded tail contains any further nonzero digit, it is not an exact tie: 2.4501 becomes 2.5.
Another common classroom convention rounds every exact half upward for positive values. Neither convention supplies extra measurement information. Follow an explicitly stated exam or laboratory rule; when a tie affects the answer, state the rule. Our examples identify such cases rather than leaving you to guess.
| Input | Target | Report under ties-to-even |
|---|---|---|
| 7.362 | 3 significant figures | 7.36 |
| 7.368 | 3 significant figures | 7.37 |
| 2.450 | 2 significant figures | 2.4 |
| 2.550 | 2 significant figures | 2.6 |
| 9.96 | 2 significant figures | 1.0 × 10¹ |
The final line needs scientific notation because the central value becomes ten. Writing simply 10 can leave its intended significant-figure count unclear. The coefficient 1.0 makes the two retained figures explicit. Do not count the exponent as another significant figure.
STOP & THINK
Using ties-to-even, how should 6.850 be rounded to three significant figures?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures / Round once, and state what happens at a tie Problem: Using ties-to-even, how should 6.850 be rounded to three significant figures? Choices: 1. 6.85 2. 6.80 3. 6.9 Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: Round once, and state what happens at a tie Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Significant figures
For addition, ask which decimal place is supported
Adding lengths combines absolute sizes. A reading reported only to tenths cannot suddenly justify the hundredths and thousandths in a sum. This is why addition and subtraction use the least precise decimal place after all values are expressed in compatible units.
Align the decimal points, calculate with the available digits, and then round the final sum to that place. Counting significant figures before adding can select the wrong answer because the sizes of the numbers may be very different.
WORKED EXAMPLE
Add readings from different resolutions
Two measured lengths are 8.2 cm and 0.367 cm. Report their sum using the usual addition rule.
Identify the last supported decimal place before rounding the arithmetic.
The first length ends at tenths; the second ends at thousandths. Tenths is the limiting place.
Compute the unrounded sum.
The hundredths digit is 6, so the tenths digit rounds upward.
What this gives usThe reported sum is 8.6 cm.
Check the result another way
The final decimal place matches the less finely reported input. The sum has two significant figures here, but decimal place—not that count—selected the rule.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Add readings from different resolutions Problem: Two measured lengths are 8.2 cm and 0.367 cm. Report their sum using the usual addition rule. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Subtraction can leave a small difference between two large readings. If the readings are 47.82 cm and 47.3 cm, their unrounded difference is 0.52 cm and the ordinary reporting rule gives 0.5 cm. Several leading digits cancel; you have not gained relative precision just because the answer is small.
If a small difference is physically important, an explicit uncertainty calculation is better than a digit-counting rule. For example, two uncertain nearly equal forces can have a resultant whose direction is not established by the nominal values alone.
STOP & THINK
What should 12.06 m + 0.8 m be reported as under the ordinary addition rule?
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures / For addition, ask which decimal place is supported Problem: What should 12.06 m + 0.8 m be reported as under the ordinary addition rule? Choices: 1. 12.86 m 2. 13 m 3. 12.9 m Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: For addition, ask which decimal place is supported Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
06 / Significant figures
For products, compare relative precision
Multiplication scales one measured number by another. The size of the last place now changes with the value, so significant-figure count is used as a rough reporting proxy for relative precision. Under the usual rule, a product or quotient retains the smallest significant-figure count among its measured inputs.
This is a reporting heuristic, not a complete derivation of propagated uncertainty. When numerical uncertainty bounds or standard uncertainties are supplied, use that information instead. Keep exact conversion factors and mathematical coefficients out of the competition for the smallest count.
WORKED EXAMPLE
Report an area, including its units
A rectangular sheet measures 2.36 m by 1.4 m. What area should be reported?
Area is a product; the second measurement has the smaller significant-figure count.
There are three significant figures in 2.36 and two in 1.4.
Multiply without rounding intermediate digits.
Round the result to two significant figures, not two decimal places.
What this gives usReport 3.3 m².
Check the result another way
The units multiply to square metres. A result of 3.30 m² would claim three significant figures, even though it has the same central value.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: Report an area, including its units Problem: A rectangular sheet measures 2.36 m by 1.4 m. What area should be reported? Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: For products, compare relative precision Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Significant figures
Mixed calculations: keep the information and the guard digits
A multi-step calculation has two tasks: track the reporting limit introduced by each operation, and preserve enough digits to avoid a second rounding error. Do not solve that tension by rounding every line. Keep the unrounded numerical intermediate value while noting the place or count that will limit the final report.
The distinction matters because rounding an intermediate value can push a later product across a rounding boundary. Extra working digits are often called guard digits. They support the arithmetic; they are not extra information to publish at the end.
WORKED EXAMPLE
An early rounding changes the final answer
Evaluate (12.4 cm + 0.36 cm) × 2.15 using ordinary reporting rules. The factor 2.15 is measured and dimensionless.
Track the addition’s tenths-place limit, but carry its unrounded value into the multiplication.
The unrounded sum is 12.76 cm. Its report would end at tenths and have three significant figures near this value.
Carry 12.76 into the product rather than replacing it by 12.8.
Both the sum’s reporting limit and the factor support three significant figures at this scale.
What this gives usThe final report is 27.4 cm.
Check the result another way
Rounding the sum early would give 12.8 × 2.15 = 27.52 and then 27.5 cm. The difference is an avoidable arithmetic rounding effect, not a new physical measurement.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Worked-example topic: An early rounding changes the final answer Problem: Evaluate (12.4 cm + 0.36 cm) × 2.15 using ordinary reporting rules. The factor 2.15 is measured and dimensionless. Guide me through reconstructing the argument myself. Ask why each step is justified before moving to the next. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
For delicate cancellation or a long nonlinear calculation, simple digit rules can become a poor guide. Carry an explicit uncertainty model instead of mechanically counting digits through every transformation. The next chapters explain what such a model means and how bounds propagate.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: Mixed calculations: keep the information and the guard digits Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
08 / Significant figures
Use the rules as a report, not as proof of accuracy
A useful final answer contains a value, a unit and an appropriate statement of precision. Read the problem for explicit instructions before choosing a convention. If the question supplies exact idealized model values, do not invent measurement errors that were not specified. If it explicitly asks for significant figures, preserve the significant zeros in the answer.
A result such as (6.42 ± 0.08) cm tells you more than 6.42 cm alone. The ± term must still be identified: it may be a bound, a standard uncertainty or something else. A decimal place is not automatically a one-standard-deviation uncertainty. That distinction will matter in experimental questions.
Before moving on, explain aloud why the sum rule and the product rule differ. Then solve the checks below without opening the examples. If your explanation is only “because that is the rule,” revisit the discussion of absolute size and relative precision; those ideas connect the rules to measurement.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Focus: Use the rules as a report, not as proof of accuracy Learning goals: - Explain why 4.7 cm and 4.70 cm can carry different measurement information even though their central numerical values are equal. - Count meaningful digits without counting place-holding zeros or powers of ten. - Choose the appropriate reporting rule for addition, multiplication and mixed calculations. - Distinguish reporting precision from accuracy, exact constants and a stated uncertainty. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BRING THE IDEAS TOGETHER
What should stay with you
- A trailing decimal zero can communicate precision even when it does not change the central value.
- Leading zeros and the exponent in scientific notation do not add significant figures.
- Addition/subtraction use the least precise decimal place; multiplication/division use the limiting measured significant-figure count.
- Exact counts and defined conversions do not limit the count. π is not exactly 22/7.
- Keep guard digits, declare a halfway rule, and use explicit uncertainty information when it is available.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
How many significant figures are explicitly written in 7.030 × 10⁻⁴ s?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Problem: How many significant figures are explicitly written in 7.030 × 10⁻⁴ s? Requested answer unit: figures Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 2
Which report unambiguously states 5200 m to three significant figures?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Problem: Which report unambiguously states 5200 m to three significant figures? Choices: 1. 5200 m 2. 5.20 × 10³ m 3. 5.2 × 10³ m Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 3
Round 3.650 to two significant figures using ties-to-even.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Problem: Round 3.650 to two significant figures using ties-to-even. Choices: 1. 3.6 2. 3.7 3. 3.65 Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 4
Report the quotient of measured 9.60 m and 4.0 s.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Problem: Report the quotient of measured 9.60 m and 4.0 s. Choices: 1. 2.400 m/s 2. 2.40 m/s 3. 2.4 m/s Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
SELF CHECK 5
A thermometer displays 21.000 °C but has an uncorrected zero error. What can you conclude?
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Significant figures Problem: A thermometer displays 21.000 °C but has an uncorrected zero error. What can you conclude? Choices: 1. The extra decimals remove the error 2. The display is finely resolved but may still be inaccurate 3. Averaging identical displayed readings guarantees accuracy Help me choose a model and explain my reasoning before checking my answer. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
Keep the last justified decimalBeginner · Free sample · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Explain why the zeros in 0.00450 do not all do the same job.
Compare your reasoning
The leading zeros locate the decimal point. The final zero records a reported digit. The value has three significant figures: 4, 5 and the final 0.
Revisit the explanation →A few days later: change the situation
A length is reported as 12.3 cm and another as 0.46 cm. How would you report their sum, and why?
Compare your reasoning
The unrounded sum is 12.76 cm. Under the usual decimal-place reporting rule it becomes 12.8 cm, because the first input is reported only to tenths. A separate uncertainty model would give a fuller account.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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