Report calculations with suitable precision and significant figures

Ontario Grade 12 Physics

Scientific Investigation Skills and Career Exploration

SPH4U expectation A1.13

A calculator can display many digits, but those digits are not automatically meaningful. In physics, a calculated result should reflect the precision of the information used to calculate it. This lesson reviews significant figures and decimal-place rules, then applies them to simple physics calculations. The goal is to report results clearly without implying more precision than the data support.

What you will learn

Explain the difference between precision and accuracy.
Identify significant figures in measured values.
Choose the correct rounding rule for calculations.
Report a calculated result with suitable digits and units.

1. What precision tells you

A measured value is a number obtained by measurement. Its digits communicate how finely the value was recorded. For example, a recorded length of

1.42\ \mathrm{m}

indicates a different recorded precision from

1.4\ \mathrm{m}

. The extra digit does not prove that the first measurement is closer to the true value; it shows that it was recorded to a finer place.

Precision describes how finely a value is stated or how closely repeated measurements agree. Accuracy describes how close a result is to the accepted or actual value. A result can be precise but inaccurate. Significant figures are the digits used to communicate the precision of a measured value.

In physics, calculations often combine measured quantities, such as distance and time. The final answer should not suggest that the measurements were more precise than they were. Keep extra digits during intermediate calculator steps, then round the reported result at the end.

Precision is not the same as accuracy.
A calculator display does not decide how many digits are meaningful.
Keep unrounded intermediate values where possible.

2. Counting significant figures

Use these rules to count significant figures. Every non-zero digit is significant. Zeros between non-zero digits are significant. Zeros at the start of a decimal number are not significant; they only locate the decimal point. Zeros at the end of a decimal number are significant when the decimal point is shown.

For example,

0.00470

has three significant figures:

4

,

7

, and the final zero. The starting zeros are placeholders. The value

2.030

has four significant figures because the zero between non-zero digits and the final decimal zero both communicate recorded precision.

A whole number ending in zeros can be unclear. For example,

1500

does not clearly show whether it has two, three, or four significant figures. Scientific notation makes the intended precision clear:

1.5\times10^3

has two significant figures, while

1.500\times10^3

has four. The power of ten changes the scale, not the number of significant figures.

N=a\times10^n

Non-zero digits and zeros between non-zero digits count.
Starting zeros do not count.
Use scientific notation when trailing zeros make precision unclear.

3. Choosing the rounding rule

For multiplication and division, round the final result to the same number of significant figures as the input with the fewest significant figures. For example, if one measured input has two significant figures and another has four, the calculated result is reported with two significant figures.

For addition and subtraction, use decimal places instead. Round the final result to the same decimal place as the input with the fewest decimal places. A decimal place is a position relative to the decimal point. For example, tenths are one place after the decimal point, and hundredths are two places after it.

When rounding, inspect the digit immediately to the right of the last place you will keep. If it is

5

or greater, increase the last kept digit by one. If it is less than

5

, leave the last kept digit unchanged. Then remove the remaining digits. This is the usual rounding rule used in these examples.

Exact quantities do not limit significant figures. An exact quantity has no measurement uncertainty in the calculation, such as a counted number of objects or a defined conversion factor. In a calculation, the measured values usually set the reporting precision.

Multiplication and division: use the fewest significant figures.
Addition and subtraction: use the fewest decimal places.
Round the final result, not each intermediate result.
Exact quantities do not restrict the result's significant figures.

4. Report a complete physics result

A physics answer is more than a number. State the relevant quantity and its unit. If the quantity is a vector, include its direction or sign convention. A scalar has magnitude only; speed is an example. A vector has magnitude and direction; velocity is an example.

Before calculating, identify the physical system, meaning the object or group being considered. State the reference frame, the viewpoint from which position or motion is described, and the positive direction when direction matters. For a scalar calculation such as average speed, a direction is not needed. These choices do not change the significant-figure rules, but they help make the result interpretable.

A final check should ask whether the unit matches the calculated quantity, whether the rounding follows the input precision, and whether the size is reasonable for the values given. Do not add digits merely because a calculator displays them.

\text{reported result}=(\text{rounded value})(\text{unit})

Report a number with the appropriate unit.
For vector results, state direction using a clear convention.
Check units, precision, and reasonableness before submitting.

Worked example

Average speed from measured distance and time

A cart travels

18.6\ \mathrm{m}

in

4.2\ \mathrm{s}

. Calculate and report its average speed with suitable precision.

Set up the situation
The system is the cart. The reference frame is the track, and the chosen positive direction is along the cart's path. Average speed is a scalar, so the final result needs no direction. The unknown is average speed.
Choose the relationship
Average speed is distance divided by elapsed time. This is a division, so the result must have the same number of significant figures as the measured input with fewer significant figures.
$v_{\mathrm{avg}}=\frac{d}{\Delta t}$
Substitute measured values
The distance has three significant figures, while the time has two. Keep the calculator value for now, but plan to report two significant figures.
$v_{\mathrm{avg}}=\frac{18.6\ \mathrm{m}}{4.2\ \mathrm{s}}=4.428571\ldots\ \mathrm{m/s}$
Round and check
To two significant figures, the result is $4.4\ \mathrm{m/s}$ . The unit is distance per time, as expected for speed. A distance of about $19\ \mathrm{m}$ in about $4\ \mathrm{s}$ gives a speed near $5\ \mathrm{m/s}$ , so the result is reasonable.
$v_{\mathrm{avg}}=4.4\ \mathrm{m/s}$

Answer: The cart's average speed is

4.4\ \mathrm{m/s}

.

Check: The answer has two significant figures, matching the least precise measured input, and its unit is metres per second.

Worked example

Adding measured lengths

Two straight segments have measured lengths of

2.35\ \mathrm{m}

and

0.8\ \mathrm{m}

. Find their combined length and report it correctly.

Identify the quantity
The system is the two connected segments, measured in the same reference frame. Length is a scalar. The unknown is the total length.
Choose the relationship
The combined length is the sum of the two measured lengths. For addition, the final answer is rounded to the least precise decimal place. The second measurement is recorded only to the tenths place.
$L_{\mathrm{total}}=L_1+L_2$
Substitute and round
Add the lengths while retaining the units. The unrounded sum is $3.15\ \mathrm{m}$ . Since the measurements support only tenths, round the hundredths digit, which is $5$ , up.
$L_{\mathrm{total}}=2.35\ \mathrm{m}+0.8\ \mathrm{m}=3.15\ \mathrm{m}\approx3.2\ \mathrm{m}$
Check the report
The result is stated to the tenths place and has units of metres. A total slightly above $3\ \mathrm{m}$ is consistent with adding lengths of about $2.4\ \mathrm{m}$ and $0.8\ \mathrm{m}$ .
$L_{\mathrm{total}}=3.2\ \mathrm{m}$

Answer: The combined length is

3.2\ \mathrm{m}

.

Check: Addition uses decimal places, not the fewest significant figures. The final result is reported to the tenths place.

Worked example

Kinetic energy from measured mass and speed

A ball has a measured mass of

0.250\ \mathrm{kg}

and a speed of

6.0\ \mathrm{m/s}

. Calculate its kinetic energy with suitable precision.

Define the system and quantities
The system is the ball, described relative to the floor. Choose the ball's direction of travel as positive; however, kinetic energy is a scalar and has no direction. The unknown is kinetic energy.
Use the course relationship
Kinetic energy depends on mass and the square of speed. Squaring the speed does not change the rule for reporting this multiplication-based calculation: the final result follows the input with the fewest significant figures.
$E_k=\frac{1}{2}mv^2$
Substitute with units
The mass has three significant figures and the speed has two. The factor $\frac{1}{2}$ is exact. Calculate with the measured values and retain extra calculator digits until the final step.
$E_k=\frac{1}{2}(0.250\ \mathrm{kg})(6.0\ \mathrm{m/s})^2=4.50\ \mathrm{kg\,m^2/s^2}$
Round and verify
The speed has two significant figures, so report two significant figures: $4.5\ \mathrm{J}$ . A joule is equivalent to a kilogram metre squared per second squared. The value is positive, as kinetic energy should be, and is consistent with a small mass moving at several metres per second.
$E_k=4.5\ \mathrm{J}$

Answer: The ball's kinetic energy is

4.5\ \mathrm{J}

.

Check: The result has two significant figures, the unit is joules, and the scalar energy is positive.

Common mistakes and how to avoid them

Keeping every digit shown by the calculator.

Correction: Retain extra digits during intermediate steps, but round the reported result according to the measured inputs.

Using the fewest significant figures for addition.

Correction: For addition and subtraction, use the fewest decimal places. Apply the significant-figure rule to multiplication and division.

Counting starting zeros as significant.

Correction: Starting zeros only locate the decimal point. In

0.00470

, only

4

,

7

, and the final zero count.

Removing a trailing zero that communicates precision.

Correction: A trailing zero after a decimal point can be significant. Keep it when it conveys the measured precision.

Reporting a number without a unit or vector direction.

Correction: Include the appropriate unit. For a vector, also state its direction or identify the positive direction used.

Lesson summary

Significant figures communicate the precision of measured values.
For multiplication and division, report the result with the fewest significant figures among measured inputs.
For addition and subtraction, report the result to the least precise decimal place.
Keep extra digits during intermediate calculations and round at the end.
Include units and, when needed, direction. Check the result's precision and reasonableness.

Check your understanding

Question 1

How many significant figures are in

0.00620

?

Two
Three
Four
Five

Show answer and explanation

Three

The starting zeros are placeholders. The digits

6

,

2

, and the final decimal zero are significant, so there are three.

Question 2

A calculation adds

1.26\ \mathrm{m}

and

0.4\ \mathrm{m}

. What is the correctly reported sum?

$1.66\ \mathrm{m}$
$1.7\ \mathrm{m}$
$2\ \mathrm{m}$
$1.660\ \mathrm{m}$

Show answer and explanation

1.7\ \mathrm{m}

The unrounded sum is

1.66\ \mathrm{m}

. Addition uses decimal places, and

0.4\ \mathrm{m}

is recorded to the tenths place, so the result is

1.7\ \mathrm{m}

.

Question 3

A measured quantity with three significant figures is multiplied by one with two significant figures. How many significant figures should the final result have?

One
Two
Three
Five

Show answer and explanation

Two

For multiplication, the result uses the fewest significant figures among the measured inputs. The input with two significant figures sets the limit.

Key terms

Accuracy: How close a result is to an accepted or actual value.
Precision: How finely a value is stated or how closely repeated measurements agree.
Significant figures: The digits in a measured value that communicate its recorded precision.
Decimal place: A digit position counted from the decimal point, such as tenths or hundredths.
Exact quantity: A quantity known by definition or counting rather than limited by a measurement.
Reference frame: The viewpoint used to describe an object's position or motion.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Physics (SPH4U), expectation A1.13. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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