How to Determine Sig Figs When Converting Units?
You determine significant figures when converting units by keeping the same number of significant figures as the original measurement. The conversion factor itself does not limit significant figures because it is considered exact. After converting, round the final value so it matches the sig figs of the starting number. This keeps the precision consistent and avoids overstating accuracy.
Understanding Sig Figs in Unit Conversions
Converting units is something you do all the time in science and math. But keeping your sig figs right during conversions trips up lots of people.
Why Sig Figs Matter During Conversions
Your original measurement has a certain level of precision. When you convert it to different units, that precision doesn’t magically improve or disappear.
If you measure a table as 3.5 feet long, that’s two significant figures. You’re certain about the 3 and reasonably sure about the 5. Converting to inches doesn’t make your measurement more or less precise. The table is still measured to two sig figs.
According to chemistry education resources, the conversion process shouldn’t add false precision or throw away real information from your measurements.
The Basic Rule
Here’s the simple version: after converting units, keep the same number of sig figs as your starting measurement had. The conversion factor usually doesn’t limit your sig figs.
Convert 45 meters to centimeters. The number 45 has two sig figs. Multiply by 100 cm/m to get 4,500 cm. Your answer needs two sig figs, so write 4,500 cm (or 4.5 × 10³ cm to make it clearer).
The 100 in the conversion factor doesn’t count. It’s an exact number.
Exact Numbers vs Measured Numbers
Not all numbers in conversions are equal. Some are exact, others are measured. This difference changes everything about sig figs.
What Makes a Number Exact?
Exact numbers come from definitions or counting. They’re known with perfect certainty. They have infinite significant figures.
Counted items are exact. If there are 12 eggs in a carton, that’s exactly 12, not 11.9 or 12.1. You could write it as 12.0000000… with infinite zeros. Those eggs don’t care about sig figs.
Defined relationships are exact. The metric system defines 1 meter as exactly 100 centimeters. Not 99.99 cm or 100.01 cm. Exactly 100. This means the conversion factor 100 cm/1 m is exact.
Educational research shows that exact numbers never limit the precision of your calculations. They have infinite sig figs, so they don’t count when determining your answer’s sig figs.
Common Exact Conversion Factors
Most metric conversions are exact because they’re defined that way:
- 1 kilometer = 1,000 meters (exact)
- 1 meter = 100 centimeters (exact)
- 1 kilogram = 1,000 grams (exact)
- 1 liter = 1,000 milliliters (exact)
All metric prefixes create exact conversions. Kilo- means exactly 1,000. Centi- means exactly 1/100. Milli- means exactly 1/1,000. These aren’t approximations. They’re definitions.
Within the US customary system, many conversions are exact too:
- 1 foot = 12 inches (exact)
- 1 yard = 3 feet (exact)
- 1 mile = 5,280 feet (exact)
- 1 gallon = 4 quarts (exact)
These are defined relationships, not measurements, so they don’t affect your sig figs.
When Conversion Factors Are Measured
Some conversion factors come from measurements, not definitions. These have a limited number of sig figs and DO affect your final answer.
The conversion between US and metric units often involves measured values:
- 1 mile ≈ 1.609 kilometers (measured, 4 sig figs)
- 1 pound ≈ 453.6 grams (measured, 4 sig figs)
- 1 gallon ≈ 3.785 liters (measured, 4 sig figs)
One big exception: 1 inch is defined as exactly 2.54 centimeters. This used to be measured but got redefined, so now it’s exact.
When your conversion factor is measured, you must consider its sig figs. If it has fewer sig figs than your measurement, it limits your answer.
Step-by-Step: Converting with Sig Figs
Let’s walk through the process of converting units while keeping sig figs correct.
Step 1: Count Sig Figs in Your Measurement
Before you start converting, figure out how many sig figs your starting number has. Understanding significant figures rules helps you count correctly.
Example: You measure 25.0 mL of water. That’s three sig figs (the zero after the decimal counts).
Step 2: Identify Your Conversion Factor
What conversion factor do you need? Is it exact or measured?
To convert 25.0 mL to liters, you use 1 L = 1,000 mL. This is exact (metric prefix), so it has infinite sig figs.
Step 3: Set Up and Calculate
Multiply your measurement by the conversion factor. Let your calculator give you all the digits.
25.0 mL × (1 L / 1,000 mL) = 0.0250 L
The calculator shows 0.025 or 0.0250 depending on settings. Don’t round yet.
Step 4: Round to Correct Sig Figs
Now apply the sig fig rules. For multiplication (which is what unit conversion is), your answer gets the same number of sig figs as the least precise number in the calculation.
Your measurement: 25.0 mL (3 sig figs) Your conversion factor: 1,000 mL = 1 L (exact, infinite sig figs)
The measurement limits your answer, so keep 3 sig figs: 0.0250 L
Those zeros after the 5 matter! They show your three sig figs. Writing 0.025 L would only show two sig figs.
Practical Examples
Real examples help this stuff stick. Let’s look at different types of conversions.
Example 1: Metric to Metric
Convert 3.5 meters to centimeters.
Starting measurement: 3.5 m (2 sig figs) Conversion factor: 100 cm = 1 m (exact)
Calculation: 3.5 m × (100 cm / 1 m) = 350 cm
Since the conversion is exact, only your measurement’s 2 sig figs matter. Answer: 350 cm (or 3.5 × 10² cm to make the sig figs clearer).
The tricky part here is that 350 looks like it might be 2 or 3 sig figs. Writing it in scientific notation removes the ambiguity.
Example 2: US System to US System
Convert 18 inches to feet.
Starting measurement: 18 in (2 sig figs) Conversion factor: 12 in = 1 ft (exact)
Calculation: 18 in × (1 ft / 12 in) = 1.5 ft
Answer: 1.5 ft (2 sig figs)
The 12 in the conversion is exact, so it doesn’t limit your precision. Your answer has the same sig figs as your measurement.
Example 3: Metric to US System (Measured Conversion)
Convert 87 kilograms to pounds.
Starting measurement: 87 kg (2 sig figs) Conversion factor: 1 kg ≈ 2.205 lb (measured, 4 sig figs)
Calculation: 87 kg × 2.205 lb/kg = 191.835 lb
Now both numbers matter. You have 2 sig figs from 87 and 4 sig figs from 2.205. The smaller number (2) limits your answer.
Answer: 190 lb (2 sig figs)
You round 191.835 to 190 (or 1.9 × 10² lb) to show 2 sig figs.
Example 4: Multiple Conversions
Convert 4.7 liters to milliliters.
Starting measurement: 4.7 L (2 sig figs) Conversion factor: 1 L = 1,000 mL (exact)
Calculation: 4.7 L × (1,000 mL / 1 L) = 4,700 mL
The calculation gives 4,700, which looks like 2, 3, or 4 sig figs. Your measurement has 2 sig figs, so your answer needs 2 sig figs.
Answer: 4,700 mL (2 sig figs), or better yet, write 4.7 × 10³ mL to make it clear.
Example 5: Chain Conversions
Sometimes you convert through multiple steps. Convert 58.2 milliseconds to megaseconds.
58.2 ms → seconds → megaseconds
Both conversions use metric prefixes (exact).
58.2 ms × (1 s / 1,000 ms) × (1 Ms / 1,000,000 s) = 0.0000000582 Ms = 5.82 × 10⁻⁸ Ms
Your starting measurement has 3 sig figs. All conversion factors are exact. Your answer gets 3 sig figs: 5.82 × 10⁻⁸ Ms.
Common Mistakes to Avoid
Lots of people mess up sig figs during conversions. Here are the big mistakes and how to avoid them.
Mistake 1: Counting the Conversion Factor
This is the most common error. People see 100 cm = 1 m and think “that 100 has lots of sig figs.”
Wrong! That 100 is exact. It has infinite sig figs. It doesn’t limit your answer.
If you measure 5.2 m and convert to centimeters, you get 520 cm (2 sig figs), not 5.2 × 10² cm because the 100 has “1 sig fig.” The 100 doesn’t count.
Mistake 2: Losing Trailing Zeros
When your answer needs trailing zeros to show the right number of sig figs, you must include them.
Convert 12.0 inches to centimeters using 1 in = 2.54 cm (exact).
12.0 in × 2.54 cm/in = 30.48 cm
Your measurement has 3 sig figs. Your answer needs 3 sig figs: 30.5 cm (rounding 30.48).
Writing 30 cm would only show 1 or 2 sig figs. That’s wrong. You measured to 3 sig figs, so report 3.
Mistake 3: Not Recognizing Measured Conversions
Some people assume all conversions are exact. They’re not.
When converting between US and metric systems (except inches to cm), check if your conversion factor is measured. If it is, its sig figs might limit your answer.
Converting 120 pounds to kilograms using 1 lb ≈ 0.454 kg (3 sig figs):
120 lb × 0.454 kg/lb = 54.48 kg
You have 2 sig figs from 120 and 3 sig figs from 0.454. The smaller number wins.
Answer: 54 kg (2 sig figs)
Mistake 4: Rounding Too Early
Never round in the middle of a conversion. Keep all digits until the very end.
If you’re converting meters to feet to inches, don’t round after the first conversion. Carry all the digits through both steps, then round once at the end.
Early rounding causes errors that pile up. The rule for significant figures in multiplication and division applies only to your final answer.
Special Cases and Tricky Situations
Some conversions are trickier than others. Here’s how to handle them.
Converting with Addition or Subtraction
Most unit conversions involve multiplication. But sometimes you add or subtract after converting.
Example: You measure two lengths in different units and want to add them.
Length 1: 2.5 feet = 30 inches (using exact conversion) Length 2: 18.7 inches
Total: 30 + 18.7 = 48.7 inches
Wait! For addition, you follow decimal place rules, not sig fig rules. The number 30 has no decimal places. Your answer should have no decimal places: 49 inches.
This is tricky. Even though 2.5 feet has 2 sig figs, once converted to 30 inches, you’re adding a whole number.
Scientific Notation Makes It Clear
When dealing with trailing zeros and ambiguous sig figs, scientific notation saves the day.
Instead of writing 2,500 (which could be 2, 3, or 4 sig figs), write:
- 2.5 × 10³ for 2 sig figs
- 2.50 × 10³ for 3 sig figs
- 2.500 × 10³ for 4 sig figs
This removes all doubt about your precision.
Density as a Conversion Factor
Density connects mass and volume, making it a conversion factor. But density is always measured, never exact.
Water’s density is about 1.00 g/mL (3 sig figs at 4°C). If you use this to convert between mass and volume, its sig figs matter.
Convert 25.0 mL of water to grams:
25.0 mL × (1.00 g / 1 mL) = 25.0 g
Both numbers have 3 sig figs. Your answer gets 3 sig figs: 25.0 g.
If you used a less precise density like 1.0 g/mL (2 sig figs), your answer would be 25 g (2 sig figs).
Temperature Conversions Are Different
Temperature conversions don’t follow the same rules. Converting from Celsius to Fahrenheit uses the formula F = (9/5)C + 32.
The 9/5 and 32 are exact (defined by the formula). But you’re doing addition, not just multiplication, so the rules change slightly.
If you have 25.0°C (3 sig figs), converting gives (9/5)(25.0) + 32 = 45.0 + 32 = 77.0°F.
The addition follows decimal place rules. Both numbers have 1 decimal place, so your answer has 1 decimal place: 77.0°F.
When to Keep Extra Digits
Sometimes it’s smart to keep extra digits temporarily during calculations. This prevents rounding errors from building up.
During Multi-Step Problems
If you need to do several conversions or calculations in a row, keep at least one extra digit in intermediate steps.
Example: Convert 15.5 m/s to miles per hour
Step 1: Convert meters to kilometers 15.5 m/s × (1 km / 1,000 m) = 0.0155 km/s
Keep this as 0.0155, not rounding to 0.016 yet.
Step 2: Convert seconds to hours 0.0155 km/s × (3,600 s / 1 hr) = 55.8 km/hr
Step 3: Convert kilometers to miles 55.8 km/hr × (1 mi / 1.609 km) = 34.677… mph
Now round to 3 sig figs (from original 15.5): 34.7 mph
If you had rounded at each step, errors would accumulate.
Final Rounding Only
Always round just once, at the very end. This is called “carrying extra digits” through your work.
Your calculator can hold many digits. Let it. Write down a few extra in your scratch work. Round only when you report your final answer.
This keeps your work as accurate as possible.
Quick Reference Guide
Here’s a simple guide to help you remember the key points.
Exact Conversion Factors (Infinite Sig Figs):
- All metric prefix conversions (km ↔ m ↔ cm ↔ mm)
- All mass metric conversions (kg ↔ g ↔ mg)
- All volume metric conversions (L ↔ mL)
- US system internal conversions (ft ↔ in, lb ↔ oz)
- 1 inch = 2.54 cm (special exact conversion)
Measured Conversion Factors (Limited Sig Figs):
- Most US ↔ metric conversions (except inch ↔ cm)
- Density values
- All approximate conversions
Basic Steps:
- Count sig figs in your measurement
- Check if conversion is exact or measured
- Do the math
- If conversion is exact: answer has same sig figs as measurement
- If conversion is measured: answer has fewest sig figs from either number
- Round only at the end
Practice Makes Perfect
The best way to get good at this is practice. Try converting different measurements and checking your sig figs.
Start with simple metric conversions where everything is exact. Once those feel easy, try US-to-metric conversions where you need to watch the conversion factor’s sig figs.
Check your work with examples. Does your answer make sense? Did you keep or lose precision appropriately?
Over time, this becomes second nature. You’ll automatically know when conversion factors count and when they don’t.
Final Thoughts
Determining sig figs during unit conversions is simpler than it looks. The main rule is this: your converted answer keeps the same precision as your original measurement because most conversions are exact. Exact conversion factors (like metric prefixes or defined relationships) have infinite sig figs and don’t limit your precision. Measured conversion factors (like many US-metric conversions) do have limited sig figs and might affect your answer.
Remember to count your measurement’s sig figs first, check if your conversion factor is exact or measured, and round only once at the end. Understanding when to use rounding decimals to three significant figures or any other precision level comes with practice.
Need help double-checking your significant figures? Use a significant figures calculator to verify your conversions and build confidence in your calculations. With practice, determining sig figs during unit conversions becomes automatic, and you’ll handle any conversion with confidence.
