A measured value states what the instrument could resolve. Every digit written down is part of that claim.
Step 1: Something to Notice
Watch first. The explanation comes later.
Measure a rectangle with a meter stick marked in centimeters and a calculator returns an area to eight decimal places.
Where did those extra digits come from, if nobody measured them?
Step 2: Find Out
Isolate the instrument and observe how many digits the same rectangle supports, then try reporting more.
Step 1 — Predict
Two measurements have 3 and 2 significant figures. How many does their product justify?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
Precision is not accuracy
Precision is how finely a value is resolved; accuracy is how close it is to the truth. An instrument can be one without the other.
Precision How finely a measurement is resolved.
Significant figures record it
The digits that were actually measured, including the last uncertain one. They are a record, not a convention.
Significant figures The digits in a value that were actually measured.
Multiplication takes the fewest
A result cannot be more certain than its most uncertain input, so the product takes the fewest significant figures.
And over-rounding is an error too
Reporting fewer digits than the instrument resolved discards work that was actually done.
Every digit is a claim
Writing 2.50 grams claims the measurement is good to a hundredth. Writing 2.5 claims only a tenth. Significant figures communicate the precision of the instrument, not the neatness of the number.
Calculations cannot create precision
Dividing measurements to nine decimal places does not make the answer more accurate than the least precise input. Rounding to the correct number of figures is honesty, not tidiness.
Step 4: A Common Mistake
Lots of people think
“Rounding to significant figures throws accuracy away, so keeping every digit is safer.”
Step 5: Precision Without Accuracy
The same idea somewhere new.
A balance that has drifted will give the same reading to four decimal places every time and be wrong by a gram. Repeatability is precision, and it says nothing about accuracy — which is why instruments are calibrated against a known standard rather than judged by how consistent they are.
Step 6: Think It Through
Practice makes it stick.
The Drifted Balance
Problem 1 of 2
A balance reads 5.0000 g every time for a sample that really weighs 7 g. What is it?
How Many Figures
Problem 2 of 2
Measurements of 3.20 cm and 4.1 cm are multiplied. How many significant figures should the answer have?
What the Digits Claim
1 of 5
What does a significant figure record?
2 of 5
How many significant figures does 0.00420 have?
3 of 5
Which describes precision?
4 of 5
Is reporting too few digits an error?
5 of 5
Why do the extra calculator digits mislead?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Does rounding to significant figures lose accuracy?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
A student argues that keeping every calculator digit is more accurate. Evaluate that argument.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- A measured value states what the instrument could resolve.
- Precision is how finely a value is resolved; accuracy is how close it is to the truth.
- A product takes the fewest significant figures of its inputs.
- Reporting too few digits is an error as well as reporting too many.