The number of decimal places on a display tells you how it was printed, not how accurate it is. An answer is only as accurate as the data behind it, and the bounds show how accurate that really is. This matters in calculated areas, speeds and costs, and it is the final skill in precision bounds and measurement.
Use it together with the interval skills from recovering an interval from a rounded measurement.
What does the number of decimal places tell you?
Decimal places say where the last digit sits. They do not say how the digit was found.
| Written | Rounded to | Interval for the true value |
|---|---|---|
| 6.3 | 1 decimal place | 6.25 ≤ x < 6.35 |
| 6.30 | 2 decimal places | 6.295 ≤ x < 6.305 |
| 0.0040 | 2 significant figures | 0.00395 ≤ x < 0.00405 |
So 6.3 and 6.30 are equal numbers, but 6.30 claims to be ten times more precise. Significant figures and decimal places measure different things: 0.0040 has four decimal places but only two significant figures.
How do you decide how accurate a calculated answer is?
- Write the interval for each measurement.
- Find the lower and upper bounds of the answer using the correct pairings.
- Round both bounds to the same number of significant figures, starting high.
- Reduce the figures until both give the same value. That value is your answer.
Worked example
A rectangle is 4.2 cm by 3.7 cm, each to 1 decimal place. The calculator shows 4.2 × 3.7 = 15.54. What accuracy can the area be given to?
Step 1, intervals: 4.15 ≤ length < 4.25 and 3.65 ≤ width < 3.75.
Step 2, bounds: lower bound = 4.15 × 3.65 = 15.1475. Upper bound = 4.25 × 3.75 = 15.9375.
Step 3, try 2 significant figures: 15.1475 gives 15 and 15.9375 gives 16. They differ, so 2 significant figures is not justified.
Step 4, try 1 significant figure: both bounds give 20. So the area is 20 cm² to 1 significant figure.
The display 15.54 looks accurate to 2 decimal places, but the data supports only one significant figure.
The mistake to watch for
A common slip is copying the display as the answer because it “looks finished”.
Mistaken answer: area = 15.54 cm² to 2 decimal places.
The student treated the number of digits on the screen as the accuracy. The bounds show the true area could be anywhere from about 15.15 to 15.94.
The correction is to test the bounds before you write a rounded answer. If the two bounds do not agree, the extra digits are not justified.
Another version of this slip is writing a zero at the end, such as 6.30, to “make it look neat”. The zero is a claim, so include it only when the data supports it.
Check yourself
Try these, then open each answer.
1. Write the interval for 6.30 and for 6.3 (1 decimal place), and compare the widths.
Show answer
6.30 to 2 decimal places: 6.295 ≤ x < 6.305, width 0.01. 6.3 to 1 decimal place: 6.25 ≤ x < 6.35, width 0.1.
The interval for 6.30 is ten times narrower.
2. The lower bound of a quantity is 2.451 and the upper bound is 2.458. To how many significant figures can the quantity be given, and what is the value?
Show answer
To 3 significant figures the bounds give 2.45 and 2.46, which differ. To 2 significant figures both give 2.5.
2.5 (2 significant figures)
3. Ali records a time as 12.0 s and Mei records 12 s (nearest second). Write each interval and say who has measured more precisely.
Show answer
Ali: 11.95 ≤ t < 12.05. Mei: 11.5 ≤ t < 12.5. Ali’s interval is narrower.
Ali has recorded the time more precisely.
Where this leads next
Now try the whole set together in the precision bounds practice set, and use the bounds and rounding explainer to check your intervals. The non-calculator working trainer helps keep the exact steps clear before rounding.
Some students get the bounds right yet still write the calculator display. A teacher in online one-to-one Mathematics tuition can help you turn the final check into a habit.