What is the implied +/- error of this measurement? (A, B, & C)

Question

A liquid sample has a reported volume of 810 mL.

A. What is the implied +/- error of this measurement? And, identify the certain digits and the uncertain digits in the measurement.

B. If the volume was instead able to be reported as 810. mL, what would be the implied +/-error? And, identify the certain and uncertain digits.

C. Which of the two volumes 810 mL or 810. mL is more precise? What would have to change to allow the reported volume to be written as 810. mL instead of 810 mL?

Answer 1

The rule behind all three parts: a measurement carries an implied uncertainty of plus or minus one unit in its last significant digit. So the whole question reduces to deciding which digit that is.

A. 810 mL

With no decimal point, a trailing zero is read as a placeholder, not as a measured digit. So 810 mL has two significant figures, and the last significant one is the 1, sitting in the tens place.

Implied error: +/- 10 mL, i.e. the true volume is somewhere between 800 and 820 mL.

Certain digit: the 8. Uncertain (estimated) digit: the 1. The trailing 0 is neither - it is only holding the ones place so the number has the right magnitude.

B. 810. mL

That trailing decimal point is doing real work. It signals that the zero was actually measured, which makes the value three significant figures with the last one in the ones place.

Implied error: +/- 1 mL, i.e. between 809 and 811 mL.

Certain digits: 8 and 1. Uncertain digit: the 0.

C. Which is more precise, and what would have to change

810. mL is more precise - its uncertainty is ten times smaller.

What would have to change is the instrument, not the notation. You are not allowed to add the decimal point to a 810 mL reading; that would be claiming precision the glassware never delivered. To report 810. mL you need a container whose smallest graduation lets you read to the ones place - graduations every 10 mL let you estimate the ones digit, whereas a vessel marked only every 100 mL leaves the tens digit as your estimate and gives you 810 mL at best.

The general rule for reading analog glassware: record every graduation you can see, then estimate one digit beyond the finest graduation. That estimated digit is the uncertain one, and it is the last digit you are entitled to write down.

One thing worth knowing beyond the assignment

Notation like 810 mL is genuinely ambiguous, and the trailing decimal point is a patch that not every reader recognizes. Scientific notation removes the ambiguity completely:

8.1 x 10^2 mL is unmistakably two significant figures
8.10 x 10^2 mL is unmistakably three

If you ever have a choice in how to report a number, that is the form to use.