# Does the addition or multiplication rule have precedence deciding the significant figures in 5×5.364? [duplicate]

How many significant figures should be present in the answer of $$5\times 5.364$$?

1. Addition rule: The result cannot have more digits to the right of the decimal point than either of the original numbers.

2. Multiplication rule: The result must be reported with no more significant figures as there are in the measurement with the few significant figures.

Using rule 1, $$5\times 5.364 = 5.364 + 5.364 + 5.364 + 5.364 + 5.364 = 26.82,$$ one would use 4 significant figures.

Using rule 2, $$5 \times 5.364 = 26.82 = 30,$$ one would use 1 significant figure.

Which one is correct?

## 4 Answers

If this is the question in its entirety, then you should not try to 'guess' what the origin, and therefore the accuracy, of the measurement 5 is. You should take it on face value, and complete the calculation using the multiplication rule, as the equation is requiring a multiplication operation. In this case, you should have one significant figure only.

The only ambiguity you need to consider in calculating your significant figures is whether the '5' is an exact number or not. Exact numbers come about either by definition (1 hour = 3600 seconds) or by counting (5 measurements were made) and have an infinite number of significant figures. Hence, they do not affect the accuracy of your measurements. Exact numbers are usually integers (speed of light and number of centimetres in an inch are a couple of exceptions). My guess is that your question is simply a case of going through significant figures calculations, using the information provided. To avoid ambiguity in the case of a single significant figure measurement, one could consider writing 5 as 5E0, or 5x100.

If you know 5 to be an exact number (ie the question gives that information), then you would still complete the calculation using the multiplication rule, but assume infinite number of significant figures for 5. In this case you would have 4 significant figures.

Using the addition rule doesn't make any sense because then you would always use the addition rule for multiplication.

You would use the multiplication rule, and if this question was given as-is, 30 would be correct. If you're told the 5 is exactly 5 (like in many/most conversions), then you would consider it infinite significant figures and answer with 4 significant digits.

If the measured width of a rectangle is 5 meters and the length is 5.364 meters, then the area would only have one significant figure, so 30 square meters by the multiplication rule.

If you measure the width of 5 bricks and the measurement of each brick is 5.364 cm, then the number "5" is not a measured value, it is an exact value. Four significant digits would be appropriate, consistent with the multiplication rule.

Contrary to the OP, literal application of the addition rule "cannot have more digits to the right of the decimal point than either of the original numbers", would mean 5 significant digits not 4.

However, when measurements are being added, the uncertainty in the square root of the sum of the squares of the uncertainties in each measurement, so the addition rule increasingly fails as more values are summed.

It really depends on what you're trying to signify, but I'd say that neither of the answers that you have given is entirely satisfactory. The "rules" you cite are good for sort of rule of thumb, but don't really work well when you're working with a single digit. 9 +/- 0.5 is about a 6% error, but 1 +/- 0.5 is a 50% error.

If you were working with a number which has 3 significant figures by a number which has 4 significant figures then the rules you cite work well enough.

The quirk here is that you can assume that 5 is +/- 0.5 so that there is an error of 10%. So 27 +/- 3 would really be a better answer.

Another thing to consider is that 30 has two significant figures. If you want to denote just one significant figure then you'd write the answer in scientific notation as 3E1 to clearly denote just one significant figure.

Of course if the 5 was an integer value then it has infinite precision. Say you're working with oxygen in the following chemical equation: $$\ce{C3H8 + 5O2 -> 3CO2 + 4H2O}$$

• Perhaps I misunderstand you, but 30 has 1 significant figure.
– long
Mar 4, 2016 at 9:15
• @long - It depends on if you know it is 30 +/- 10 (which has 1 significant figure), or if you think it is 30 +/- 0.5 (which has two significant figures). I pass a speed limit sign that says 30 mph. I don't think the law means +/- 10 mph.
– MaxW
Mar 4, 2016 at 18:58