# Can adding two numbers increase the significant figures?

I am having a question regarding significant figures in calculating average of two masses. For example adding two masses 7 kg and 8 kg gives 15 kg, both inputs 7 and 8 have only one significant figure but final answer 15 has two significant figures.

Now if I take average of the masses 15/2 gives 7.5, (Note: the number 2 has infinite significant figures because it represent a whole quantity, so it is 2.0000... in actual)

Hence the calculated average 7.5 kg has two significant figures obtained from the sum 15 kg (which has 2 significant figures and it determines the significant figure of the result of division)

Is this logic correct? Can addition (or taking average) increase the significant figure of the result? Can I write my final average as 7.5 kg or should I write only 8 kg (after rounding off 7.5 to one significant figure because both inputs have only one significant figure)?

Consider what the number of significant figures indicates:

The number written as $$7$$ has $$1$$ significant figure; the number is somewhere between $$6.5$$ and $$7.5$$

The number written as $$8$$ has $$1$$ significant figure; the number is somewhere between $$7.5$$ and $$8.5$$

So, when you add these two numbers;$$7+8=15$$

the answer, $$15$$, lies somewhere between $$6.5+7.5 =14$$, and $$7.5+8.5 =16$$

$$15 ± 1$$ is not two significant figures...

• Basically correct, but a more sophisticated treatment of the experimental errors would get a somewhat different estimate of the final error. Commented Sep 9, 2021 at 8:08

The question which was asked was Can adding two numbers increase the significant figures?
In this context the question is misleading in that when values range over a value of . . . . . $$0.1$$ or $$1$$ or $$10$$ . . . . etc the use of the term significant figures can give a false impression.
The accuracy of the value $$9.99$$ is not really any different from the accuracy of the value $$10.01$$ even though the second value is quoted to four figures whilst the first value is quoted to only three figures.

It is not clear if the quoted masses, $$7\,\rm kg$$ and $$8\,\rm kg$$, are actually more accurate than the one figure quoted. For example you might expect a hanger mass that is found in the laboratory with a label of $$100\,\rm g$$ on it to be accurate to at least $$\pm 0.1\,\rm g$$.

Suppose that the device which measured these two masses was graduated in $$1\,\rm kilogramme$$ steps and so any mass between $$6.5\dot0 \,\rm kg$$ and $$7.4\dot9\,\rm kg$$ is registered as being of mass $$7\,\rm kg$$ but note that there are measuring devices where the range for a reading of $$7\,\rm kg$$ might be $$7.\dot0\,\rm kg$$ to $$7.\dot 9\,\rm kg$$.

Thus the mass of $$7\,\rm kg$$ could have a mass between $$6.5\,\,\rm kg$$ and $$7.5\,\,\rm kg$$ and the mass of $$8\,\rm kg$$ could have a mass between $$7.5\,\,\rm kg$$ and $$8.5\,\,\rm kg$$.
When these two masses are combined one could say that the combined mass is $$15\,\rm kg$$ with a range between $$14\,\rm kg$$ and $$16\,\rm kg$$.
In practice this range would probable be an overestimate as the likelihood of both masses being on the very light side (eg $$6.5$$ and $$7.5$$) or both masses being on the very heavy side ($$7.5$$ and $$8.5$$) is not as great as the masses being closer to their stated values and possibly one lighter and the other heavier
Perhaps then a better estimator of the error in the sum is $$\pm\sqrt{0.5^2 + 0.5^2} \approx \pm 0.7\,\rm kg$$ with the percentage error for the sum being smaller than the percentage error for the individual masses.

The question which was asked was Can adding two numbers increase the significant figures?
In this context the question is misleading in that when values range over a value of . . . . . $$0.1$$ or $$1$$ or $$10$$ . . . . etc the use of the term significant figures can give a false impression as the accuracy of the value $$9.99$$ is not really any different from the accuracy of the value $$10.01$$ even though the second value is quoted to four figures whilst the first value is only quoted to three figures.