### How old do you think this calculator is?

Calculators are great devices. Their invention has allowed for quick computation at work, school, and other places where manipulation of numbers needs to be done rapidly and accurately. But they are only as good as the numbers put into them. The calculator cannot determine how accurate each set of numbers is, and the answer given on the screen must be assessed by the user for reliability.

## Uncertainty in Addition and Subtraction

Consider two separate mass measurements: \(16.7 \: \text{g}\) and \(5.24 \: \text{g}\). The first mass measurement, \(\left( 16.7 \: \text{g} \right)\), is known only to the tenths place, or to one digit after the decimal point. There is no information about its hundredth place and so that digit cannot be assumed to be zero. The second measurement, \(\left( 5.24 \: \text{g} \right)\), is known to the hundredths place, or to two digits after the decimal point.

When these masses are added together, the result on a calculator is \(16.7 + 5.24 = 21.94 \: \text{g}\). Reporting the answer as \(21.94 \: \text{g}\) suggests that the sum is known all the way to the hundredths place. However, that cannot be true because the hundredths place of the first mass was completely unknown. The calculated answer needs to be rounded in such a way as to reflect the certainty of each of the measured values that contribute to it. For addition and subtraction problems, the answer should be rounded to the same number of decimal places as the measurement with the least number of decimal places. The sum of the above masses would be properly rounded to a result of \(21.9 \: \text{g}\).

##### Example \(\PageIndex{1}\)

Determine the combined molecular mass of a glucose molecule and a maltose molecule.

Glucose molecule = \(180.156\frac{g}{mol}\)

Maltose molecule = \(342.3\frac{g}{mol}\)

###### Solution

\(180.156+342.4=522.456\)

When adding and subtracting we know to look at the least number of decimals in our starting values; in this case 342.3 has only 1 digit after the decimal, so we need to round our answer to the same place.

\(522.456\to 522.5\frag{g}{mol}

When working with whole numbers, pay attention to the last significant digit that is to the left of the decimal point, and round your answer to that same point. For example, consider the subtraction: \(78,500 \: \text{m} - 362 \: \text{m}\). The calculated result is \(78,138 \: \text{m}\). However, the first measurement is known only to the hundreds place, as the 5 is the last significant digit. Rounding the result to that same point means that the correct result is \(78,100 \: \text{m}\).

##### Example \(\PageIndex{2}\)

What is \(4200 + 540\) = ?

###### Solution

\(4200 + 540 = 4740\)

To determine where to round our answer, we look at our starting numbers to see which has the fewest number of decimal places. They both have 0 so we round to the nearest whole number, 4740.