# Significant Figures

Significant figures (sig. figs.) identify which digits in a number are known reliably, and the first digit that is doubtful.  Rules have been developed for how to display a final answer after mathematical computations.

## Short Recap

• All non-zero digits are significant.
• Zeros that simply “fix” the decimal point are not significant.
• Zeros within a number are always significant.
• Trailing zeros to the right of the decimal point are always significant.
• Final answers contain the smallest amount of significant figures of the numbers used in the calculations.
• When rounding:
• if the first insignificant figure is <5, leave the last significant figure unchanged, or
• if the first insignificant figure is ≥5, increase the last significant figure by one.

## Long Version with Explanation

When you divide numbers and get many decimals, where should you round the answer off? When you calculate an average value, for instance, an average temperature from measurements you and two classmates took using three different thermometers, where should you round the answer off? If you leave many decimals you are implying that you have confidence in this temperature measurement down to tenths, hundredths, thousandths…perhaps to billionths of one degree Celsius. But the thermometers you used aren’t that sensitive.

Scientists have worked out a method for deciding how many figures to leave – how many significant figures to leave. The practice is to include all figures that are reliably known, plus one more that is an estimate, or approximation, (the first uncertain figure).

If three people in the class measured the air temperature in the room with three different thermometers and got values of 22.1°C, 22.3°C and 22.4°C, the average value would be 22.26666667°C. You know with certainty that the temperature is 22°C, so you would include those two figures, plus one more—the first uncertain figure. Your final temperature, then, should be rounded to three digits: 22.3°C. These three digits are said to be significant figures.

Identifying the number of significant figures in a number takes a trained eye.  Zeros need careful examination because sometimes it’s not obvious whether they are indicating a value or are merely showing the location of the decimal point. Again, some conventions have been worked out to communicate when zeros are significant.

### Rules for Zeros

1. All non-zero digits are significant.
There are four significant figures in 9,273 and also in 3.284. There are three significant figures in 461 and 7.99.
2. Zeros that simply “fix” the decimal point are not significant.
There are three significant figures in 24,700. ‘Why is that, when five digits are involved?’ you may wonder. The number 24,700 tells us that we know this value reliably to twenty-four thousand. The first uncertainty creeps in at the seven hundred mark. We do not have enough resolution to know the value down to the 10s level or lower with any reliability. So the two zeros are not significant. They are important in telling us about the magnitude of the number (it is twenty-four thousand and seven hundred, not two hundred and forty-seven), but they are not significant. In the same way, the number 0.000523 also has three significant figures. The zeros just ‘fix’ the decimal point; in other words, the zeros tell us where the decimal is and how big the number really is (the order of magnitude).
3. Zeros within a number are always significant.
There are four significant figures in 2,047 and in 0.09105. In the latter number the four significant digits are 9105. The previous two zeros simply fix the decimal place.
4. Trailing zeros to the right of the decimal point are always significant.
The number 5.1920 has five significant figures. The zero is not needed to tell us where the decimal is, therefore, the only reason it would have been included is to indicate significance down to the 1/10,000th level. The number 67.0000 has six significant figures because, again, these zeros are not needed to fix the decimal. They are significant.

Expressing numbers in scientific notation is often helpful to avoid confusion about when zeros are significant. The number 5,974,000 seems to contain four significant figures. But what if the scientist who measured this value actually knows it reliably down to the hundreds, tens, and ones? If he or she had written it as 5.974000 x 106 then we would know that there are seven significant figures.

### Basic Math with Sig. Figs.

Geographers often need to perform calculations and leave their answers to a reasonable number of significant figures. The conventions for multiplying and dividing, and then for rounding off are given below.

#### Multiplying and Dividing

When multiplying or dividing numbers, the results can only be as accurate as the least number of significant figures used in the calculations. For example, if you multiplied 9.84321 (6 sig. figs.) by 230 (2 sig. figs.), initially, you would get 2263.9383. This initial answer implies a high degree of precision. The data we used in our calculation were not that precise, so it would be dishonest of us to claim that we know the answer to a level of 7 significant figures. Properly expressed in scientific notation, our answer would only have 2 significant figures and would be expressed as 2.3 x 103.

#### Rounding Off

When you need to reduce the number of figures in an answer, round off using the following rules:

1. If the first ‘insignificant’ figure (in other words, the first figure you’ll drop) is less than 5, drop off all the trailing ‘insignificant’ figures, and leave the last significant figure unchanged. Forexample, leaving the following values to two significant figures would change 1.93 to 1.9, change 7.822 to 7.8 and 943.7 to 940.
2. If the first insignificant figure is equal to or greater than five, drop off all the trailing‘insignificant’ figures and increase the last significant figure by one. For instance, leaving the following values to two significant figures would change 1.652 to 1.7, 927.2 to 93.0, and 5.85 to 5.9.