#### The topic of assigning significant figures is nearly every student’s least favorite. The rules may seem excessive and arbitrary, but they serve a purpose. The foundation of significant figures is preserving the degree of uncertainty of the instruments we used to make the measurements. Assigning significant figures and using the rules to round the result of computations gives us a systematic way of indicating the degree of trustworthiness of the result.

In a measurement, all of the certain digits (those read directly from the scale) and one uncertain digit (one number made by estimating between marks on the scale) are significant. Regardless of the rules and whether or not some of the digits are zeroes, measured numbers are significant. The difficulty is in preserving the degree of uncertainty once you start performing mathematical operations on the numbers, such as diving mass by volume to calculate the density.

## Rules for determining the number of significant figures

The rules for assigning significant figures are based on deducing which digits could not have been the result of rounding off another number. When you round a number, you replace digits with zeroes, so zeroes are what we have to scrutinize. The following rules are a systematic scrutiny to determine if the digits in a number reflect actual measurements with their associated degrees of uncertainty.

- Non-zero digits are significant.
- Zeroes that lie between two non-zero digits cannot be the result of a rounding operation, so these are significant.
- Zeroes that serve only to hold the decimal place are not significant. None of the zeroes in the numbers 550, 0.003 or 10,000,000 are significant.
- Zeroes to the right of non-zero digits
**and**right of the decimal point are significant. The zeroes in 1.200 are significant. They had to have been measured, because, if the number was rounded to the tenths place, we would write it as 1.2. - The rules for determining if a zero is significant do not apply if one knows for certain that the number was measured as a zero rather than rounded. I have ten fingers measured to the nearest whole finger (not nine or eleven rounded to ten). In that context, the number 10 has two significant figures, even though, by the rules, the number 10 has only one significant figure.
- If a zero is measured yet it may appear to anyone reading the data to have been rounded, the data should be written in scientific notation. If a number is written in scientific notation, all zeroes are significant.
- Some important numbers have values by definition. An example is that one minute is sixty seconds long. These numbers are not measurements and have never been rounded. They are known as
**exact numbers**. When we decide where to round the result of a calculation, we ignore the exact numbers because there is no uncertainty associated with them.

**Concept Check:**

How many significant figures are there in the number in the title **1001 Arabian Nights**?

**Answer:** There are four significant figures in the number 1001. The zeroes between non-zero digits count as significant figures because they cannot be the result of a rounding operation. The author understood the ideas behind the rules for assigning significant figures. ‘1000 nights’ implies a pretty long time, but the number ‘1001 nights’ implies that we know how many nights to the nearest night.

## Rounding the result of addition/subtraction:

When you add and subtract using pencil and paper (rather than a calculator), you line up the decimals so that you are adding digits from the same decimal place to one another. Decimal places shared by all of the numbers contribute to the result. Higher powers of ten contribute, even if they aren’t shared by all of the numbers. Lower powers of ten don’t have much impact on the result. **The number that results from addition or subtraction should be rounded at the lowest decimal place that the original numbers have in common. **If we add the numbers 2.2 + 41.066 + 19.11 we get the result 62.376. This implies that we measured to the thousandths place, but one of the numbers was only measured to the tenths place. The sum should be rounded to 62.4 at the tenths place, which is the lowest decimal place that all three numbers share.

## Rounding the result of multiplication/division:

When you multiply or divide numbers, you change their scale. The decimal point may shift as a result, so it does not give us an indication of the uncertainty of the original measurements. **The number that results from multiplication or division should be rounded so that it has the same number of significant figures as the original number with the fewest number of significant figures.** For example, if we multiply the numbers 22 * 30.1 * 1.4 we get the result 927.08. The lowest digit is the one with uncertainty, so this implies that we made measurements that had only one in ten thousand parts uncertainty! We need to round the result to 930, which has two significant figures, since the original numbers had two and three significant figures.

**Concept Check:** Translate the title **20,000 Leagues Under the Sea** into metric units.

**Answer:** The metric title would be **100,000 Kilometers Under the Sea**. One league is three miles, so 20,000 leagues is 60,000 miles. 1.000 mile is 1.609 km, so 60,000 miles is 96,540 km which rounds to 100,000 km. The number 20,000 has one significant figure by the rules. Context can always over-turn the rules, but, in this case, the context is “really deep” rather than a measurement to the nearest league.

## Rounding more complex calculations:

When calculations involve different kinds of operations, apply the rules sequentially. In this example, a student measures the volume and mass of a sample of metal pieces. She measures the mass on a balance using a beaker to contain the pieces. She measures the volume by adding the pieces to some water in a graduated cylinder and reading the volume displaced. She then uses the data to calculate the density of the metal using the formula D = M/V.

Lab Experiment Data Table | |

Description |
Measurement/Result |

1. Mass of empty beaker | 113.229 g |

2. Mass of beaker + metal | 158.954 g |

Mass of metal (2 -1) | 46.725 g |

3. Volume of water | 100.5 mL |

4. Volume of water + metal | 117.5 mL |

Volume of metal (4 – 3) | 1.70 x 10^{1} mL |

Density | 2.69 g/mL |

The result of subtracting the mass of the beaker from the beaker + metal is rounded to the third decimal place by the addition/subtraction rule. The result of subtracting the volume of water from the volume of water + metal is rounded to the first decimal place by the subtraction rule. This gives 17.0 with three significant figures. The number is written in scientific notation to make it clear that the zero is significant. The density is the mass of the metal divided by the volume of the metal. When we divide a number with five significant figures by one with three significant figures, the answer is correctly rounded to three significant figures.