**Core Concepts**

In this **tutorial about significant figures**, we will cover their definition, relevant guidelines, and historical context.

**Topics Covered in Other Articles **

- Balancing Chemical Equations
- Definition of Lewis Acid and Base
- Percent by Weight Calculation
- Oxidation Number and State
- Protons, Neutrons, and Electrons
- How to Write Electron Configurations

**What are Significant Figures?**

Significant Figures, often referred to as “Sig Figs,” are specific digits that denote the degrees of precision exemplified by different numbers. We can classify certain digits as significant figures; others, however, we cannot. A given digit’s status as either significant or non-significant stems from a checklist of criteria.

**Rules for Determining Significant Figures **

**What Constitutes a Significant Figure?**

First, let’s review these criteria that define sig figs. We can classify numbers as significant figures if they are:

- Non-zero digits
- Zeros located between two significant digits
- Trailing zeros to the right of the decimal point
- (For digits in scientific notation format, N x 10
^{x})

- All digits comprising N are significant in accordance with the rules above
- Neither “10” nor “x” are significant

Specific amounts of precision, designated by significant figures, must appear in our mathematical calculations. These appropriate degrees of precision vary, corresponding to the type of calculation being completed.

To determine the number of sig figs required in the results of certain calculations, consult the following guidelines.

**Rules for Addition and Subtraction Calculations:**

- For each number involved in the problem, quantify the amount of digits to the right of the decimal place–these stand as significant figures for the problem.
- Add or subtract all of the numbers as you normally would.
- Once arriving at your final answer, round that value so it contains no more significant figures to the right of its decimal than the LEAST number of sig figs to the right of the decimal in any number in the problem.

### **Rules for Multiplication and Division Calculations:**

- For each number involved in the problem, quantify the amount of significant figures using the checklist above. (Look at each whole number, not just the decimal portion).
- Multiply or divide all of the numbers as you normally would.
- Once arriving at your final answer, round that value so that it contains no more significant figures than the LEAST number of significant figures in any number in the problem.

**Origination of Significant Figures **

We can trace the first usage of significant figures to a few hundred years after Arabic numerals entered Europe, around 1400 BCE. At this time, the term described the nonzero digits positioned to the left of a given value’s rightmost zeros.

Only in modern times did we implement sig figs in accuracy measurements. The degree of accuracy, or precision, within a number affects our perception of that value. For instance, the number 1200 exhibits accuracy to the nearest 100 digits, while 1200.15 measures to the nearest one hundredth of a digit. These values thus differ in the accuracies that they display. Their amounts of sig figs–2 and 6, respectively–determine these accuracies.

Scientists began exploring the effects of rounding errors on calculations in the 18th century. Specifically, German mathematician Carl Friedrich Gauss studied how limiting sig figs could affect the accuracy of different computation methods. His explorations prompted the creation of our current checklist and associated rules.

## Further Thoughts

We appreciate our advisor **Dr. Ron Furstenau** chiming in and writing this section for us, with some additional thoughts on the subject.

##### From Dr. Ron Furstenau

It’s important to recognize that in science, almost all numbers have units of measurement and that measuring things can result in different degrees of precision. For example, if you measure the mass of an item on a balance that can measure to 0.1 g, the item may weigh 15.2 g (3 sig figs). If another item is measured on a balance with 0.01 g precision, its mass may be 30.30 g (4 sig figs). Yet a third item measured on a balance with 0.001 g precision may weigh 23.271 g (5 sig figs). If we wanted to obtain the total mass of the three objects by adding the measured quantities together, it would **not **be 68.771 g. This level of precision would not be reasonable for the total mass, since we have *no idea* what the mass of the first object is past the first decimal point, nor the mass of the second object past the second decimal point.

The sum of the masses is correctly expressed as 68.8 g, since our precision is limited by the least certain of our measurements. In this example, the number of significant figures is *not* determined by the fewest significant figures in our numbers; it is determined by the least certain of our measurements (that is, to a tenth of a gram). The rules for addition and subtraction is necessarily limited to quantities with the same units.

Multiplication and division are a different ballgame. Since the units on the numbers we’re multiplying or dividing are different, following the precision rules for addition/subtraction don’t make sense. We are literally comparing **apples to oranges**. Instead, our answer is determined by the measured quantity with the least number of significant figures, rather than the precision of that number.

##### Example from Dr. Ron Furstenau

For example, if we’re trying to determine the density of a metal slug that weighs 29.678 g and has a volume of 11.0 cm^{3}, the density would be reported as 2.70 g/cm^{3}. In a calculation, carry all digits in your calculator until the final answer so as not to introduce rounding errors. Only round the final answer to the correct number of significant figures.

## Accuracy vs Precision

Accuracy is how close your measured number is to the “real” or actual answer. Precision is how close together repeated measurements are. If you measure something (e.g. weighing something) several times, and each measurement is almost the same, your precision is high. The accuracy, however, may or may not be high, depending on whether the scale was recently calibrated. This video explains the difference well:

## Significant Figures – **Practice Problems**

Determine the amount of significant figures in the following values and problems. Use the checklist and rules discussed in this article.

1. 0.00784** **g

2. 1.056 mm

3. 500 K

4. 700. °C

5. 0.0114 x 10^{4 }J

6. 8.9568 g + 13.75 g

7. 33.85 g x 806.5988 g/mol

Answer Key: 3, 4, 1, 3, 22.71 (round to 2 significant figures), 7. 29672.6 (round to 6 significant figures) |