# Significant Figures

Significant figures are very simple to work with, yet they are extremely important in communicating numbers effectively in science. The way you write an answer communicates how precisely the data was collected. If you measure the volume of a liquid that has divisions for tenths of milliliters, you would be able to estimate between lines and get a reading in the hundredths, such as 42.36ml or 42.37. However, if your container only has lines for milliliters, you would only be able to estimate in the tenths place, such as 42.3 or 42.4. The last digit written is always the one you had to estimate.

Side note: accuracy of a measuring device (like a balance) is whether it gives you the correct mass, whereas precision is the place to which the balance measures (to the .001g vs. to the .01g).

## Counting significant figures:

The number of significant figures in a number is simply the number of digits in the number (32.734 has 5 sig figs). The only exception is that sometimes zeros don't count. Zeros will count when they are trapped by nonzero digits like in 4509 (4 sig figs) or when the zeros follow both the decimal and the nonzero digits like in 37.0 or .370 (both with 3 sig figs). The number 370 only has 2 sig figs because the zero follows the nonzeros but it does not follow the decimal. The number .0037 only has 2 sig figs because the zero follows the decimal but does not follow the nonzero digits.

Sometimes you want a zero to be significant when it wouldn't normally look like it is. For example, if you want the number 300 to imply 2 significant figures, you have a couple of choices. You can put a bar over the first zero (the second digit) or you can put the number in exponential notation (3.0 x 10^2).

Let's try some to make sure you have it. Jot your answers down on a piece of scratch paper and then check your answers on the answer page. Just click on the colored "answer page" and it will take you there. Then click on the back button to come back to this page.

### How many significant figures are there in each of the following numbers?

1. 648
2. 90
3. 307
4. 2.408
5. 3.00
6. .0082
7. .00790
8. 407.0050
9. 8.900 x 10^6

How did you do? If not so well, remember that you count all nonzero digits then count only the zeros that are between nonzeros and zeros that are behind the decimal point and behind the nonzeros.

If you need another self-quiz, try these. If you don't need more practice, go down to the section below on math with significant figures.

### How many significant figures are there in each of the following numbers?

1. 8097
2. .0056
3. .04500
4. 360.80
5. 5000

If you are still shaky on this, go back to the paragraph with the original rules in it. To do this, click here. ## Math and Significant Figures:

Scientists are not asked to state how many significant figures there are in a number. What they do have to do is express answers to calculations using the correct number of significant figures. A measurement can't all of a sudden gain precision just because it was entered in a calculator.

Rules for expressing answers to calculations:

### Multiplying or dividing:

When multiplying or dividing, look at the number of significant figures in each of the numbers you used in your calculation. The least number of significant figures among those is the number of significant figures that your answer can have.

Example: What is the density of an object that has a mass of 34.5g and a volume of 50ml?

D = M/V = 34.5g/50ml = .69g/ml

Now look at the original data. 34.5 has 3 sig figs, 50 has 1 sig fig. The least of these is 1 so the answer has 1 sig fig and is rounded to .7 (not .70 because that would have 2 sig figs).

I have made the division, addition, etc signs in red so that it looks less cluttered.

### Adding or subtracting:

When adding or subtracting, look at the number of decimal places in the original numbers, not the significant figures. The answer will have the least number of decimal places.

Example: Temperature in degrees Celsius is converted to Kelvin by adding 273.15. Convert 45.6C to K:

45.6 + 273.15 = 318.75

Now look at the numbers in the calculation. 45.6 has 1 decimal place and 273.15 has 2 decimal places. The least is 1, so the answer has 1 decimal place and is rounded to 318.8 Notice that this number does not have the number of significant figures of either of the original numbers. It doesn't have to. It all goes by decimal places.

Okay, try this quiz. Again, the answers are on the answer page.

### Express the answers to these calculations in the correct number of significant figures:

1. 98.00 * 34 =
2. 11.3 - 4.56 =
3. (8.75 + 3)/4.0 =

For more information and practice, try this web site on significant figures.

Now that you are an expert at significant figures, you can go back to the introduction page and review another topic. Author: Jennifer Caskey, Mahtomedi High School, Minnesota
Created: June 12, 1999 Updated: June 28, 1999
URL: /lincon/w99/projects/apchem/apchem/significant.html