Because different measurements in chemistry have different levels of precision, care must be taken when reporting numbers.
For example, suppose you are measuring the density of a liquid compound that you synthesized in chemistry lab. Density is a measure of the ratio of an object or substance’s mass divided by its volume.
First, you might measure the mass with a fairly sophisticated scale like this one and get 10.47 g.
Next, you would measure the volume of your liquid with a graduated cylinder like this one and get 7.3 ml.
Now divide 10.47 g by 7.3 ml. According to your calculator, the density of your liquid would be 1.434246575342466 g/ml.
10.47g / 7.3ml = 1.434246575342466 g/ml
Do you believe that number?
Is it really valid to report the density to 15 decimal places when you only know the volume to 1 decimal place?
Best case, the final answer can only be known to the same level of precision as the least precise measurement that went into the calculator. Otherwise the calculator is adding information to your measurement that isn’t really there!
What if the scale was set to kilograms (kg) instead of grams? Instead of reporting 10.47 grams, it would say 0.01047 kg.
What if the scale was set to milligrams (mg) instead of grams? Instead of reporting 10.47 g or 0.01047 kg, it would say 10,470 mg
Do you see how all three measurements are the same except that the decimal moved left or right depending on the units?
All three measurements contain 4 pieces of measured information, 1-0-4-7.
These numbers that contain real information about a measurement are called Significant Figures.
The leading zeros in 0.01047 kg are not considered to be significant since they were not measured, they are only there to change the answer to kg.
Likewise, the trailing zero in 10,470 mg is not considered to be significant since it was not measured, it is only there to change the answer to mg.
Significant Figures are always measured values.
Here are some simple rules that you need to memorize for determining significant figures:
- Non-zero digits are significant, no exceptions.
- Zeros between non-zero digits are also significant.
- Leading zeros (to the left of the measured value) are not significant.
- Trailing zeros (to the right of the measured value) with no decimal point are not significant.
- Trailing zeros before or after a decimal point are significant.
Significant figures also help us to identify if a number is exact (measured) or estimated. Think about the distance from the Earth to the sun. Most astronomers would answer 93,000,000 miles. How many significant figures are in that number? 2!
The trailing zeros are not significant. This is because 93,000,000 is an approximate value. No one believes that if you sent a rocket to the sun that it’s odometer would read precisely 93,000,000 miles when it reached the front door of the sun.
But what if someone really did want to express the exact value of 93 million? In chemistry, we use decimals a little differently than in other areas of study. In chemistry, you’ll sometimes see a decimal point at the end of a number without any other numbers after it. This is how we can express a number that has trailing zeros exactly. In this case, it would look like this:
93,000,000.
This version of 93 million has 8 significant figures because the trailing zeros have a decimal point.
Let’s practice.
How many significant figures are in the number 20354?
- There are 5 significant figures in 20354. 4 are non-zero numbers and the 0 falls between two non-zero numbers.
How many significant figures are in the number 20300?
- There are 3 significant figures in 20300. 2 are non-zero numbers and the 0 falls between two non-zero numbers. The trailing zeros are not significant because there is no decimal.
How many significant figures are in the number 0.0035?
- There are 2 significant figures in 0.0035. The leading zeros are not significant.
How many significant figures are in the number 0.003500?
- There are 4 significant figures in 0.003500. The leading zeros are not significant but the trailing zeros are because they come after a decimal.
How many significant figures are in the number 4,020,000.?
- There are 7 significant figures in 4,020,000.. In this case, the trailing zeros are significant because of the decimal point.
HOW TO USE SIGNIFICANT FIGURES IN A CALCULATION
Let’s go back to the original calculation of density. What would be the best way to report that number?
10.47g / 7.3mL = 1.434246575342466 g/mL
The rule for multiplication and division problems is to report the final answer to the fewest number of significant figures that went into the calculation.
In this case, 10.47 has 4 significant figures and 7.3 has 2 significant figures. Thus, we can only report the density to 2 significant figures. We would use the first non-significant figure to round the last significant figure, so 1.43 rounds to 1.4 g/ml.
The rule for addition and subtraction problems is a little different. In that case, we just report our answer to the fewest decimal places that went into the calculation. For example, try adding 20413.1 and 22.988.
20413.1 has 1 decimal place
22.988 has 3 decimal places
Thus, the final answer will be rounded to 1 decimal place.
Your calculator says 20436.088. You’ll round that answer to 20436.1.