Significant figures

Definitions & Key takeaways

Significant figures are the digits in a number that carry meaning in terms of the precision of the measurement. Significant figures are used in scientific measurements and calculations to communicate their level of precision and accuracy. They include all non-zero digits and any leading zeros that come before the first non-zero digit in a number, and exclude trailing zeros that come after the last non-zero digit unless they are explicitly indicated to be significant.

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Significant figures are important for calculations involving measurements. For example, we can look at a thermometer that has markings of 5 degrees Celsius.
So if we zoom in, and see a measurement that is between 25 and 30, you have to guess on the second digit. It might look like it’s 26 or 27, it’s hard to say.
So if it is 27 degrees, then that measurement has two digits, the first digit we know for certain, and the second digit is estimated and therefore uncertain.
But both are considered significant figures. Now let’s look at a different thermometer, which has markings of 1 degrees Celsius.
So this time, the measurement is between the markings for 26° C and 27° C. So you don’t have to guess on the second digit, you know that it’s 27.
But you have to guess on the third digit, which looks like it might be 27.2 degrees Celsius. In this case, the uncertain digit is the last two.
But all three digits are considered significant figures for this measurement.In general, when reporting measurements, the precision of your measuring device determines the number of significant digits.
If any estimation is involved in reading a number from your device --- like a thermometer or a graduated cylinder --- then you record a measurement rounded to one decimal place beyond the decimal place of the markings.
If you use a digital measurement, like a digital scale or pH probe, then there’s no estimation involved. You can just record however many significant figures are reported by the measurement device.
So the number of significant figures is affected by the instrument used to perform a measurement. As another example, for finding the volume of a liquid, the most common way would be to use a graduated cylinder.
In order to maximize the precision of our measurement, we would pour liquid into our graduated cylinder and then locate the bottom of the meniscus.
We can draw a line there, and then read it off our measurement device. So in this case, the meniscus is between two markings on the graduated cylinder, 4.5 milliliters and 4.6 milliliters.
The last digit we estimate, which makes it the final significant figure. The meniscus looks evenly spaced between the markings, and so we can record 4.55 milliliters as our measurement, for a total of three significant figures.Now that we know how to record our measurement, we want to figure out which of our recorded digits are significant figures.
There are seven rules to remember when keeping track of significant figures:Rule 1 is that any non-zero digit is significant.
For example, if the measurement is 5.76 meters, then the three nonzero digits: five, seven, and six, are all significant, meaning that there are three significant figures.
Rule 2 is that included zeros are always significant. So if the measurement is 1.076 meters, then we have an included zero in the middle of the measurement, and the surrounding digits are nonzero - in this case the 1 and the 7.
So there are four significant figures. Another example, is the number 2035.
The included zero is significant because the surrounding digits are nonzero - in this case the 2 and the 3. So there are four significant figures.
Rule 3 is that trailing zeros in a decimal number are significant. So if we had a measurement on a balance of 6.100 grams, the two trailing zeros, since they are in a decimal number, are significant.
So there are four significant figures. Next, rule 4 is that leading zeros are not significant.
Let’s say we had 0.000620 grams as our measurement, then we could count these first four zeroes as leading zeros. The zero at the end is considered to be significant, going back to rule 3 which says that the trailing zeros in a decimal number are significant.
So there are three significant figures.Rule 5 says that when scientific notation is used, all digits are significant. So, if we put the previous measurement into scientific notation, then 0.000620 becomes 6.20 x 10-4 grams.
Here, all of the digits are considered significant, so there are three significant figures.Rule 6 says that trailing zeros in a non decimal number are not significant.
Say we had “94000 seconds” as our measurement. The trailing zeros would be the three zeros on the right side.
But for this measurement there are only two significant figures. According to rule 6, the nine and the four are nonzero digits which are significant.
And these three zeros to the right are not significant. The way to make them significant would be to put a decimal point at the end.
So “94000.” with a decimal point at the end would give you five significant figures overall.Rule 7 says that exact numbers are infinitely significant.
An example of that would be a calculation for kinetic energy, which has the formula “one half M V squared”. In this equation an exact number would be the 1/2 in front of this formula.
You don't have to worry about significant figures when you're dealing with an exact number so you could just do your calculation like it's not even there.It is important to keep track of significant figures when performing calculations that involve combining multiple measurements.This ensures that your final answer doesn’t appear to be more or less precise than any of the measurements that went into it.
You can use significant figures in calculations by remembering a few more rules.Let’s start with multiplication or division.
Suppose you are trying to find the area of a rectangle, and one side measures 6.21 centimeters, and another side measures 5.2 cm.
Then to find the area we multiply those two numbers together, and we get 32.292 centimeters squared. When multiplying or dividing you are limited in significant figures by the smallest number of significant figures.
From our measurements, 6.21 cm has three significant figures, but 5.2 cm has only two significant figures. So you are limited to two significant figures by 5.2 cm, because that is your least precise measurement.
So our answer should be rounded to 2 significant figures. In this situation, 32.292 centimeters squared, rounds to 32 centimeters squared.
Now, if the answer had been 32.592, then we would round that up to 33 centimeters squared. Now, let’s look at the rules for adding and subtracting.
Suppose we’re adding two measurements together. The first measurement is 102.02 meters, but we don't know the tenths or the hundredths place for the second measurement, which is six meters.
So the second measurement limits the precision of the final answer to the ones place. Since we don’t know any decimal places for one of the measurements, we can’t know it for the final answer either.
So we add the two measurements together to get 108 meters.Now that we know how to combine significant figures in calculations, let’s look at an example of how we would use significant figures when performing a real world measurement.
Density equals mass divided by volume. For example, water has a density of 1.00 grams per centimeter cubed at 25 degrees Celsius.
Since one centimeter cubed equals one milliliter, you can also say that water has a density of 1.00 gram per milliliter.
So, going back to our formula, if we have 10.0 milliliters of water, then multiplying 1.00 grams per milliliter and 10.0 milliliters gives us 10.0 grams of water.
10.0 milliliters of water has three significant figures and 1.00 grams per milliliter has three, so our final answer of 10.0 grams should also have three significant figures.So now let's use our knowledge of significant figures to calculate the density of an unknown metal.
Suppose we take a piece of metal and put it on a balance. The balance gives us a mass of 80.01 grams.
That is four significant figures. Next let’s say that we fill a graduated cylinder with water to a level that reads 20.00 milliliters.
Then, we then put the unknown piece of metal into our graduated cylinder, causing the water level to rise up to 27.61 milliliters.
The volume of the object can be identified by subtracting the two readings 27.61 - 20.00 milliliters, to get a volume of 7.61 milliliters for our unknown metal.
To find the density of our unknown metal, we take the mass divided by the volume, so that’s 80.01 grams divided by 7.61 milliliters.
We have four significant figures in our numerator and we have three significant figures in our denominator. Because this is a division calculation, we are limited by the smaller number of significant figures for the accuracy for our answer.
So our answer is limited to 3 significant figures. The final answer should be 10.5 grams per milliliter.
Now, we can look this value up in a reference table, which would tell us that our unknown metal is probably silver.##SummaryAs a quick recap, significant figures tell us about the uncertainty inherent in any measurement.
They are affected by how precise our measuring devices are, as well as how carefully we read measurements from our measurement devices.
When combining measurements using multiplication and division---such as when we measure density---we determine the significant figures of our results using whichever of our measurements has the fewest significant figures.
When adding or subtracting measurements, we determine the significant figures of our result by tracking the decimal places of the significant figures in our measurements.This method ensures that the number of significant figures in the final answer has a precision that reflects the precision of the original