In this video, we’re going to talk about significant figures, what they are, why they’re important, and how to use them practically.
Imagine for a moment that you’re an astronaut and you’re planning to be the very first person ever to travel from Earth to Mars. Suppose that through a very careful set of laser-based measurements, the point of closest approach between the Earth and Mars was determined to be 54628600 kilometers. So, that’s as close together as these two planets ever get.
Imagine further that someone designed a rocket ship for this journey. And they told you that the range that this rocket could travel was 55 times 10 to the sixth kilometers. The question is, would you be willing to ride that rocket to Mars? If you thought about it for a moment, you might compare the two numbers one to another, the range of the rocket and the precisely measured distance from Earth to Mars.
The range of the rocket seems to be far enough. But then again, that might be a result of rounding. For example, if the real range of the rocket is 54.5 times 10 to the sixth kilometers, that result might round to the range that you heard. But it wouldn’t be enough to actually get you to Mars from Earth. This is a case where having more significant figures than we’re given is vitally important to figuring out this question. So, this gets at the question of why significant figures are important.
Significant figures enable us to gain a better understanding of accuracy and precision in the numbers and measurements we work with. In particular, significant figures, we also sometimes call them sig figs for short, let us properly maintain precision so our calculated result is more accurate.
For example, let’s say that you were part of a team designing a fast new sports car. And your role in the project team was to design high-performance tires that the car would drive on. Imagine that the design specification for these tires was that they would have a diameter of one meter. With a design specification that imprecise, you might easily end up making tires that had very different sizes one from another. If you put a set of four on them on a car, the car wouldn’t drive straight.
The parts and pieces of nearly any machine that we might manufacture rely on significant figures for the parts of that device to work well together. When it comes to significant figures, there are some very concrete and clear rules that we can learn about them. And once we’ve learned them, we’ll become expert at working with them.
Here are four rules to keep in mind when working with significant figures. Number one, all nonzero digits are significant. This rule is simple. It just means that any time you see a digit one through nine in a number, that digit is significant. So, for example, if we saw the number 342.146, we would see that all of these digits are nonzero. And therefore, all are significant. You can see we’ve already covered a lot of ground because we’ve dealt with all numbers one through nine. The only remaining digits we might find in a number are zeros. And that’s what the other sig fig rules address.
Rule number two is that zeros that appear between nonzero digits are significant. For example, in the number 4004, we have two zeros that appear between two nonzero digits. So, overall, this number has four significant figures.
The third sig fig rule talks about zeros that appear at the end of a number, also called final zeroes. Final zeros on a number with a decimal point are significant. For example, in the number 160.00, the three final zeros, or trailing zeros, are significant because the number has a decimal point. On the other hand, if we had a number that does have trailing zeros but no decimal point, then those trailing zeros are not significant. The number 17000 only has two significant figures.
The fourth, and final, sig fig rule for us to remember is that leading zeros are not significant. For example, in the number 0.0521, the two zeros that lead this number are not significant. So, the number, overall, has three sig figs. These four rules, which are very helpful to keep in mind, apply to individual numbers only. Now, let’s talk about sig fig rules when combining numbers.
When it comes to combining numbers, either through addition, subtraction, multiplication, or division, it turns out that there are just two rules we’ll want to keep in mind. The first rule says that when we’re adding or subtracting, we round our answer to the least number of decimal places.
For example, let’s say we were adding together the numbers 807.124 and 4.6. Adding these numbers directly, we would get a result of 811.724. But because we recall our sig fig addition rule, that we only wanna keep our answer to the least number of decimal places, we round our result to a number with just one decimal place in it. So, using this rule, after adding these two numbers together, we would report a result of 811.7. That’s the rule when we’re adding or subtracting. Now, let’s look at the second rule which has to do with multiplying and dividing.
This rule tells us that when we’re multiplying or dividing numbers, we’ll round our answer to the least number of significant figures. For example, say we had two numbers 8.6741 and 12.5 and we were multiplying these numbers together. If we directly multiply them together, we’d get a result of 108.42625. But we know we’ll want to apply a significant figures rule to this result.
This is where the difference between multiplying/dividing and adding/subtracting comes in. If we were using the adding/subtracting rule, where we keep the least number of decimal places, then we would report a result of 108.4. But since we’re multiplying or dividing, we know we’ll want to around our answer to the least number of significant figures.
Looking at our two values, we see the first one has five sig figs. And the second one has three. This means that when we multiply them together, the result we would report only includes three significant figures. It would be 108. When we’re able to keep in mind the four rules we saw earlier along with these two for combining numbers with significant figures, we’ll be able to solve nearly any significant figure question. Let’s look at a couple of those questions now.
Determine the number of significant figures in the following measurement, 0.007. Determine the number of significant figures in the following measurement, 14540.00. Determine the number of significant figures in the following measurement, 8.0 times 10 to the fifth. Determine the number of significant figures in the following measurement, 57.0320. Determine the number of significant figures in the following measurement, 600.24.
Starting with our first number, 0.007, we identify seven as significant because it’s a nonzero digit. The other digits, based on the rule that leading zeros are not significant, are not significant figures themselves. So, this number has one significant figure.
Looking at our next number, 14540.00, we know the one, four, five, and four are significant since they’re nonzero. And the three trailing zeros, because this number has a decimal point, are also significant. This means that all seven figures in this number are significant.
In the third number, 8.0 times 10 to the fifth, eight is significant because it’s nonzero. And the zero is significant because it trails a number that has a decimal. The reported exponent, 10 to the fifth, does not add precision to this number. So, it has two significant figures.
In the fourth value, 57.0320, we highlight the nonzero numbers as significant. The first zero is also significant because it’s surrounded by nonzero numbers. And the last zero is significant as well because it’s a trailing zero in a number with a decimal point. So, all six of the figures in this number are significant.
Looking at our last number of 600.24, the six, two, and four are significant because they’re nonzero numbers. And the two zeros are also significant because they have nonzero numbers on either side. This means there are five significant figures in this number. Now, let’s get some practice combining numbers, including their significant figures.
A can contains 375 milliliters of soda. How much is left after 308 milliliters is removed?
This exercise involves combining two numbers and keeping track of the significant figures involved for our answer. We’re told that we start with 375 milliliters of soda in a can. We then remove 308 of those milliliters. And we want to know how much is leftover. What we’re doing is combining two numbers through subtraction, that each have three significant figures in them.
If we perform the subtraction directly, we get a result of 67 milliliters. But the question is, since each of the numbers we added together has three significant figures, does our answer need to have three as well? In other words, would we report 67.0 milliliters as our result?
We can recall the rule that when we’re adding or subtracting numbers, the way to keep track of significant figures, is to round our answer to the least number of decimal places involved. The two numbers we’re working with, both have zero decimal places. So, our final answer will as well. The amount of soda leftover is 67 milliliters.
Let’s recall what we’ve learned about sig figs. First, sig figs ensure that measurement precision is maintained. There are four rules for determining how many significant figures a number has and two rules for combining numbers together. And finally, significant figures and keeping track of them lets us design and build precisely-working machines, such as cars, planes, clocks, rocket ships, and so on. Overall, significant figures are great enablers of precisely performing technology.