In this lesson, we’re going to learn about orders of magnitude, what they are, why they’re important, and how to use them in exercises. To start out, imagine that your little sister comes to you with a problem. Today is show and tell at her school. And she wants to bring her bead collection to show the class. The problem is, she has lots and lots of beads, more than we can show here. In fact, because you’ve seen her purchase them by the pack of 500, you know that your sister has 10000 beads in her collection. And she wants to find some way of containing all 10000 of those beads so she can carry them to school and show them to her classmates.
When your sister comes to you with this request, you yourself are trying to get out the door. But you look at the back of one of the packs of her beads to see just how big one of her 10000 beads is. According to the manufacturer, each one of these 10000 tiny beads has a volume of 10 to the negative ninth cubic meters. Then the question becomes, what size container will be needed to carry all of your sister’s beads?
This is a problem involving orders of magnitude. When we talk about orders of magnitude, we’re speaking of numbers that have been rounded to the nearest power of 10. So examples of numbers expressed this way are 10 to the one — that’s simply 10 — 10 to the fifth or 100000, 10 to the negative sixth — that’s one one millionth — and so on. One nice thing about using orders of magnitude is that it allows calculations that move across very large scales. For example, a question like “How many drops of water are in the Pacific Ocean?” is an orders of magnitude question. Or, similarly, we might want to know how many stars are there in the universe. That’s an orders of magnitude question.
Another advantage of using numbers in orders of magnitude is that they enable estimates of results that are hard to compute otherwise. Here’s an example. Say we were asked the question, “How many pizza restaurants are there in Rome, Italy?” Now, we may not know much at all about Rome, Italy. We may not have an idea of how many people live there or how much they like pizza or how many restaurants are there to make pizza. But using orders of magnitude estimates, we can find an answer to this question that will probably not be far off.
Let’s start off in finding an answer by estimating the number of people who live in Rome, Italy. Say we estimate that there are 10 to the sixth or 1000000 residents of this city. Now, of course, we don’t know if that’s exactly right. It may be a little bit more or a little bit less. But probably, Rome, Italy, has fewer than 10 million. That would be 10 to the seventh. And it probably has more than 10 to the fifth. That would be 100000 people. So even if this estimate is not spot-on, to the order of magnitude, it’s fairly good.
Now, let’s further say that each of these 1000000 people eats pizza every 10 to the one or ten days. So this means that slightly less often than once a week, each of the 1000000 residents eats pizza. And let’s further say that every time a person eats pizza, that is, once every 10 days, that person eats 10 to the zero or one pizza. Now, that’s a lot of pizza. It’s an entire pizza pie. But recall that we’re using order of magnitude estimates. An entire pizza being consumed is probably closer to the truth than having a tenth of a pizza or 10 pizzas.
So we’re now ready to understand how much pizza is consumed in Rome, Italy, on a daily basis. We bring together our three estimates that there are a million people in the city, that each one eats pizza every 10 days, and each time a person eats pizza, the person eats one whole pie. If we combine these orders of magnitude. The number of people times how much pizza they eat each time they eat pizza, divided by the frequency at which they eat pizza. We find in the city of Rome, Italy, we estimate 10 to the fifth or 100000 pizzas are eaten every day. So that’s our estimate for the demand of pizza in Rome, Italy.
Now, let’s consider the supply. Let’s imagine a standard pizza restaurant. And we can imagine that this standard restaurant might have one pizza oven. And in this oven, of course, the restaurant will create pizzas. We can further imagine that each oven will be able to produce 10 to the one or 10 pizzas every hour. They may be able to produce more, may be able to produce less. But that’s a pretty good estimate to the nearest order of magnitude. And let’s imagine if this pizza oven that makes 10 pizzas per hour is in operation for 10 hours a day. We’ll call that the time that each pizza restaurant is open daily.
Bringing our three estimates together. That each restaurant will have one pizza oven. Each oven will be able to make 10 pizzas an hour. And each restaurant will be open for 10 hours a day. We find that each pizza restaurant will be able to produce 10 to the two or 100 pizzas daily. So if that’s the number of pizzas per day that a single pizza restaurant could make, then to find the total number of pizza restaurants required to meet the demand, we simply divide that amount for the demand, 10 to the fifth, by the ability of a single restaurant to meet it, 10 to the two. That comes out to 10 to the three or 1000.
So based on a series of approximations we’ve made, we’ve come to a fairly accurate estimate of how many pizza restaurants there are in the city of Rome, Italy. This answer won’t be exactly right of course. But it’s close enough to being correct that it’s still useful. This is another advantage of using numbers in orders of magnitude. Now, let’s get a bit of practice with an orders of magnitude example question.
Determine how many hydrogen atoms placed end-to-end in a line would stretch across the diameter of the Sun. Use a value of 10 to the negative 10th meters for the diameter of a hydrogen atom. And use a value of 10 to the ninth meters for the diameter of the Sun.
We can name the value of 10 to the negative 10th meters, the diameter of a hydrogen atom, 𝐷 sub 𝐻. And we’ll call the diameter of the Sun, 10 to the ninth meters, 𝐷 sub 𝑆. We want to solve for a number, how many hydrogen atoms will be needed end-to-end to stretch across the diameter of the Sun. We’ll call that number capital 𝑁.
This is an exercise involving orders of magnitude. To solve for 𝑁, we’ll divide the diameter of the Sun, 𝐷 sub 𝑆, by the diameter of a single hydrogen atom. This fraction will tell us how many hydrogen atoms are needed to stretch across that Sun’s diameter. 10 to the ninth meters divided by 10 to the negative 10th meters is equal to the pure number 10 to the 19th. This number slightly more than a million trillion is the number of hydrogen atoms that will be needed to stretch across the Sun’s diameter. Let’s try another orders of magnitude exercise.
Mount Everest in the Himalayas is the highest mountain on Earth above sea level. The mountain is approximately nine kilometers tall. The diameter of Earth is roughly 10 to the fourth kilometers. Approximately, what fraction of Earth’s diameter is the height of Everest?
We can call the height of Mount Everest, nine kilometers, 𝐻 sub 𝑚. And the approximate diameter of the Earth, 10 to the fourth kilometers, we’ll call 𝐷 sub 𝐸. As a fraction of the Earth’s diameter, the height of Everest will be equal to 𝐻 sub 𝑚 over 𝐷 sub 𝐸. That is, the height of Everest divided by the Earth’s diameter. When we plug in for these two values and calculate the fraction, we find that the answer is unitless. And it’s nine times 10 to the negative fourth. That’s the height of Mount Everest expressed as a fraction of the Earth’s diameter.
Now, we come back to our original question of how your little sister can carry her bead collection to school. Recall that there are 10 to the fourth or 10000 beads in the collection and that the volume of each individual bead is 10 to the negative ninth cubic meters. With our newfound knowledge of orders of magnitude, we can calculate the volume it would take to carry all 10000 beads. It’s the number of beads, 10 to the fourth, multiplied by the volume of each one.
When we calculate this product, we find it’s equal to 10 to the negative fifth cubic meters. That’s the volume of storage your sister would need to carry her collection. Looking around, we see an empty film canister on the countertop. That will work. That has enough volume to carry her 10000-bead collection.
In summary, orders of magnitude numbers are typically approximate values. They enable accurate estimations and also enable calculations that go across large scales: distance or time or some other measured value. And finally, they’re normally expressed as powers of 10, for example, 10 to the zero, 10 to the negative one, 10 to the 10th, and so on. Using orders of magnitude numbers is a good, shorthand way of making calculations.