How can we make reasonable approximations without tedious calculations?

Ordinarily, when we do estimates, we tend to say things like “about a dozen,” or “around four thousand.” If we look closely, we can see that not all estimates, even verbal ones, are of the same accuracy. The vaguest sorts of estimates are essentially how many zeros are in any measurement. This is known as an order of magnitude estimate – finding the closest power of ten to the number you’re looking for.

Suppose you’re at a crowded hockey stadium. An order of magnitude estimate means that, instead of saying, “There are 4335 people at this hockey game”, you can say, “There are a few thousand people at this hockey game.” You don’t have enough information to say with any precision how many people there are, or even how many thousands of people there are. But you do know enough to say with certainty that there are too many to count in the hundreds, and not enough to count in the tens of thousands.

**How is this useful?**

While this may seem vague, order of magnitude calculations are actually quite a powerful tool because they allow you to compare values to see how well they measure up to each other, even without access to precise information about those values. For example, you may not know exactly how much energy is put into manufacturing a paper plate or a car, but you probably do have a reasonable guess at arriving at an order of magnitude estimate. Using this estimate, you can quickly conclude that making one paper plate is insignificant when compared to making a car.

**Sample Calculation**

Suppose you want to calculate how much fuel your car uses each month. You could start an estimation by saying, “Well, I drive around 50 km a day” (an invaluable tool is Google Earth’s ruler function; use it to trace out your path). Then you can search for your car’s fuel economy on the Internet, say 11 L/100 km. Now you just multiply (Eqn.1):

$\dfrac{50 \text{ km}} {1 \text{ day}} \times \dfrac{11 \text{ L gas}}{100 \text{ km}} \times 30 \text{ days} = 165 \text{ L gas}\tag{1}$

Notice, however, how many things we didn’t take into account – tire pressure, passengers, road conditions, stoplights… there are a whole host of factors that could change this number. However, we do know that many of these factors are insignificant, because they may be an order of magnitude smaller (i.e. weight of passengers – around 100 kg vs. weight of the car – around 1000 kg), so we know they can be neglected. However, some of them are significant and will reduce the trust we have in our number. The more assumptions we make, the less sure we are, so we may have to say the fuel consumption is 20 000 L, or even “on the order of 10 000 L.”

**Verifying Results**

It is often a good idea to make an order of magnitude estimate even if you are planning on calculating a more precise value afterwards. Your order of magnitude estimate can help you check your results. If your order of magnitude estimate is on the order of 10 000 L and you calculate the value to be 14 231 L, your value is reasonable. However, if you calculated 3 L, there is a good chance you made a mistake somewhere – 3 L is not on the order of 10 000 L.

**Guidelines**

Here are some rough guidelines on when and when not to use order of magnitude approximations:

#### Use order of magnitude when:

- You don’t know any of your measurements for sure
- You want to “play it safe” and make a conservative estimate
- You have calculated a more precise result but want to know if your answer is reasonable

#### Don’t use order of magnitude when:

- You’re comparing two things that are close in size (say within an order of magnitude of each other)
- You have access to complete information that is very precise

**Summary:**

Order of magnitude calculations are useful for making estimates when incomplete data is available. They are also useful for checking if answers obtained in more precise calculations are reasonable.

**Revised (CEW) 2019-10-19**