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Order of Magnitude: Analysis & Problems

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

It is a wonderful skill to be able to estimate a numerical result without having to use a calculator or computer. In this lesson we show how to do this using the order of magnitude idea.

What is Order of Magnitude?

If you live in North America and look up at the sky during the late fall, you may see geese migrating south for the winter. Their iconic 'V' flying pattern is very distinctive. In one view of the sky, how many birds do we see? Could we use this information to estimate the population size of geese in North America?

Migrating geese in the sky.
Migrating geese in the sky.

In this lesson we look at determining the order of magnitude and using it to estimate related quantities. The order of magnitude is the exponent of a power of 10. For example, the number 100 can be written as 10 to the power of 2. The order of magnitude of the number 100 is 2. The short way to write this uses a capital O with the order number in parentheses: the number 100 is O(2). Out loud we would say: 'the number 100 is order 2.' This O is sometimes called the big O notation and it's not zero but the capital letter O.

Let's Get Started

Using the order of magnitude to take the place of a number approximates the number. For example, 100 is O(2) but so is 45 and so is 449. That's a big range of numbers! Informally, you can look at a group and say: 'there are definitely more than 1 of those'. Like the geese you see in the sky, the order of magnitude is bigger than O(0). By the way, O(0) means 10^0 which equals 1. On the other hand, when looking at one 'V' pattern of geese, we don't see 100 geese! It's not O(2). Probably O(1). This approach means using your common sense and your observations to bracket your number. Bracketing is being able to say the order is greater than something but less than something else. The next time you are in a group, just for fun, see if you can bracket your order of magnitude estimate for the number of people present. Like a lot of things, estimating the order of magnitude gets better with practice. But is there a more formal way to get this number? I'm sure those migrating geese would like to know.

It's all about scientific notation. In scientific notation we write a number as digit decimal-point digit digit … X 10^power. For example, 110 is 1.10x10^2. The number 45 is 4.5x10^1. The next step in finding the order of magnitude is to round so there is no decimal point. In our examples, 1.10 rounds down to 1 and 4.5 rounds up to 5. We are going to drop the rounded number entirely but before we do, check to see if the number is 5 or greater. If the number is 5 or greater, it is closer to the next higher order of magnitude, so add 1 to the exponent. Meaning 45 is 4.5x10^1 which rounds to 5x10^1 which is closer to 10^2 than 10^1. Meaning 45 is O(2). What about 110? Well, 110 is 1.10x10^2 which rounds to 1x10^2. The 1 gets dropped and there is no change to the exponent because the 1 is less than 5. The number 110 is O(2). So far so good? My estimate of the number of geese flying overhead in one 'V' is O(1).

Calculating with Orders of Magnitude

Here's how to use order of magnitude estimates to calculate related information. It's all based on operations with exponents. When we multiply exponent numbers with the same base, we add the exponents. For example, 10^2 x 10^1 = 10^(2+1) = 10^3. Let's get back to our migrating friends.

How many flocks traverse the sky in one 10-minute interval? Estimate: O(1). Remember, that would be a number like 10. How many intervals in one day? O(2). How many days in the migrating season? O(2). Great!

What is the size of the Canadian geese population? It's the multiplication of four numbers: the number of geese in one 'V' flock O(1), the number of flocks we see during one 10-minute interval O(1), the number of 10-minute intervals in one day during which the birds are flying O(2), and the number of days in the migrating season O(2). None of these numbers is known precisely and it would be impossible to count all the individual birds. In multiplying those four numbers together, we are adding their respective orders of magnitude: O(1) O(1) O(2) O(2) = O(1+1+2+2) = O(6). Now, order of magnitude 6 is 10^6 which is 1 million. The published population number for geese in North America is a number between 1 and 2 million. We are very close with these order of magnitude estimates! The geese would be proud.

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