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In this lesson, you will learn the definition of order of magnitude and how it is closely associated with scientific notation. You will also learn how to determine the quantity of orders of magnitude in numbers with interesting real-life examples. Following the lesson will be a brief quiz.

## What Is Order of Magnitude?

When we think of the word ‘magnitude,’ we think of something very large.

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So we can assume that order of magnitude has something to do with large numbers, right? Long ago, scientists and mathematicians were working with very large numbers for certain values, like the speed of light or the distance from Earth to the sun, and they decided that they needed a simpler way to write and refer to these large numbers. That’s when they came up with scientific notation. It’s important to know about scientific notation when we speak about order of magnitude, so here are a few examples in this chart to get you quickly acquainted: ## Examples

What does scientific notation have to do with order of magnitude?Order of magnitude is the quantity of powers of 10 that there are in a number, or the number of powers of 0.1 in a negative number.

Order of magnitude is usually written as 10 to the nth power. The n represents the order of magnitude. If you raise a number by one order of magnitude, you are basically multiplying that number by 10. If you decrease a number by one order of magnitude, you are basically multiplying that number by 0.

1.Let’s go through some examples so you can see order of magnitude in action.

#### Problem #1:

How many orders of magnitude are in 1,000,000 (one million)?For the number 1,000,000, we will shift the decimal to the left, stopping just before the first digit of the number. The number of moves you make to the left is the order of magnitude. Since we moved it six times, there are six orders of magnitude of 1,000,000, meaning that you can multiply 10 six times and get 1,000,000.

#### Problem #2:

Find the order of magnitude for the earth’s mass, which is equal to 6,000,000,000,000,000,000,000,000 kilograms.

Whoa! That’s a lot of zeros! I’ll make it easier for you and tell you that there are 24 zeros there. When scientists and mathematicians write such big numbers, it would be just silly for them to write out all of those zeros. That’s why they would write it in scientific notation: Since there are 24 zeros, we would have to move to the left 24 times to shift the decimal point from where it is now to just after the first digit in the number.

The order of magnitude for the earth’s mass is 24. So we can say that the earth’s mass is 6 times 10 raised to 24 orders of magnitude. Remember that orders of magnitude are how many powers of 10 there are in a number. Let’s get some practice.

## How Many Orders of Magnitude?

How many orders of magnitude are in the following examples? The solutions are in scientific notation, but the order of magnitude (circled in red) is what we are looking for.

#### Example #1:

A classroom of 35 people? #### Example #2:

An auditorium of 502 people?

There are two orders of magnitude. Example #3:

A school of 4,054 people?

There are three orders of magnitude. Example #4:

A city of 1,987,000 people?  So how many orders of magnitude are there here? That’s right! There are 22.

This is telling us that we would have 7 times 10 times 10 times 10, and so on, 22 times! That would give us the estimated number of stars in the universe.Let’s look at another one. The mass of a neutron is approximately 0.000000000000000000000000001675 kilograms.

Whoa! Again, a lot of zeros and that number is certainly very small. How would we write this in scientific notation? Well, because the number is still less than 1, the order of magnitude would be negative. Here is the scientific notation:

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png” alt=”Scientific notation for the mass of a neutron” />

So how many orders of magnitude are there here? That’s right! There are negative 27. This is telling us that we would have 1.675 times 0.1 times 0.

1 times 0.1, and so on, 27 times! That would give us the estimated mass of a neutron.It is now clear to see why orders of magnitude are used in science and math.

## Lesson Summary

Order of magnitude is the number of 10s that are in a number. When speaking of order of magnitude, we usually have a base 10 and an exponent, n, which is our order of magnitude. If you increase an order of magnitude, you are basically just multiplying by 10. If you decrease an order of magnitude, you are basically just multiplying by 0.

1. Scientists and mathematicians use order of magnitude so they can make estimates of big numbers.

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