Someone told me that on the Richter Scale for earthquakes, adding 1 gives an earthquake 10 times as big. Question 1: is this true?

Then they told me that adding 0.5 does not give an earthquake 5 times as big. Question 2: is that true, and why?

Doreen

Hi Doreen,

To both of your questions, the answer is "Yes."

The Richter Scale is called a "logarithmic" scale, which - for our purposes - just means that a 4.0 earthquake is 10 times as "big" as a 3.0 earthquake. And if you go from a 4.0 earthquake to a 6.0 earthquake, that's an earthquake 100 times as big (multiply by 10 to get to 5.0, and multiply by 10 again to get to 6.0).

("bigness" of an earthquake has to do with the amplitude of its seismic waves).

Regarding your second question, it is tempting to think that if a change of 1 means "ten times as big," a change of 0.5 means "five times as big." But it doesn't work that way.

Let me show you why. Let's talk about a 3.0 earthquake, a 3.5 earthquake, and a 4.0 earthquake.

You know that if you go from a 3.0 earthquake to a 4.0 earthquake, that's 10 times as big.

But if a change of 0.5 meant 5 times as big, then we'd calculate:

From 3.0 to 3.5 is 5 times as big;
From 3.5 to 4.0 is 5 times as big;
Therefore, from 3.0 to 4.0 is  5x5 = 25 times as big! That's not right!

So let's just break down the math a bit.

If you want to go from a Richter scale reading of x to a Richter scale reading of y, you would take the difference of x and y, raise 10 to that number, and that's your "bigness" ratio.

For example, if you wanted to go from 2.0 to 7.0, your ratio is 10^7-2 = 10⁵ = 100,000.

So let's talk about a Richter Scale difference of 0.5:

Suppose you wanted to go from 2.0 to 2.5. Your ratio is 10^{2.5 - 2} = 10^0.5. What is 10^0.5? It's the square root of 10, which is approximately 3.1623.

So if one earthquake has a Richter value of 4.0, and another had a Richter value of 4.5, the 4.5 earthquake is 3.1623 times as big as the 4.0 earthquake.

By the same reasoning, we could figure out how much bigger a 4.2 earthquake is compared to a 4.1 earthquake:

10^{4.2 - 4.1} = 10^0.1 = 1.2589.