Orders of Magnitude in the Earth sciences

An introduction to the "order of magnitude" of numbers

Have you heard your instructor refer to one earthquake being "an order of magnitude" larger than another? Or that your estimate of a lake's volume is "within an order of magnitude" of the published data? Earth scientists are often interested in comparing data; i.e., are they the same, bigger or smaller? How much bigger? How much smaller? Earth scientists also don't always know every exact measurement, and sometimes we just need a rough sense of the size of some parameter. Estimated and calculated orders of magnitude provide values that describe the sizes or scales of geologic parameters and processes. Orders of magnitude present numbers relative to a scale based on powers of 10 and are particularly useful when we compare very large or very small values or ones that differ greatly in their scale. For example, when we compare the diameter of the Sun (~1.4 billion meters) to the diameter of a single hydrogen atom (1 x 10^-10 m), the Sun's diameter is many orders of magnitude larger (19 orders of magnitude different or 10¹⁹ times different in their magnitude of size). Imagine what 19 orders of magnitude difference suggests for the number of hydrogen atoms that could fit within the Sun.

Orders of magnitude are used in many aspects of geology from values on the pH scale to the mass and length needed to describe our world and universe!

This module will share how to estimate orders of magnitude and how to calculate them.  Calculated orders of magnitude are especially important if you intend to use the values in a research paper.  They are also important when you are comparing values that are close enough that they round into the same order of magnitude.  For example, when two people estimate the weight of a rock sample as either ~9 grams or ~11 grams, these two values have the same calculated order of magnitude.

When are orders of magnitude useful?

Both calculated and estimated orders of magnitude help geoscientists simplify data for the purposes of comparing one value to another.  We can even use these values in understanding numerical models because we can see how changing one value by an order of magnitude impacts a calculation, so it helps us understand variable sensitivity.

Estimating orders of magnitude

Estimation allows Earth scientists to talk about values without knowing exact numbers.  When estimating orders of magnitude, we rely on powers of 10.  The table below is a good reference for these values and also shows how select powers of 10 are written as exponents and in scientific notation:

Name	Common Notation	Power of 10 notation	Scientific notation
Quintillion	1,000,000,000,000,000,000	1018	1 x 1018
Quadrillion	1,000,000,000,000,000	1015	1 x 1015
Trillion	1,000,000,000,000	1012	1 x 1012
Billion	1,000,000,000	109	1 x 109
Million	1,000,000	106	1 x 106
Thousand	1,000	103	1 x 103
Hundred	100	102	1 x 102
Ten	10	101	1 x 101
One	1	100	1 x 100
One tenth	0.1	10-1	1 x 10-1
One hundredth	0.01	10-2	1 x 10-2
One thousandth	0.001	10-3	1 x 10-3
One millionth	0.000001	10-6	1 x 10-6
One billionth	0.000000001	10-9	1 x 10-9
One trillionth	0.000000000001	10-12	1 x 10-12
One quadrillionth	0.000000000000001	10-15	1 x 10-15
One quintillionth	0.000000000000000001	10-18	1 x 10-18

As a geological example, the average diameter of a coarse sand grain is ~1 mm whereas something that is 100 times larger would be a cobble (~100 mm).  Both of these grain size measurements are a power of 10.  The sand grain's size is equal to 10⁰ (or 10 to the zero power).  The cobble's size is 10² (or 10 to the second power).  The exponents used in these two measurements are orders of magnitude.

If you would like to determine an estimated order of magnitude, such as the estimated annual amount of microplastic that goes into Lake Michigan, follow these steps:

Step 1:  Start with an estimate that is a power of 10 or is written in scientific notation.  If you have an estimate in scientific notation, skip to step 3.

Step 2:  Write your estimate in an exponent form that shows the power of 10.  (no adjustment is needed for scientific notation)

Step 3:  Identify the exponent used for your power of ten or for your scientific notation.  That value is the estimated order of magnitude.  (Make sure that if you have a negative exponent that you don't lose the negative sign-it is ok to have a negative order of magnitude value)

Comparing estimated orders of magnitude

Having one estimated order of magnitude is a good start, but very often we want to see how one estimate compares to another.

Colonial corals such as brain corals are made up of 10's to 1000's of small organisms called polyps.  A single coral polyp is ~ 1 mm in diameter.  In 2024, the largest known brain coral was roughly ~5 m in diameter.  To estimate the difference between the estimated orders of magnitude for these measurements, we follow these steps:

Step 1. Check that the two values being compared have the same measurement units.  If they are different, convert one of the numbers so that both values have matching units. If you are using non-metric units (like inches to feet), please be very careful with the conversions.

Step 2.  Determine the estimated order of magnitude for both measurements.

Step 3.  Subtract the estimated order of magnitude of the smaller number from the larger estimated order of magnitude.  If the result is zero, it means that the two values have the same order of magnitude.

Step 4. If you would like to describe the comparison as a magnification value (or demagnification), use the result from Step 3 as an exponent for 10 (such as "10ⁿ", or "10 to the power of n", where n is the result from Step 3).

 

How do I calculate the order of magnitude for any number?

If you need to calculate the mathematical value for the order of magnitude, you will need to involve a component of rounding that is not a part of the estimated order of magnitude values described above.  By definition, a number's order of magnitude is the power of 10 that most closely approximates the value of the number.  For example, the number 20 is closest to 10 or 10¹.  The number 90 is best approximated by 100 or 10².  The exponents are the orders of magnitude values.

There are various ways to calculate order of magnitude.  For this module we will focus on one, which is to take the base-10 log (often written as log₁₀) of the number and round the result.  The result is the order of magnitude for your original number and will always be a whole number.

Let's say you would like to calculate the order of magnitude for 4.57 × 10²³ which happens to be the approximate number of carbon atoms in the Hope diamond.

Step 1. Take the base-10 logarithm (log₁₀) of the number. A handy reference for taking a base-10 logarithm is here.

You can use a calculator that has a log button or use the Google browser calculator to find the value.

Step 2. Round the answer to the nearest whole number (integer).  This is the calculated order of magnitude.

Examples of Calculated Order of Magnitude:

Starting number (written in standard notation and scientific notation)	Log10 value	Calculated order of magnitude
0.00000876 (8.76 x 10-6)	-5.1	-5
0.00876 (8.76 x 10-3)	-2.1	-2
8.76 (8.76 x 100)	0.9	1
876 (8.76 x 102)	2.9	3
87,600,000 (8.76 x 107)	7.9	8

*Note that the calculated order of magnitude doesn't always match the exponent in the scientific notation.  For the values used in the table, the calculated orders of magnitude do not match the estimated orders of magnitude and that is ok.  Just make sure that you always know if you are determining an estimated or a calculated order of magnitude.

How do I compare numbers using calculated orders of magnitude?

In some cases, you might be asked to compare the calculated orders of magnitude. Below are steps for this.

Let's say you would like to determine the difference in the calculated orders of magnitude between the depth of the lithosphere at approximately 100 km and a borehole drilled in the shallow subsurface to a depth of 100 m:

Step 1. Check that the two values being compared have the same measurement units.  If they are different units, convert one of the numbers so that they have the same units. If you are using non-metric units (like inches to feet), please be very careful with the conversions.

Step 2.  Determine the order of magnitude for two numbers.

Step 3.  Subtract the order of magnitude of the smaller number from the larger order of magnitude.

Step 4. If you would like to describe the comparison as a magnification value (or demagnification), use the result from Step 3 as an exponent for 10 (such as "10ⁿ", or "10 to the power of n", where n is the result from Step 3).

 

Where do you use Orders of Magnitude in Earth science?

Estimated and calculated orders of magnitude are valuable when we want to describe or compare the scale of data, or if you don't need to know exact numbers.     Some common scales that we encounter in the Earth sciences include:

• Time Scales -- how long do certain geological processes take? (e.g., how long did dinosaurs exist)
• Spatial Scales -- how big are geological materials? (e.g., how thick is a soil horizon)
• Quantities and Concentrations -- how much of a geological material exists? (e.g., how much iron is in the Earth's mantle)
• Rates and Processes -- what is the speed of a geological process? (e.g., how fast is a river flowing)
• Magnitude of Events -- what is the intensity of a geologic event? (e.g., how intense was an earthquake)

Applying orders of magnitude helps Earth scientists analyze data efficiently and helps us compare values of geologically relevant numbers that can cover large ranges.

Also, measurements using the metric system's prefixes are an ideal place to use orders of magnitude because the metric system is based on powers of 10. e.g., 1 m vs. 1 km vs. 1 mm.  Each of these units represents a different order of magnitude.  The same is true when comparing concentrations in ppb, ppt, ppm units (parts per billion, parts per trillion, parts per million).

Next steps

I am ready to PRACTICE!

If you think you have a handle on the steps above, click on this bar to try practice problems with worked answers.
Or, if you want even more practice, see 'More help' below.

More help (resources for students)

• Example exercise determining order of magnitude from Khan Academy

Pages written by Dori Farthing (SUNY Geneseo) and Susan Bilek (New Mexico Institute of Mining and Technology).