What Is An Order Of Magnitude? (3 Key Concepts To Know)


Order of magnitude is an idea that is used often in physics, mathematics, computer science, and other disciplines.  It helps us to get a grasp of how large quantities are – but it still helps to know the nuts and bolts of this concept.

So, what is an order of magnitude?  An order of magnitude is the relative size of a number. A base 10 logarithm is often used to find an order of magnitude (in that case, multiplying by 10 increases order of magnitude by 1). For example, 100 has order of magnitude 2, (102 = 100) while 1000 has order of magnitude 3 (103 = 1000).

We can use order of magnitude to compare two numbers to see if they are in the same “neighborhood” (e.g. 2 million and 3 million are both in the same “neighborhood”, since one is 2×106 and the other is 3×106).

In this article, we’ll talk about order of magnitude, what it means, and why we use it.  We’ll also look at some examples of how to calculate orders of magnitude.

Let’s get started.

What Is An Order Of Magnitude?

An order of magnitude is a category that tells us the relative size of a number.  When taken together, all orders of magnitude classify the set of real numbers.

The following table shows orders of magnitude for various sets of numbers.  Note that for higher orders of magnitude, the range of numbers is larger.

Range Of
Numbers
Order Of
Magnitude
0.01<=x<0.1-2
0.1<=x<1-1
1<x<100
10<=x<1001
100<=x<10002
1000<=x<100003
This table shows various ranges of numbers
and the corresponding order of magnitude.

How Do You Calculate Orders Of Magnitude?

Usually, we use powers of 10 to rewrite numbers and find their order of magnitude.  An alternative method is to take the logarithm (base 10) of the number and round down to the nearest whole number.

For a real number R, we can find the order of magnitude by writing it in this form:

  • R = A*10B

where 1 <= A < 10 and B is the order of magnitude (B is always a whole number, but it can be positive, negative, or zero).

Example 1: Finding An Order Of Magnitude

The number 400 has an order of magnitude 2.  To see this, we first factor 400 in the form above:

  • 400 = 4*102 [A = 4, B = 2]

Since B = 2, we can say that the number 400 has an order of magnitude of 2.

We can also use the logarithm base 10 method to find the order of magnitude:

  • First, log10(400) = 2.602…
  • Now, round down (truncate the decimal) to get 2.
  • So, 400 has order of magnitude 2.

This agrees with what we found above.

Example 2: Finding An Order Of Magnitude

The number 6000 has an order of magnitude 3.  To see this, we first factor 6000 in the form above:

  • 6000 = 6*103 [A = 6, B = 3]

Since B = 3, we can say that the number 6000 has an order of magnitude of 3.

We can also use the logarithm base 10 method to find the order of magnitude:

  • First, log10(6000) = 3.778…
  • Now, round down (truncate the decimal) to get 3.
  • So, 6000 has order of magnitude 3.

This agrees with what we found above.

Example 3: Finding An Order Of Magnitude

The number 78 has an order of magnitude 1.  To see this, we first factor 78 in the form above:

  • 78 = 7.8*101 [A = 7.8, B = 1]

Since B = 1, we can say that the number 78 has an order of magnitude of 1.

We can also use the logarithm base 10 method to find the order of magnitude:

  • First, log10(78) = 1.892…
  • Now, round down (truncate the decimal)to get 1.
  • So, 78 has order of magnitude 1.

This agrees with what we found above.

Example 4: Finding An Order Of Magnitude

The number 9 has an order of magnitude 0.  To see this, we first factor 9 in the form above:

  • 9 = 9*100 [A = 9, B = 0]

Since B = 0, we can say that the number 9 has an order of magnitude of 0.

We can also use the logarithm base 10 method to find the order of magnitude:

  • First, log10(9) = 0.954 …
  • Now, round down (truncate the decimal)to get 0.
  • So, 9 has order of magnitude 0.

This agrees with what we found above.

Example 5: Finding An Order Of Magnitude

The number 0.56 has an order of magnitude -1.  To see this, we first factor 56 in the form above:

  • 0.56 = 5.6*10-1 [A = 5.6, B = -1]

Since B = -1, we can say that the number 0.56 has an order of magnitude of -1.

We can also use the logarithm base 10 method to find the order of magnitude:

  • First, log10(0.56) = -0.251…
  • Now, round down to get -1.
  • So, 0.56 has order of magnitude -1.

This agrees with what we found above.

Example 6: Finding An Order Of Magnitude

The number 0.002 has an order of magnitude -3.  To see this, we first factor 0.002 in the form above:

  • 0.002 = 2*10-3 [A = 2, B = -3]

Since B = -3, we can say that the number 0.002 has an order of magnitude of -3.

We can also use the logarithm base 10 method to find the order of magnitude:

  • First, log10(0.002) = -2.698…
  • Now, round down to get -3.
  • So, 0.002 has order of magnitude -3.

This agrees with what we found above.

Why Do We Use Orders Of Magnitude?

We use orders of magnitude to categorize and compare numbers.

As noted earlier, we can find out which “range” of values a number falls into, and assign it an order of magnitude.  For example, the number 500 is between 100 and 1000, so its order of magnitude is 2.

We can also compare two numbers by finding their orders of magnitude to see if they are in the same “neighborhood”.  For example, 800 is also between 100 and 1000, so its order of magnitude is 2, meaning it is “comparable” to 500 (both are in the range 100 to 1000).

There are several applications of the concept of orders of magnitude in real life, including:

  • We can use orders of magnitude to compare the best, worst, and average case run times for algorithms in computer science (this can help us to benchmark speed and optimize code).
  • We use orders of magnitude (base 10 logarithms) to convert concentrations of hydrogen ions to a pH (power of hydrogen), which is useful in chemistry to test acids and bases, for soil testing in farming and gardening, etc.
  • We use orders of magnitude (base 10 logarithms) to find the strength of earthquakes (Richter scale).
  • We use orders of magnitude to measure sound.  Take the decibels, divide by 10, and use the result as the exponent of 10.  This tells you how much louder a sound is than a baseline noise level of 0 decibels.
chemistry
We use orders of magnitude (pH is the negative of the logarithm base 10 of hydrogen ion concentration) to compare acids and bases in chemistry

Can Order Of Magnitude Be Negative?

Order of magnitude can be negative in some cases.  When a number is between 0 and 1, its order of magnitude will be negative.

This is because the logarithm (base 10) of any number between 0 and 1 is negative.

Any number between 1 and 10 has a zero order of magnitude.

Any number 10 or above has a positive order of magnitude.

Note that the order of magnitude of zero is undefined.

Range Of
Numbers
Sign Of
Order Of
Magnitude
x=0Undefined
0<x<1Negative
1<=x<10Zero
x>=10Positive
This table shows the sign of the
order of magnitude for various
ranges of numbers.

How Many Orders Of Magnitude Is 1000?

The number 1000 is 3 orders of magnitude (its order of magnitude is 3).  This is because we can rewrite 1000 as 103.

Another way of seeing this is to take the logarithm (base 10) of 1000 and round down:

  • First, log10(1000) = 3
  • Now, round down to get 3.
  • 1000 has order of magnitude 3.

Conclusion

Now you know what orders of magnitude are and why we use them.  You also know how to do calculations to find the order of magnitude for a given number.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

Don’t forget to subscribe to my YouTube channel & get updates on new math videos!

~Jonathon

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