# Order of Magnitude Calculator

By Maciej Kowalski, PhD candidate
Last updated: Dec 30, 2020

Welcome to Omni's order of magnitude calculator, where we'll learn how to determine a number's order of magnitude estimate using scientific notation. In essence, it tells us roughly how large (or tiny) the value is without mentioning it in any way. Don't worry if that sentence seemed confusing at first; we'll soon see a proper order of magnitude definition to make it more precise.

So, what is an order of magnitude? Well, let's jump straight into the article and find out, shall we?

## What is an order of magnitude?

With scientific progress, people are able to study objects and structures that seemed inaccessible centuries ago. For instance, chemistry can use precise equipment to look at particles, while astronomers are able to find out more and more about planets and galaxies many light-years away.

Whichever side you're on, it may prove tiring or even difficult to make calculations about objects that are so tiny or so huge. Of course, we can always convert the units to their smaller/larger equivalents, but there's a far more elegant way of dealing with this problem - scientific notation.

Formally, expressing a number in scientific notation (also called standard form, especially in the UK) means writing it as:

`a = b * 10ⁿ`,

where:

• `a` is the number we want to have in scientific notation;
• `b` is (including a sign) a value between 1 and 10 (including 1 but excluding 10); and
• `n` is some integer (possibly negative).

Essentially, the process boils down to writing `a` as something times `10` to some power. That exponent `n` is called the order of magnitude of `a`.

So what is an order of magnitude? It is the largest power of `10` that is smaller or equal to our value when the number is positive (we reverse the inequalities for negative entries). If you'd like to sound fancier, we could say that it corresponds to rounding down the logarithm with base `10` of the number's the absolute value.

Arguably, such symbolic order of magnitude definition is nice to have, whichever description you choose. Nevertheless, how about we see how it works in practice on some examples?

## Order of magnitude estimate examples

We explain how to find the order of magnitude of a number in detail in the next section. For now, let's just see how the thing looks and what information it gives.

Let's have two simple scientific notation examples:

`1,370 = 1.37 * 10³` and `0.082 = 8.2 * 10⁻²`,

so their orders of magnitude are `3` and `-2`, respectively.

Firstly, note that although

`1,370 = 1.37 * 10³ = 13.7 * 10²`,

only the first representation is in scientific notation since, in the second, the number `13.7` is larger than `10`. That is why we've established such strict rules in the order of magnitude definition on the value that appears before "times 10 to some power." Otherwise, we wouldn't know whether to pick `3` or `2`.

Also, note how the order of magnitude tells us nothing about the number in front of the multiplication by `10` to some power. When someone tells us that a number's order of magnitude is `-2`, then the value could be equal to `0.082`, but it could just as well be `0.01093`. Arguably, the numbers are quite far apart. Therefore, remember that the order of magnitude estimate is... well, an estimate. It only gives us a rough idea of how tiny or large a number is.

These examples were very simple, and although some would call the two numbers large and small, respectively, let's get back to the atom- and planet-sized objects that we've mentioned in the first section. What is more, we'll take them as an opportunity to see how to find the order of magnitude ourselves or by using the order of magnitude calculator.

## Using the order of magnitude calculator

Let's talk mass. On one end of the spectrum, let's take that of the Earth, which is approximately:

`5,972,000,000,000,000,000,000,000 kg`,

and on the other, that of a helium atom:

`0.0000000000000000000000000066423 kg`.

For our imperial friends out there, feel free to use Omni's weight converter to have the two numbers in pounds or any other unit of your choice.

If we input the values into the order of magnitude calculator under "Number", it will spit out their scientific notation and order of magnitude underneath. Nevertheless, let's see how to get them ourselves.

In both cases, we begin by looking for the first non-zero digit from the left. We take it and put a decimal dot after it, followed by consecutive digits of our number until we reach the last non-zero digit. For the Earth, this gives the number `5.972`, while for helium, we get `6.6423`. These will be the numbers that come before the "times `10` to some power." It remains to figure out the some power part.

If the initial number was larger than `1` (like the mass of the Earth most certainly is), we count all digits, but the first - in this case, it's `24`. If it were smaller than `1` (like the mass of a helium atom), we count the zeros that come before the first non-zero digit, plus that digit - in our case, it's `26 + 1 = 27`. Additionally, for the second case, we take the opposite of the number, i.e., `-27` instead of `27`, which gives us the order of magnitude. In other words, we have the scientific notation:

`5,972,000,000,000,000,000,000,000 kg = 5.972 * 10²⁴ kg`

and

`0.0000000000000000000000000066423 kg = 6.6423 * 10⁻²⁷ kg`.

To sum up, the order of magnitude estimates are `24` for the mass of the Earth and `-27` for the mass of a helium atom.

Hopefully, with these two examples, you've come to appreciate the simplicity of using the order of magnitude (or at least the scientific notation) over having the numbers in their full form. Surely, you didn't enjoy counting the zeros above, did you? Or did you just take it for granted that we gave you the right number?

Well, someone has to do the monotonous part to have you appreciate the work we do here at Omni!

Maciej Kowalski, PhD candidate
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