Scientific Notation

Scientific notation is a method we use to write extremely large or tiny numbers without getting a headache. Instead of writing a never-ending string of meaningless digits, we express it as a simple number multiplied by a power of 10.

I know, you probably enjoy math about as much as I enjoy explaining the obvious. But let’s stay focused. A number in scientific notation is written like this:

$$ N \cdot 10^x $$

Here, $N$ is a single number between 1 and 9, either positive or negative.

$$ 1 \le N \lt 10 $$

No matter how many times I explain it, someone always tries to sneak in an 10. Don’t be that person. The $x$ is the exponent, which tells you how many times to move the decimal point.

How does it work?

I can already see some confusion, so here’s a quick step-by-step guide:

If the number is less than 1
1. Move the decimal point to the right until you hit the first non-zero digit.
2. Count how many places you moved it. That number will be your negative exponent.
Example: \[
0.0017 = 1.7 \cdot 10^{-3} \] We moved the decimal point 3 places to the right. The exponent is -3.
If the number is greater than 1
1. Move the decimal point to the left until the number is between 1 and 10.
2. Count how many places you moved it. That will be your positive exponent.
Example:
\[ 540000 = 5.4 \cdot 10^{5} \] We moved the decimal point 5 places to the left. Simple as that.

Don’t think this is just some trick to make textbooks look fancy. Scientific notation is genuinely useful, even if it doesn’t always seem that way at first.

When should you use scientific notation?

It comes in handy when you’re dealing with numbers so huge or tiny that trying to read them makes your head spin.

Small numbers, like the mass of an atom or the charge of an electron. You know, those things that make you feel like a giant.
Huge numbers, like the distance between galaxies or the mass of a planet, which make you realize just how small we are in the grand scheme of things.

What are the benefits of scientific notation?

First off, it simplifies calculations. You can multiply and divide numbers just by handling the exponents of 10 and forget about the rest. It also improves readability. Huge or tiny numbers finally become easy to understand, unless you’re the kind of person who enjoys counting zeros for hours.

For example, the mass of a hydrogen atom is a very small number:

$$ m_H = 0.00000000000000000000000000000000167 \, \text{kg} $$

In scientific notation, it becomes much easier to read:

$$ m_H = 1.67 \cdot 10^{-27} \, \text{kg} $$

We moved the decimal point 27 places to the right. That’s right—27! Hopefully, you now see just how useful this is.

Let me give you another example to really drive the point home.

The mass of the Earth is a very large number:

$$ m_T = 5 \, 970 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, \text{kg} $$

In scientific notation, we write it as:

$$ m_T = 5.97 \cdot 10^{24} \, \text{kg} $$

Here, the decimal point moved 24 places to the left. The value is the same, but it’s written in a much more manageable way, and you’re spared from counting all those zeros.

Finally, scientific notation is a standard in science because scientists use it across nearly every field (physics, chemistry, etc.). So, it’s not something you can just ignore... especially if you want to sound smart during conversations.

Example

The average distance between the Earth and the Sun is roughly $ d = 149,600,000 $ kilometers.

Let’s express this using scientific notation.

First, we focus on the significant part of the number $ 149,600,000 $, which can be rewritten as:

$$ 1.496 \times 100,000,000 $$

Now, express 100,000,000 as a power of 10:

$$ 100,000,000 = 10^8 $$

So, $ 149,600,000 $ becomes $ 1.496 \times 10^8 $.

Therefore, the distance between the Earth and the Sun in scientific notation is:

$$ d = 1.496 \times 10^8 \ \text{km} $$

And that’s it! Now you know everything you need to about scientific notation. Don’t say I never taught you anything useful!