Learn how scientific notation makes it easier to work with large numbers.
By the end of elementary school, most people are comfortable with reading and writing numbers. Well, sort of. In truth, most people are really only comfortable with reading and writing numbers in the standard format that we learned in school. That is, numbers written like 7, -20.2, 355,000, and so on. But that isn’t the only way to write numbers! In fact, I’m willing to bet that you’ve seen (and perhaps been confused by) another way of writing numbers—in particular large numbers—called “scientific notation.”
How does scientific notation work? And why is it useful? Keep on reading to find out.
Review: What Are Exponents?
Before we get into the details of reading and writing scientific notation, we first need to review the idea of an exponent. As we talked about in What Are Exponents? one way to think about an exponent is as a number that tells you how many copies of a base number to multiply together. For example, 2^4 means the same thing as 2 x 2 x 2 x 2. In other words, 2^4 is the number you get when you multiply 4 copies (that’s the exponent) of 2 (that’s the base number) together.
Of course, you can use other bases besides 2 when writing numbers with exponents. In fact, you can use any number you want for the base. Of particular interest to us right now—since we’re interested in understanding scientific notation—is what happens when we use the number 10 as the base.
How to Write Powers of Ten Using Exponents
So what happens? Well, let’s start by looking at what we get when we raise 10 to various positive integer powers. Since 10^1 tells us to multiply one copy of 10, we see that 10^1 = 10. And since 10^2 (aka, “10 squared”) tells us to multiply two copies of 10 together, we see that 10^2 = 10 x 10 = 100. Continuing this pattern, we find that:
10^3 = 10 x 10 x 10 = 10 x 100 = 1,000
10^4 = 10 x 10 x 10 x 10 = 10 x 1,000 = 10,000
10^5 = 10 x 10 x 10 x 10 x 10 = 10 x 10,000 = 100,000
10^6 = 10 x 10 x 10 x 10 x 10 x 10 = 10 x 100,000 = 1,000,000
and on and on in an ever-growing sequence of larger and larger powers of ten.
But why is this sequence useful? Because it’s the key to reading and writing numbers in scientific notation. And we’re now ready to talk about exactly how that works.
What Is Scientific Notation?
The idea behind scientific notation is to write all numbers as decimal numbers times multiples of ten. To see what this means, notice that any number that’s greater than or equal to 1 can be written as:
some number of 1s (meaning 1, 2, 3, 4, 5, and so on up through 9), plus…
some number of 10s (meaning 10, 20, 30, 40, 50, and so on up through 90), plus…
some number of 100s (that is 100, 200, and so on through 900), plus…
some number of 1,000s, plus…
some number of 10,000s,
and on and on up to some multiple of as large of a power of ten as needed. For example, a number like 4,250,000 can be thought of as:
4,000,000 + 200,000 + 50,000 = (4 x 1,000,000) + (2 x 100,000) + (5 x 10,000)
And, as we found earlier, all of these powers of 10 can also be written using exponents. Which means we can save ourselves the trouble of writing all those zeros and instead write a number like 4,250,000 as
4x10^6 + 2x10^5 + 5x10^4
Believe it or not, if you understand everything up to this point, you now understand scientific notation—because this expression is written using scientific notation!
How to Write Large Numbers Using Scientific Notation?
But we’re not quite done yet because this way of writing numbers is still kind of cumbersome and definitely more than a little long-winded. Which is not a good thing since one of the main reasons to use scientific notation is to make working with large numbers easier and more efficient.
What if instead of writing 4,250,000 as 4x10^6 + 2x10^5 + 5x10^4, we write it as 4.25x10^6? Do you see what we’ve done here? We’ve taken the largest power of 10 in the sum (in this case 10^6) and multiplied it by a decimal number (in this case 4.25) that gives us back the entire number. In other words: 4,250,000 is equal to 4.25-million. Pretty clever, right?
Certainly this makes things more concise, but is it really an improvement over just writing 4,250,000? It’s true—the benefits are borderline when working with numbers of this size. But what about when we’re working with really big numbers? For example, it’s much easier to express the fact that the Sun puts out as much energy as 3.846x10^24 100-Watt lightbulbs rather than writing the number out in full:
Don’t you agree? And the benefits go beyond just making it easier to read and write large numbers. As it turns out, scientific notation is useful when working with really small numbers too. And it makes doing many calculations easier. All of which are things that we’ll be talking about more in future articles.
Number of the Week
Before we finish up for today, it’s time for this week’s featured number from my post on QDT’s blog The Quick and Dirty. This week’s number—1.608x10^18 km—is a big number, which makes sense since we’ve been talking about writing big numbers using scientific notation. So what is it? It’s the distance the Solar System travels as it completes one lap around the Milky Way galaxy. Just for comparison, that’s nearly 2x10^9 (which is 2 billion) times further than the Earth travels around the Sun in a year. How long does it take the Solar System to travel around the galaxy? Well, moving at about 220 km/s (which is really fast), it takes a total of about 225 million years! So enjoy the ride!
Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at firstname.lastname@example.org.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!