Episode 96: March 1, 2012
by Jason Marshall
As we’ve talked about before, by the end of elementary school most of us are pretty comfortable with reading and writing numbers—as long as those numbers are written in the standard form that we were taught…like 10, -200.5, 30,250, and so on. But as we learned when we first talked about scientific notation, this standard way of writing numbers isn’t the only way to do it. And that’s actually a very good thing because sometimes the standard way is a really terrible way! We talked about just such a case before when we looked at how scientific notation makes it easier to read and write really large numbers. And today we’re going to see another example of this as we talk about the easiest way to read and write really (really) small numbers.
Review: What Is Scientific Notation?
Before we start worrying about how to read and write all those tiny numbers, let’s remind ourselves about how scientific notation works. The idea behind scientific notation is to write a number as some decimal number times a multiple of ten. What exactly does that mean? Well, as we’ve learned in our previous articles about working with exponents, 10^0 = 1, 10^1 = 10, 10^2 = 100, 10^3 = 1,000, and so on. The exponents here tell you how many copies of the number 10 to multiply together. For example, we find that 10^3 = 1,000 by multiplying 10x10x10.
But there’s another way to look at the exponent in a number like 10^3 that’s very useful when it comes to scientific notation. And that way is to think of the 3 in 10^3 as the number of times you need to move the decimal point in the number 1.0 to the right. So 10^2 is the number we get when we move the decimal point in 1.0 two positions to the right to get 100, 10^3 is the number we get when we move the decimal point in 1.0 three positions to the right to get 1,000, and so on. So a large number written in scientific notation such as 3x10^8 is just 3x100,000,000 = 300,000,000. As this example clearly shows, one of the main advantages of using scientific notation when writing large numbers is that you don’t have to worry about reading, writing, and trying to understand the meaning of all those zeros!
What Are Really Tiny Numbers?
So that’s how scientific notation works for really large numbers, but what about today’s main topic: really small numbers? Well, the first thing we should do is clarify what we mean by “really small.” In terms of the number line, since “big numbers” are all the numbers extending way out towards infinity in the positive direction of that line, you might be inclined to think that “small numbers” are all the numbers extending way out to infinity in the negative direction. And while those numbers are certainly small in that sense, it’s not the sense in which in which I mean “small” today. Instead, when I say small number I’m talking about a number with a small absolute value—that is, a number that’s much smaller in magnitude than the number 1. For example, in this sense the diameter of a hydrogen atom—which is about 0.000000005 cm—is a really small number.
What Are Negative Powers of 10?
Do you notice anything problematic about this really small number? Perhaps the fact that it has a rather unwieldy number of 0s in to keep track of? Indeed, just as in the case of really big numbers, it’s pretty cumbersome to write really small numbers using the standard notation we learned in school since we have to write all those 0s. So what’s the solution? Scientific notation.
In order to write really large numbers with scientific notation, we needed to know what 10^2, 10^3, and so on mean. Now, to extend our use of scientific notation into the realm of tiny numbers, we also need to know what negative powers of 10 mean. In other words, we’re going to need to use the fact that 10^–1 = 0.1, 10^–2 = 0.01, 10^–3 = 0.001, and so on. But how do we know this? Well, as we learned before, a negative exponent tells you how many copies of the base you need to divide by. So 10^–2 means to divide 1 by 10x10. In other words, 10^–2 is equal to the number 1/100 = 0.01.
How to Use Scientific Notation to Write Small Numbers
Writing small numbers using scientific notation is just like writing big numbers except the exponents are negative.
Just as we found that when working with large numbers in scientific notation it’s very useful to think of the exponent 3 in 10^3 as the number of positions you have to move the decimal point in 1.0 to the right, when working with small numbers in scientific notation it’s very useful to think of a negative exponent such as the –3 in the number 10^–3 as telling you the number of positions you need to move the decimal point in 1.0 to the left. So in the problem 10^–3, the exponent –3 tells us to move the decimal point in 1.0 three positions to the left…giving us 0.001.
Once you understand this, you also understand how to write small numbers with scientific notation. Take for example the diameter of a hydrogen atom that we talked about before: 0.000000005 cm. Since this number is just equal to 5 x 0.000000001 cm, we can write it much more simply without all those zeros using scientific notation as 5x10^–9 cm. Which, once you know how to read it, really is a lot simpler and easier to understand…don’t you think? To sum this all up, just remember that writing very small numbers using scientific notation is just like writing very big numbers except the exponents are negative.
Okay, that’s all the math we have time for today…except, of course, for a few practice problems. Can you figure out how to write these numbers in scientific notation?
0.005 = _____ x 10^_____
0.00072 = _____ x 10^_____
0.0000000823 = _____ x 10^_____
You can find the answers I come up with in my post this week on The Quick and Dirty blog.
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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!
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