Today we're going to talk about scientific notation. Scientific notation is a shorter way of writing numbers that have a lot of digits, numbers that are really big or numbers that are very small. So an example of a number that's very large is the world population is approximately 7 billion. In scientific notation, I could write it as 7.0 times 10 to the 9th power. This is a lot quicker than this. Another example of a number that is very small is this diameter of a red blood cell in meters is 0.000007. In scientific notation, it's just 7.0 times 10 to the negative 6.
So we'll start by looking at the idea behind scientific notation. We'll look at the rules for writing a number in scientific notation. And then we'll do some examples. So we can see from our first two examples that scientific notation uses a decimal multiplied by a power of 10. And that power of 10 or, 10 to some exponent, can either be a positive exponent or have a negative exponent.
So before we get into writing numbers in scientific notation, let's quick review what those powers of 10 look like. So 10 to the positive 1 exponent is just going to be equal to 10. 10 to the positive 2, or 10 times 10, is going to be equal to 100. And 10 to the 3rd is going to be equal to 1,000. So an important having to see here is that our exponent is the same as the number of digits after our 1.
OK, so let's look and see what happens when we have a negative exponent. 10 to the negative 1 we know could be written as a fraction as 1 over 10. As a decimal, that's going to be 0.1. 10 to the negative 2 would be 1 over 10 to the positive 2 or 1 over 100, which is going to give me 0.01. And 10 to the negative 3 will be 0.001. So the important pattern to see here is that the number of digits after our decimal is the same as our exponent.
So before we do our examples, let's look at a couple of rules for writing a number in scientific notation. So here's an example of a number in scientific notation. The first rule is that you can only have one non-zero digit before the decimal. So if you have a number that's a 0 or if you have more than one digit in front of the decimal, you have to rewrite it by changing the exponent so that you only have one non-zero digit in front of the decimal. And I'll show you how to do that in an example.
The second rule is that you can have any number of digits after the decimal. So the more digits that you have after the decimal, the more accurate the number will be. For example, 3.51 times 10 to the 3rd is a better approximation or more accurate number that just 3.5 times 10 to the 3rd.
[INAUDIBLE] first example, I've got the number 36,500,000. And I want to convert that into scientific notation. So I know that following my rule, I can only have one number in front of my decimal. So I want my decimal point to be right here between the 3 and the 6.
So my decimal is going to start at the end. And I want to move it to between the 3 and the 6. And I need to count how many times I'm moving it, because that's going to tell me what my exponent is.
So starting at the end, I'm going to move it-- 1, 2, 3, 4, 5, 6-- 7. So I can rewrite this number as 3.65 times 10 to the 7th power. So the thing that you want to make sure that you get into your notes with this example is that when we move our decimal to the left, we are increasing our exponent.
[INAUDIBLE] example two, I've got 0.0004362. So this is a pretty small number. So we know we're going to have a negative exponent. And we also know that using our rules of scientific notation, I want my decimal to be right in between the 4 and the 3 so I only have one digit in front of my decimal.
So I'm going to move my decimal and count how many times it's moved-- 1, 2, 3, 4. So I can rewrite this in scientific notation as 4.36. Remember, I can have any number of digits after my decimal. And my exponent is going to be a negative 4.
So here we moved our decimal to the right. So the important thing to get into your notes along with this example is moving your decimal to the right will decrease your exponent. It decreases your exponent.
So for this example, I've got a number that is not quite in scientific notation, because I know that my decimal needs to come where there's only one digit to the left of the decimal. So I'm going to move my decimal point over to the left so that it becomes right after the 1. So now my decimal is 1.274.
And because I moved my decimal to the left, I know that my exponent is going to increase. Since I moved the decimal twice, I'm going to increase my exploited by 2. So this will become times 10 to the 5th.
All right, so for this example, I'm starting with a number in scientific notation, 5.6634 times 10 to the 6th. And I want to write it in standard form. So I can see here that my exponent is positive. So I know that that's going to give me a very big number. But it will be helpful for you to get into your notes that a positive exponent-- we need to move the decimal to the right.
All right, so let's see what that will look like. My exponent tells me how many times I need to move my decimal-- so 1, 2, 3, 4. And I still need to move it two more times, but there aren't any other digits. So I'm going to move it twice. And I'm going to need to use two zeroes here as placeholders, because I don't have any other digits. So this is going to become 5663400, which is 5,653,400 written in standard form.
So for the last example, we'll look at one other number that's written in scientific notation and convert that to standard form. So here I've got a negative exponent, which I know means it's going to be a very small number. But again, let's write in our notes that a negative exponent means that you move your decimal to the left.
All right, so let's see what that looks like. My decimal is starting after the 1. I'm going to move into the left 3 times-- 1, 2, 3. And it's going to end up here. Again, I ran out of digits. But I can use zeros as placeholders in these two empty spots. So this is going to become 0.0012 written in standard form.
So let's go over our key points for today. The first is that when you're writing in scientific notation, you can only have one non-zero digit to the left of the decimal, but you can have any number of digits to the right. The second thing we talked about is that when you're writing in scientific notation, moving the decimal to the left is going to increase your exponent and moving the decimal to the right is going to decrease your exponent.
And lastly, we talked about that when you're writing in standard form, a positive exponent indicates moving the decimal to the right and a negative exponent indicates moving the decimal to the left. So keep on using these notes and practicing your examples, and soon you'll be a pro writing scientific notation. Thanks for watching.
A way to express numbers as the product of a decimal number and a power of 10.