Source: All images created by Anthony Varela
Hi. Today, I'd like to talk about writing numbers in scientific notation. So we're going to start by talking about what scientific notation is, and then we're going to practice writing numbers in scientific notation, and then we'll practice writing numbers from scientific notation. So the good thing about what scientific notation is. I want you to imagine the sun and the Earth.
Now, there is a huge distance between these two objects in outer space, and on average, this distance is about 150 billion meters. That's a long distance. And scientific notation allows us to express this as simply 1.5 times 10 to the 11th meters, and we'll talk more about how we can do that in a bit.
Similarly, I want you to imagine a red blood cell running through your veins. This is a very small object, and if we were to measure the length of a red blood cell, this might be expressed as six millionths of a meter, and in scientific notation, we can write this as 6.0 times 10 to the negative 6 meters. So scientific notation, what I know about it so far is that it's a way to express numbers as the product of a decimal number and a power of 10, and you also notice that you're using fewer digits.
So it's a simpler way to write a numbers that would ordinarily have lots of digits. And in general, we can write this then as a times 10 to the power of b. So we have the product of a decimal number and a power of 10. So let's talk then about these powers of 10.
So looking at 10 to the first power, I know that this is just using the number 10 in a chain of multiplication one time, so this is 10. I know that 10 of the second power is 10 times 10, which is 100. Well, there's a pattern here-- a shortcut to remember how to write 10 to the third and that I'm going to write the number one and three zeros after that. So 10 to the fourth power would be the number one with four zeros after that.
How can I remember the negative powers of 10? So 10 to the negative first. I know that that is 1 over 10, so that's 110. 10 to the negative 2 would be 1 over 100 or one 100th. And a shortcut to remembering how to write these negative powers of 10, so 10 to the negative third one. What I'm going to do is write 0. and then I'm going to write a total of three digits, all of them are zero, but the last one is one, so 0.001.
Three digits, all of them are zero, but the last one is one. So what would 10 to the negative fourth look like? Well, that would be 0. and then four digits following that. All of them are zero, except the last one is 1. So 0.0001. Now, let's talk about then positive powers of 10 would correspond to large numbers and negative powers of 10 would correspond to small numbers.
So we talked about the powers of 10. Now, let's talk about there are rules about this decimal component to numbers in scientific notation. That decimal number can only have a single non-zero digit to the left of the decimal. So here, we see one digit that's not a 0 to the left of the decimal. Here, again, we see one digit to the left of the decimal, and it's not a 0.
There can be as many digits as you want to the right of the decimal, however. And the more digits you include, the more accurate your number is. So for example, I can express this distance between the sun and the Earth as 1.5 times 10 to the 11th meters. I can also write this more accurately as 1.4960 times 10 to the 11th. That would be a more accurate description of the distance.
So only one single non-zero digit to the left, but as many as you want to the right of that decimal. So let's take a look at these two numbers here, and we notice that they're not written in scientific notation. Why not? Well, we see more than one digit to the left of the decimal here. We have an invisible decimal right here, and here, we have a zero that's to the left of the decimal, and that isn't allowed in proper scientific notation.
So we need to rewrite these numbers that they do fit proper scientific notation. So what I need to do is shift my decimals, and then adjust my exponent. So to get only one non-zero digit to the left of my decimal, I have to move the decimal place over one space to the left, and I have to adjust my exponent. So no longer to the 10th power up here but to the 11th power.
So notice that moving my decimal one place to the left means increasing my exponent by 1. Let's take a look at this number over here. I need to move my decimal so that that single digit to the left is not a zero. So I can move my decimal over to the right. So I moved it over one place to the right, and moving my decimal over to the right means that I have to adjust my exponent, and I'm going down one from negative 5 to negative 6.
So moving the decimal one place to the right means decreasing your exponent by 1. So we're going to write that down. Important to remember a shift to the left is an increase in your power of 10. A decimal shift to the right is a decrease in your power of 10.
So now, let's practice writing a number that's written in standard form in scientific notation. So what I like to do as a starting place is write in a power of 10 that is 10 to the zero power. That just equals 1, so I'm not changing anything by multiplying by that power. And what I need to do then is count how many places my decimal is going to move to fit my rules for what that decimal component can look like.
So I'm going to count three, six, seven places so that I have a single non-zero digit to the left of my decimal. So I have moved my decimal over seven places, and that means I need to do what? I need to increase my power of 10 by seven. So I have 10 to the 7th there. I don't need to write all of those zeros, so this is simply 1.78 times 10 to the 7th.
Let's try this one. Here, I have 0.000042, and I'd like to write this in scientific notation. So I have to move my decimal place, but first, I want to put it in my power of 10, which is going to be zero. So I need to move my decimal place, and let's see. I need to move it over five places so that I have a single non-zero digit to the left of that decimal.
So 4.2, and then since I have moved my decimal five places to the right, I have to decrease my power of 10 by five, and I don't need to write all those zeros. So 4.2 times 10 to the negative 5th is proper scientific notation. Now, remember, we cannot have more than one digit to the left of the decimal or that digit cannot be 0.
So this is not scientific notation, and just as a reminder, we have to shift our decimal and adjust our exponent in this case. So I have moved my decimal one place to the left, so I have to increase my exponent from negative 5 to negative 4. That's an increase of 1.
All right, now, let's go the other way. Let's take a number already written in proper scientific notation and turn it into a standard number. So here, I have 1.45 times 10 to the negative 4th. And once again, I like to write my power of 10, because that's really my goal is to get a power of 10 that has tended to 0, and at the very end, I don't even need to write that.
So what I notice is to go from negative 4 to 0, I have to increase my power of 10. So increase means moving my decimal to the left, and I've increased it by 4, so I have to move my decimal over four places. So started right here, moved over one, two, three, four, there it is. So as a standard number, this is 0.000145.
Let's do another one. Here, we have 2.776 times 10 to the 6th power. I'd like to write this as a standard number, so first, I'm going to put my power of 10, which is 10 to the 0, and I notice to get from 6 to 0, I have to decrease that power. So decrease means moving my decimal to the right, and because I went down six, I have to move six places.
So taking my decimal and moving it over one, two, three, four, five, six. There we go, and I don't need to write 10 to the 0, so this number as a standard number is 2,766,000. So let's review our notes. Scientific notation is a way to express numbers as the product of a decimal number and a power of 10.
So we can write this very generally as a times 10 to the b. Positive powers of 10 are large numbers and negative powers of 10 are small numbers. Remember that the decimal component can only have one non-zero digit to the left, so that means if you ever move your decimal to the left, you increase your power of 10, and if you move your decimal to the right, you decrease your power of 10. Thanks for watching.