Writing Numbers in Scientific Notation | Math with Mr. J

[Music] Welcome to Math with Mr. J. In this video, I'm going to cover how to write numbers in scientific notation. We will start with large numbers, move to small numbers, and then end with some practice problems that you can try on your own. Now, remember, scientific notation is a way for us to write really large numbers or really small numbers in a shorter, more condensed way. This helps make really large and really small numbers more manageable and easier to work with. For example, we have this really large number right here. This number goes to the trillions. And this is actually the number of miles light travels in one year. This is what we call a lightyear, which a lightyear is again the distance light travels in one year. Lightyears are used to measure very far distances in space. Now, working with numbers like this and writing numbers like this can be difficult, tedious, and take up a lot of space with all of those digits, all of those zeros. So, we use scientific notation to condense them to write them in a simpler form. Here's that number in scientific notation. So, we can see how that number was condensed. it was shortened, but it's still equal to the number above. Now, when it comes to scientific notation, we're going to have a number that is greater than or equal to 1 but less than 10 and we're going to multiply that by a power of 10. Now, keep in mind these are equal 5.88 * 10 to the^ of 12. So the scientific notation is equal to this number. So how do we go from this to this? Well, let's jump into our examples. Starting with number one, where we have 6,500. We're going to start smaller than the example above and work our way up. Now, basically all we need to do here is move the decimal to the right of the first nonzero digit. So, anything 1 through 9 and we count how many places the decimal moved. That will give us everything we need to write the number in scientific notation. So, looking at this number and reading it from left to right, what's the first nonzero digit that we come across? Well, this six right here. So, we need the decimal to the right of that. So right here in between the six and the five. So let's move the decimal there. And the decimal comes after the ones place to the right of the ones place. Basically just put it after a whole number. So right here. And we need to move it once, twice, three times. So the decimal is going to go right here. So what we do? we write 6.5. Now we need to leave any zeros to the right off. We don't need those. We are condensing this number. And then we need to multiply this by a power of 10. Now we need an exponent. And our exponent is going to be how many places the decimal moved. And that was three. So we need an exponent of three here. And this is our scientific notation. 6.5 * 10 to the power of 3. This is equal to 6,500. We have a number greater than or equal to 1 but less than 10 times a power of 10. Now I do want to mention before moving on to number two that when we have scientific notation with a positive exponent we are working with large numbers. When we have a negative exponent we are working with really small numbers and we will get to those next. Let's move on to number two where we have 19 million. So we need to start by moving the decimal to the right of the first nonzero digit. So looking from left to right here, our first nonzero digit is this one. So we need to move the decimal right here in between the one and the nine. The decimal goes right here. And we need to move it once, twice, three times, four times, five times, six times, seven times. Now we can write this in scientific notation. So we write 1.9 and we leave all of the zeros to the right off times and now we need our power of 10. So how many places did the decimal move? Well, seven. So that is our exponent and this is our scientific notation for number two. 1.9 * 10 to the power of 7. Moving on to number three. Let's write this number in scientific notation. So we need to find that first nonzero digit which that's going to be this seven right here. And we need the decimal to the right of that. So in between the seven and the zero, the decimal goes right here. And now let's move it. So we have once, twice, three times, four, five, 6, 7, 8, 9, and 10. So we need to write this as 7.0 0 8 2 and then we can leave all of those zeros to the right off times our power of 10 which is going to be 10 to the power of 10 since we moved that decimal 10 times and this is our scientific notation 7.082 * 10 to the power of 10. Lastly, let's move on to number four. So, let's find that first nonzero digit, which is going to be this four right here. And we need to move the decimal to the right of that. The decimal goes right here after the ones place after the whole number. And now, let's move it. And since we have a lot of zeros here, one thing we can do is move the decimal in groups of three. We don't have to do it this way, but I at least wanted to show this. It can be helpful and quicker when working with numbers with a lot of zeros. For example, three, six, 9, 12, 13, and 14. So, we can write this as four. And then we leave all of those zeros to the right off. And we just have a four here, a whole number. We don't have any decimal digits to include. We just have zeros to the right. So this is fine. Times 10 to the power of 14. And this is our scientific notation. So there you have it. There's how to write large numbers in scientific notation. Let's move on to writing small numbers in scientific notation. Now let's take a look at writing small numbers in scientific notation. For example, we have this number right here. Let's say that this is the diameter of a cell. So 0.00000072 m. So we have a measurement of something that is very small here. So here is that number written in scientific notation. We have a number to start with right here 7.2 that is greater than or equal to 1 but less than 10 times a power of 10 and this is equal to the number above. So how do we go from this to this? Well, let's jump into our examples starting with number one. Now, this is going to be very similar to when we write large numbers in scientific notation. We move the decimal to the right of the first nonzero digit. So, anything 1 through 9 and we count how many places the decimal moved. That will give us what we need to write something in scientific notation. With small numbers though, we will move the decimal to the right and use negative exponents which we will talk about here shortly. So those two things do differ from when we have large numbers. So looking at our number here, we are reading it from left to right. What's the first nonzero digit that we come across? Well, it's going to be this one right here. So we need to move the decimal to the right of that one. So we need to move the decimal once, twice, three times, four times, five times, and six times. So we moved the decimal six places to the right. So what we do now, we write one, and we just have a whole number here. We don't have any decimal digits for this one. So we just have one. And all of those zeros to the left of one, we leave off. We don't need those. And then we multiply that by a power of 10. So 1 * 10. And now we need an exponent. So what power of 10 here? Our exponent is going to be how many places we moved the decimal. And since we are working with a small number here, this is going to be a negative exponent. Well, we moved the decimal six places. So our exponent here is going to be -6. So we have 1 * 10 to the power of -6. And that's our scientific notation. Now we're using a negative exponent here. We want to multiply by that negative exponent because this is a way to represent division or taking a fraction of something which gives us that smaller number. Remember a negative exponent just means we can turn it into a fraction with a numerator of 1 with a positive exponent in the denominator. So 10 to the power of -6. Let me write this up here. So 10 to the power of -6 equ= 1 over 10 ^ of positive 6 and 10 ^ of 6 equ= 1 million. So this is actually 1 over 1 million 1 millionth. So looking at our scientific notation, we have 1 * 10 ^ of -6, which this just means 1 * 1 / 10 ^ of 6 or 1 * 1 over 1 million, which all of this is the same as dividing by 1 million. Multiplying by 1 over 1 million is the same as dividing by 1 million. And that gives us our original decimal. So with small numbers, we use a negative exponent because that's going to give us those small numbers. Keep in mind this is equal to this. Let's move on to number two. And we need to start by finding the first nonzero digit. So looking from left to right, this three is the first nonzero digit. So we need to move the decimal to the right of that. So we need to move it once, twice, three times, four times, five times. So the decimal goes right here. So all we need to do now is write three. 8 times. And now we need our power of 10. So our exponent is going to be how many places we moved the decimal. That was five. And we're working with a small number here. So we need a negative. So this is 10 to the power of five. So for our scientific notation, we have 3.8 * 10 to the power of -5. Let's move on to number three. And we can start by looking at this number from left to right and finding that first nonzero digit. So what's the first nonzero digit that we come across? Well, this four right here. So we need to put the decimal to the right of that. Let's move the decimal. So once, twice, three times, four times, five times, six times, seven times, eight times, nine times. So the decimal is to the right of that first nonzero digit now. So we write out 4.0 2 5. And now we need to multiply that by a power of 10. So time 10 to the power of well how many places did we move the decimal? Nine. So we need a negative 9 here for our exponent. So 4.025 * 10 to the power of -9. And that's our scientific notation. Lastly, moving on to number four, we need to start by finding that first nonzero digit. So looking from left to right, that's going to be this nine right here. So, let's move the decimal to the right of that. We need to move the decimal once, twice, three times, four times, 5, 6, 7, 8, 9, 10, 11, 12, 13. So we have 9.7 * 10 to the power of -3. And that's our scientific notation. So there's how to write small numbers in scientific notation. And now that we've covered large and small numbers, let's try some practice problems. So here are our practice problems. Have a piece of paper and pencil ready and you can try these on your own. Feel free to pause the video and give these a try or let the video play and I'll give you 3 minutes. Whenever you're ready, click ahead to the answers and see how you did. So, write each of these in scientific notation. Go ahead and start. [Music] I'm a I'm hey. [Music] [Music] I'm a Hey, hey, hey. [Music] Hey, Okay, so let's take a look at the answers here and see how you did. So here are our answers. For number one, we have 1 * 10 to the^ of 7. For number two, we have 8.2 * 10 to the power of -5. For number three, 2.095 * 10 to the power of 11. For number four, 5 * 10 to the power of -8. For number five, 9.94 * 10 to the power of -9. And lastly for number six, 6.73 * 10 to the power of 14. So there you have it. There's how to write numbers in scientific notation. I hope that helped. Thanks so much for watching. Until next time, peace.