Scientific Notation | Writing Large Numbers in Scientific Notation | Math with Mr. J

Welcome to Math with Mr. J. In this video, I'm going to cover how to write large numbers in scientific notation. 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 light year, which a light year is, again, the distance light travels in one year. Light years 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 one, 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 times 10 to the power 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 non-zero digit. So anything one through nine. 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 non-zero 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 times 10 to the power of three. This is equal to 6,500. We have a number greater than or equal to one, 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 non-zero digit. So looking from left to right here, our first non-zero 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 times 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 non-zero 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, six, seven, eight, nine, and 10. So we need to write this as 7.082. 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 times 10 to the power of 10. Lastly, let's move on to number four. So let's find that first non-zero 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, 3, 6, 9, 12, 13, and 14. So we can write this as 4. And then we leave all of those zeros to the right off. And we just have a 4 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. Next, we'll take a look at writing small numbers in scientific notation. I hope that helped. Thanks so much for watching. Until next time. Peace.