Scientific notation examples

It always helps me to see a lot of examples of something so I figured it wouldn't hurt to do more scientific notation examples. So I'm just going to write a bunch of numbers and then write them in scientific notation. And hopefully this'll cover almost every case you'll ever see and then at the end of this video, we'll actually do some computation with them to just make sure that we can do computation with scientific notation. Let me just write down a bunch of numbers. 0.00852. That's my first number. My second number is 7012000000000. I'm just arbitrarily stopping the zeroes. The next number is 0.0000000 I'll just draw a couple more. If I keep saying 0, you might find that annoying. 500 The next number -- right here, there's a decimal right there. The next number I'm going to do is the number 723. The next number I'll do -- I'm having a lot of 7's here. Let's do 0.6. And then let's just do one more just for, just to make sure we've covered all of our bases. Let's say we do 823 and then let's throw some -- an arbitrary number of 0's there. So this first one, right here, what we do if we want to write in scientific notation, we want to figure out the largest exponent of 10 that fits into it. So we go to its first non-zero term, which is that right there. We count how many positions to the right of the decimal point we have including that term. So we have one, two, three. So it's going to be equal to this. So it's going to be equal to 8 -- that's that guy right there -- 0.52. So everything after that first term is going to be behind the decimal. So 0.52 times 10 to the number of terms we have. One, two, three. 10 to the minus 3. Another way to think of it: this is a little bit more. This is like 8 1/2 thousands, right? Each of these is thousands. We have 8 1/2 of them. Let's do this one. Let's see how many 0's we have. We have 3, 6, 9, 12. So we want to do -- again, we start with our largest term that we have. Our largest non-zero term. In this case, it's going to be the term all the way to the left. That's our 7. So it's going to be 7.012. It's going to be equal to 7.012 times 10 to the what? Well it's going to be times 10 to the 1 with this many 0's. So how many things? We had a 1 here. Then we had 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 0's. I want to be very clear. You're not just counting the 0's. You're counting everything after this first term right there. So it would be equivalent to a 1 followed by 12 0's. So it's times 10 to the twelfth. Just like that. Not too difficult. Let's do this one right here. So we go behind our decimal point. We find the first non-zero number. That's our 5. It's going to be equal to 5. There's nothing to the right of it, so it's 5.00 if we wanted to add some precision to it. But it's 5 times and then how many numbers to the right, or behind to the right of the decimal will do we have? We have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and we have to include this one, 14. 5 times 10 to the minus 14th power. Now this number, it might be a little overkill to write this in scientific notation, but it never hurts to get the practice. So what's the largest 10 that goes into this? Well, 100 will go into this. And you could figure out 100 or 10 squared by saying, "OK, this is our largest term." And then we have two 0's behind it because we can say 100 will go into 723. So this is going to be equal to 7.23 times, we could say times 100, but we want to stay in scientific notation, so I'll write times 10 squared. Now we have this character right here. What's our first non-zero term? It's that one right there, so it's going to be 6 times and then how many terms do we have to the right of the decimal? We have only one. So times 10 to the minus 1. That makes a lot of sense because that's essentially equal to 6 divided by 10 because 10 to the minus 1 is 1/10 which is 0.6. One more. Let me throw some commas here just to make this a little easier to look at. So let's take our largest value right there. We have our 8. This is going to be 8.23 -- we don't have to add the other stuff because everything else is a 0 -- times 10 to the -- we just count how many terms are after the 8. So we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 8.23 times 10 to the 10. I think you get the idea now. It's pretty straightforward. And more than just being able to calculate this, which is a good skill by itself, I want you to understand why this is the case. Hopefully that last video explained it. And if it doesn't, just multiply this out. Literally multiply 8.23 times 10 to the 10 and you will get this number. Maybe you could try it with something smaller than 10 to the 10. Maybe 10 to the fifth. And well, you'll get a different number but you'll end up with five digits after the 8. But anyway, let me do a couple more computation examples. Let's say we had the numbers -- let me just make something really small -- 0.0000064. Let me make a large number. Let's say I have that number and I want to multiply it. I want to multiply it by -- let's say I have a really large number -- 3 2 -- I'm just going to throw a bunch of 0's here. I don't know when I'm going to stop. Let's say I stop there. So this one, you can multiply out. But it's a little difficult. But let's put it into scientific notation. One, it'll be easier to represent these numbers and then hopefully you'll see that the multiplication actually gets simplified as well. So this top guy right here, how can we write him in scientific notation? It would be 6.4 times 10 to the what? 1, 2, 3, 4, 5, 6. I have to include the 6. So times 10 to the minus 6. And what can this one be written as? This one is going to be 3.2. And then you count how many digits are after the 3. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So 3.2 times 10 to the 11th. So if we multiply these two things, this is equivalent to 6 -- let me do it in a different color -- 6.4 times 10 to the minus 6 times 3.2 times 10 to the 11th. Which we saw in the last video is equivalent to 6.4 times 3.2. I'm just changing the order of our multiplication. Times 10 to the minus 6 times 10 to the 11th power. And now what will this be equal to? Well, to do this, I don't want to use a calculator. So let's just calculate it. So 6.4 times 3.2. Let's ignore the decimals for a second. We'll worry about that at the end. So 2 times 4 is 8, 2 times 6 is 12. Nowhere to carry the 1, so it's just 128. Put a 0 down there. 3 times 4 is 12, carry the 1. 3 times 6 is 18. You've got a 1 there, so it's 192. Right? Yeah. 192. You had them up and you get 8, 4, 1 plus 9 is 10. Carry the 1. You get 2. Now, we just have to count the numbers behind the decimal point. We have one number there, we have another number there. We have two numbers behind the decimal point, so you count 1, 2. So 6.4 times 3.2 is equal to 20.48 times 10 to the -- we have the same base here, so we can just add the exponents. So what's minus 6 plus 11? So that's 10 to the fifth power, right? Right. Minus 6 and 11. 10 to the fifth power. And so the next question, you might say, "I'm done. I've done the computation." And you have. And this is a valid answer. But the next question is is this in scientific notation? And if you wanted to be a real stickler about it, it's not in scientific notation because we have something here that could maybe be simplified a little bit. We could write this -- let me do it this way. Let me divide this by 10. So any number we can multiply and divide by 10. So we could rewrite it this way. We could write 1/10 on this side and then we can multiply times 10 on that side, right? That shouldn't change the number. You divide by 10 and multiply it by 10. That's just like multiplying by 1 or dividing by 1. So if you divide this side by 10, you get 2.048. You multiply that side by 10 and you get times 10 to the -- times 10 is just times 10 to the first. You can just add the exponents. Times 10 to the sixth. And now, if you're a stickler about it, this is good scientific notation right there. Now, I've done a lot of multiplication. Let's do some division. Let's divide this guy by that guy. So if we have 3.2 times 10 to the eleventh power divided by 6.4 times 10 to the minus six, what is this equal to? Well, this is equal to 3.2 over 6.4. We can just separate them out because it's associative. So, it's this times 10 to the 11th over 10 to the minus six, right? If you multiply these two things, you'll get that right there. So 3.2 over 6.4. This is just equal to 0.5, right? 32 is half of 64 or 3.2 is half of 6.4, so this is 0.5 right there. And what is this? This is 10 to the 11th over 10 to the minus 6. So when you have something in the denominator, you could write it this way. This is equivalent to 10 to the 11th over 10 to the minus 6. It's equal to 10 to the 11th times 10 to the minus 6 to the minus 1. Or this is equal to 10 to the 11th times 10 to the sixth. And what did I do just there? This is 1 over 10 to the minus 6. So 1 over something is just that something to the negative 1 power. And then I multiplied the exponents. You can think of it that way and so this would be equal to 10 to the 17th power. Or another way to think about it is if you have 1 -- you have the same bases, 10 in this case, and you're dividing them, you just take the 1 the numerator and you subtract the exponent in the denominator. So it's 11 minus minus 6, which is 11 plus 6, which is equal to 17. So this division problem ended up being equal to 0.5 times 10 to the 17th. Which is the correct answer, but if you wanted to be a stickler and put it into scientific notation, we want something maybe greater than 1 right here. So the way we can do that, let's multiply it by 10 on this side. And divide by 10 on this side or multiply by 1/10. Remember, we're not changing the number if you multiply by 10 and divide by 10. We're just doing it to different parts of the product. So this side is going to become 5 -- I'll do it in pink -- 10 times 0.5 is 5, times 10 to the 17th divided by 10. That's the same thing as 10 to the 17th times 10 to the minus 1, right? That's 10 to the minus 1. So it's equal to 10 to the 16th power. Which is the answer when you divide these two guys right there. So hopefully these examples have filled in all of the gaps or the uncertain scenarios dealing with scientific notation. If I haven't covered something, feel free to write a comment on this video or pop me an e-mail.