We've looked at some exponential functions. But, now we're going to take a short break from them to look at representing very small and very large numbers. Imagine you want to travel to the far, far regions of the known universe. Out past Mars, right past Pluto, out past where Voyager is sending back images, far beyond our own galaxy, past our nearest neighbor, Andromeda. On, and on, and on, traveling over. Well, traveling a very long distance, a distance so long that it's hard to express with words. We could write this out as 5, another 5, and then 22 zeros if you want to talk about it in miles. In kilometers, this would be 8, 8, and then 22 zeros, or perhaps you want to examine some bacteria. How big is a tiny organism like this. If we wanted to talk about how long the bacteria is, we could write it like this, 0.0000005 m. Apart from it being really boring to write out all those zeros, it's easy to make a mistake when we write out numbers like this, and it's also hard to read. We really need a better way to write quantities like this. And guess what, there is one. It's called scientific notation. And as you've already seen, it's very useful, especially when we're talking about very big and very small quantities. Let's get ready to talk about it.
Scientific notation is written in a very specific form. So people all over the world can compare numbers that are really large or really small. That form is a times 10 to the b. Note this x here denotes mulitiplication. It is not a variable. This formula requires that this first number here is a number between power of 10. So an example of something like this would be 5.6 times 10 to the need to be very careful about. And it's easy to make a mistake. For example, we can make one very small change. If instead, we have 56 times 10 to the 2, we are no longer writing in scientific notation. 56 is not between 1 and 10. Now I'm going to write down several numbers. And I'd like you to tell me, which of them are not written in the proper form for scientific notation? We have 5 times 10 to the 12th, 6.543 times 10 to the negative 4, 6 times 10 to the 6.5, but not least, 325 times 10 to the negative 7.
The numbers here that I've put boxes around are the ones that do not fit in the form of numbers in scientific notation. In the next question, we will look at why.
Here are all of the numbers that we selected in the last question. For each of these numbers, I would like you to check off the reason why it is not in scientific notation. Your choices are, it does not have a between 1 and 10, including 1, and not including 10, the base is not 10. And lastly, b is not an integer. Remember that general form for scientific notation, a times 10 to the b.
Let's go through these one by one to see what the problem with each of them is. Remember, we're going for this form right here. For 0.5 times 10 to the 3rd, the first number here, which is a, is not between 1 and 10, so that is our problem. The base is 10 and b is an integer. For 6 times 10 to the 6.5, b is not an integer since it's 6.5. In 4 times 2 to the 6th, the base of our exponential part of our expression is 2, not 10, so that is our problem. Next, we have 325 times 10 to the negative 7. Once again, the issue is with a. 325 is definitely greater than 10. And finally, 1.2 times 100 to the 4th. Again, we have a value of the base that is not 10, 100 is not equal to 10.
Next, we are going to look at how to enter the numbers we've been talking about into a calculator and convert from scientific notation. For example, 3 times 10 the 3rd to a notation we usually use 3000. Now, your calculator may differ a bit from the one that I'm going to use. And I would encourage you to use your own calculator, or choose Google calculator, like I'm going to. Let's think about the number, 5 times 10 to the 2nd. If you take this into a scientific calculator, and then pressed Enter, what number do you think the calculator would show you?
what my calculator would show. Let's check, here's my calculator, so let's enter this number. We have 5 times 10 to the second power. That looks right. Let's press e equals sign. Sure enough, we get 500.
Now I'd like you to take a turn using Google's scientific calculator, just like I've been using. There's a link straight to this in the instructor comments for this video. In that calculator I'd like you to enter each of these three numbers. When you press enter or the equal sign, what do you get? To be clear I'd like you to type in here exactly what the calculator tells you.
For the first one, we just get 31,000. For 3.1 times 10 to the 10, we get this much longer number. It's 3, 1, and then 9 0s. This is 31 billion, in fact. For this last one, 3.1 times 10 to the 21, we get a kind of funny answer though. When I type this in, and then press Enter, we get 3.1 I don't know what this means. Well this is actually just how the calculator shows, 3.1 times 10 to the just stands for exponent. And it means the exponent of 10 is 21. And if there's a positive sign here, that's just to distinguish this from negative 21. Some calculator show this differently. So check yours to see what it displays, when you enter the same number. Also as a challenge, see if you can guess how you'd write this number in the same format as these other two we already saw. How many digit's do you think it has?
To figure out what was happening in the last question, let's just look at one of those numbers as an example to explore more. How about that first one? 3.1 times 10 to the 4th. We can keep 3.1 as is, but we know that 10 to the 4th can be written differently. This is just 10,000. So, 3.1 times 10 to the 4th is equal to 3.1 times 10,000. What do we do now? Well, you may remember that to multiply a number by 10, you add a 0. But in reality, what you're actually doing is moving the decimal point of the number one place to the right when you multiply by 10. And in some cases, this requires the addition of a 0. For example, if we look at the number 31 and we want to multiply it by 10, we need to note that the decimal point is invisible, but it's technically right here. We need to move it from being right after the 1, the one place to the right. And since there's nothing here right now, we need to add a 0 in, then we put a decimal point after that 0. In this case though, we're dealing with 10 to the this number is going to get ten times bigger, so the decimal needs to move one step to the right in each of those factors of 10. The first time we move it, we end up with 31. The next time we move it, the decimal point would be out here, and here we have one of those invisible spaces. We know we're going to need to put a 0 there, but let's not derail ourselves at this point. We've already moved over two times, so that takes care of two of our four factors of 10. We need two more. There's our third and there is our fourth. So our decimal point started out right here. And in the end, it's all the way out here to the right. We went from having 3.1, to having 31, to having 310, to 3,100, and finally, to in. We know it's generally invisible though, for numbers that don't have decimals written outside of them. To give you a chance to try out what we've been talking about, see if you can write out that number that we had at the end of the last question. 3.1 times 10 to the 21. How would you write this using our normal number notation, not scientific?
We know that we need to multiply 3.1 by 10, 21 times. So that means we need to move this decimal point to the right 21 times. Let's do it. Okay. I'm going to start with our number we have originally, 3.1, and let's get ready to move. Okay, count with me. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, and twenty one. well that was a lot of work. The exponent is these empty ones needs a 0 in it. So let me write those in. There we go, this is our final number. We've moved the decimal point all the way out here. Turns out 3.1 times 10 to the 21. Is a pretty big number.
In all the examples we've looked at so far, the exponent has been positive. Let's see what happens if it's negative instead. Now we have 3.1 times 10 to the negative 4. What is this equal to? You can use your calculator to figure this out.
Let's use our calculator to plug in this number and see what happens. We have hit the equal sign. We get 0.00031. Interesting.
So, why do we get such a small number for this calculation? Well, if you think way back to when we did exponent notation at the beginning of the course, we talked about negative exponents and what they are. x to the negative 1 means 1 divided by x. x to the negative 2 means 1 over x squared. So if we multiply something by x to the negative 1, we really want to divide that by x. And if we multiply something by x to the negative 2, we really want to divide it by x squared. So if we multiply something by 10 to the negative 4, what we really want to do is divide it by 10 to the 4th. With multiplication by 10 to the positive 4, the answer we got was bigger than the number we started with. But, when we divide by 10th to the 4, it gets smaller. For each factor of 10 we're dividing by, the decimal point moves one step to the left. So, we start with up with these blank spots, we need to fill in zeros. Our decimal point is right here now. So what we actually have is, as we saw before, 0.00031. Now, note that I put a 0 in front of our decimal point, to the left of it. Now this is the technically correct way to write decimal numbers in general. If you miss out on writing this, it's kind of like using slang for everyday speech. People will still know what you mean but its not quite as correct. Its not quite as proper in mathematics. So now that we've seen one example of using a negative exponent in scientific notation, I'm going to give you another one to try out. Here is 3.1 times 10 to the negative 7. This time try without using a calculator to write this in normal number notation.
To write out 3.1 times 10 to the negative 7th, we need to divide 3.1 by 10, seven times. So that means we need to move the decimal point seven times to the left. Let's do it. One, two, three, four, five, six, seven. So this is where my decimal point belongs now. Let's fill in our zeros. We have one to the left of the decimal point, and then we have six more zeros before we get to the 3. So our answer is 0.00000031. That's a pretty tiny number.
Here are some more numbers written in scientific notation. Thinking about what you know about exponents, try to figure out which of these quantities matches each of these distances. You might want to think about the order of the sizes in each list to figure this out. Here are the numbers that are your options. to the 1st meters, 8.8 times 10 to the 5th meters, and 2 times 10 to the negative 4th meters. Each of these matches one of these distances and here are those distances. The distance from the earth to the sun, the height of a human, the distance from San Francisco, California to Paris, France, the distance to the Milky Way's nearest neighboring galaxy Andromeda, and the size of an ant. Now, just in case you didn't know, a meter is a little bit shorter than a yard. It's about the length from your nose to your fingertip if you hold your arm out sideways.
Here are our solutions. 1.5 times 10 to the 11th meters is the distance from the earth to the sun. 2.4 times 10 to the 22nd meters is the distance to the Milky Way's nearest neighboring galaxy, Andromeda. 1.7 times 10 to the 1st meters is the height of a human. 8.8 times 10 to the 5th meters is the distance from San Francisco to Paris. And 2 times 10 to the negative 4 meters is the size of an ant.
What is the world population written in scientific notation? Use Google to find the number. Round your answers so that this first box contains an integer between one and nine, including one and nine.
There are seven billion people in the world, about. So that's 7 times 10 to the
Now I'd like you to add two numbers in scientific notation. What is three times times 10 to the 8th, 36 times 10 to the 4th, 306 times 10 to the 4th, 9 times sure that the answer you pick is also in scientific notation.
adding these together, we would get 3,060,000. Written in scientific notation, that is, 3.06 times 10 to the 6th. So what can you learn from this? Well you can't add things that have different powers of ten. That's like saying $400 plus $30 equals $700. That would be really great if I was being paid that. But it would also be pretty horrible if you were shopping and all of a sudden everything cost a ton of money. Luckily, this is not actually an equality. We need to be careful about when we are adding numbers that are in scientific notation.
We have 3 times 10 to the 3rd small ants. And each one is of the length 2 times Here's a hint. Think about multiplying. You don't need to give your answer in scientific notation. Just write in a normal number here.
One way that it's helpful to approach word problems is to start off by thinking about what information we already have. In this problem, we know that the number of ants is equal to 3 times 10 to the 3rd, and the length of one of those ants is 2 times 10 to the negative 4 meters. Once we know what information we have to work with, we can think about what we want to find. In this case, that's the length of the entire line of ants. All these ants lined up in one line. To get from this information to this information, we need to multiple these two numbers together. So, we need to calculate 2 times 10 to the negative 4th, times 3 times 10 to the 3rd. Now the way we would normally write those numbers, would be 0.0002 and 3,000. So two 10,000ths times 3,000 is equal to 0.6. So now we can conclude, that the whole line of ants is 0.6 meters long.
For ant questions, it was fine to convert our numbers from scientific notation to being normal decimal numbers, and then multiply them together. But what if the exponents were much bigger? We'd probably get lost in all the zeroes we'd have to write, or at least I would. I think we need a better strategy. When we're multiplying two numbers in scientific notation together, we can use something we learned about very early on in this course. It's the commutative property of multiplication. Let's see how this works. Let's say that we want to multiply 2 times 10 to the negative 4th, by 3 times 10 to the 3rd again. Well, the commutative property of multiplication says that we can rearrange the order of any of these things we're multiplying together. What we actually have over here, even though we have some grouping by parentheses right now, Is just four numbers multiplied. Either ignore the parentheses and just say 2 times 3 and then multiply that by the 10s but they're exponents. We can rearrange this so that instead we multiply the 2 and the 3 together. And then we multiply the two factors with the bases of 10. Now what is this equal to in scientific notation? Think back to the start of the course again, when we talked about how to combine exponents, when the base is the same.
power. Now I know that a negative 1 belongs here because we multiply terms that have the same bases, we just need to add the exponents. Negative 4 plus 3 is negative 1. This gives us that same answer of 0.6 that we had before.
Now, here's another problem for you to try using the same method. Multiply 1.2 times 10 to the 8th, by 4 times 10 to the 7th. Fill in the numbers you need here, for scientific notation.
There's one important thing we need to note here. Using this method, your answer is not always going to end up in scientific notation, since sometimes, depending what two numbers we have multiplied here, this number won't end up being between 1 and 10. So, make sure that if you decide to use this method, you look at the final answer and double check that it is in fact in scientific notation. If it's not, adjust it properly.
One thing that has fascinated me for literally my entire life, is stars, and how far away from us they are. Did you know that when we look at a star, like the sun for example, we're looking at something from the past because of the time it takes the light to reach us. That's pretty weird and pretty awesome, I think. So let's work out how long it takes the light to reach us from the sun. As you saw before, the distance from the earth to the sun is approximately 1.5 times 10 to the 11th meters. Light travels at a speed of 3 times 10 to the 8th meters per second. As we can see from the units of speed here, meters per second, speed is equal to some distance, divided by the time it takes to go that distance. Now thinking about this equation, what do we need to do to calculate time? Do we need to multiply speed by distance, divide distance by speed or divide speed by distance?
We need to divide distance by speed to find time. Let's look at this equation to figure out why. Time is in the denominator, so let's multiply both sides by time. Then we have that speed times time equals distance. Now we just need to divide both sides by speed. And we get time equals distance over speed.
The distance from the earth to the sun is approximately 1.5 times 10 to the eleventh meters. And light travels at about 3 times 10 to the eight meters per second. We just found that time is equal to distance over speed. So can you tell me how many seconds it takes for light to travel from the sun to the earth? You can use your calculator or you can try to think of another way to do this.
To find the number of seconds this is going to take, we need to divide 1.5 times 10 to the 11th by 3 times 10 to the 8th. Let's use our calculator to do this. We have 1.5 times 10 to the 11th. You want to divide that by 3 times 10 to the 8th. And look at that. We get 500. 500 seconds is just over 8 minutes. So when we look at the sun, we're seeing 8 minutes into the past. That's some crazy stuff.
Now, again, there is a way we can do this without a calculator. We can just divide 1.5 by 3 and then multiply that by 10 to the 11th over 10 to the 8th. If you do that, what numbers do you end up with? Think about how you divide these terms when the base is the same.
divide terms that have the same base, you just subtract the exponents. This number is actually not in scientific notation, but we're not going to worry too much about that.
Now I'd like you to divide, 8 times 10 to the 20 by 2 times 10 to the 8th.
Let's rewrite this just like we saw before. We can write it instead as 8 over 2 times 10 to the 20 times 10 to the 8. This gives us 4, and here, we have 10 to the 20 minus 8 or 10 to the 12th. So scientific notation gives us a great way to deal with very small and very large numbers. Numbers expressing the vastness of the universe, numbers expressing tiny cells in the body. There're are even smaller quantities around us, the building blocks of everything. A scientific notation is also very useful for describing.