 Greetings and welcome to math help for science courses. In this video we are going to go through and talk about the uses of scientific notation and why it is important in science to be able to express the numbers using this notation. And a quick answer is primarily it is because the numbers that we look at are often very large or very small. And the traditional way of writing numbers is not convenient for expressing numbers in those type of formats. So let's go ahead and get started here. First of all as an introduction, what do we mean by scientific notation itself? So what is scientific notation? And what we call it is a way to more conveniently express very large and very small numbers. And in sciences we often deal with numbers that are extremely large and extremely small. It is all based on powers of 10. So essentially what we can do is get rid of lots of the zeros in those very big and very tiny numbers to express the number in something a little more reasonable. So what we do is we split the number into two parts that are multiplied, then multiplied together. There is the digits term and the exponential term. And let's look at an example of this. If we want to write the number 93 million, and we would write that in scientific notation, we would write that as 9.3 times 10 to the 7th power. So the digits term is 9.3, the exponential term is 10 to the 7th. So we can put them together. And we're going to look at some examples as how we go about doing this. But if you start looking at numbers, 93 million has all of those zeros there, and we can write that concisely as 9.3 times 10 to the 7th power. So let's look at how this works. First of all, to convert a number into scientific notation, what do we do? First step here is to move the decimal point so that there is only one nonzero number to the left. This will give you the digits term. So we're going to take that number and move to the left. So if we had a number of, start with an easy one, let's just say 345, the decimal point and that would be right here. We want to move that so that there is only one decimal point to the left. So we have to move it here and change 345 into 3.45. Now we have the digits term. Second step, that's not the number though because we've changed the number. We cannot just have 345 and 3.45 are quite different numbers. So the second step is to count how many places you moved the decimal point. This will give you the exponent. So in this case, we moved the decimal point two places. So that's going to tell us that it's times 10 to the second power. Now the only thing we're missing that we want to look at here is that you have to remember moving to the left as we did here will give you a positive value for the exponent. So yes, this is 10 to the second. If we had moved to the right, that would give a negative value for the exponent and that would have been a minus power. We do this as I said because most many numbers in science are far too large or small to be conveniently expressed in traditional notation. Certainly we can write them out but that doesn't mean that they are convenient and some of them would be very, very tedious to write out some numbers that we use in a traditional notation. So let's look at a couple of examples here. First, let's look at some big numbers. And in big numbers, we first have 150 billion meters. So let's write it out again, 150 and all of these zeros. So we have to write, if we want to write this number over and over again, we constantly have to write all of those zeros. How can we go ahead and convert that? Well remember the decimal point is here and we're going to move that decimal point until there is just one non-zero digit to the left of the decimal point. So we move it 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 spaces. So we had to move that 11 spaces to the left. Now if you recall, that means that the exponent will be positive. So we moved it to the left so the exponent is going to be positive. So our digits place is now just 1.5 and the exponent is going to be 11 so this is times 10 and remember we moved to the left so it is times 10 to the 11th power. So we have now converted 150 billion into scientific notation and you should see this is a much more concise way of being able to write this number and that would be meters. So let's look at one more example which is 300 million meters per second. So let's look at what this is. Again we do this exactly the same. Our decimal point would be right here at the end if it's not specified in the number and you count how many spaces you move. 1, 2, 3, 4, 5, 6, 7, and 8. So we moved 8 spaces and again we moved to the left. So that means we would write this as 3 times 10 to the 8th. Again in both cases we moved to the left. That means the exponent is going to be positive. Now those were a couple of examples with a big numbers. Let's look at a couple of examples with small numbers. So here are a couple of very small numbers that we could use and let's go ahead and do an example with these. Again a very small number of meters. In this case we can write that out as 0.0000000000000529. So now we see our decimal point. We don't have to imply that one is there. We can see our decimal point right here and now we're going to be moving it to the right. So we count how many spaces. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Again we stop when there is one non-zero number to the left of the decimal point. So our digit portion with this case would be 5.29 and times 10. Now we moved it 11 places in this case to the left. To this case we moved it to the right and that means our exponent is going to be negative. So this is 10 to the negative 11th power. So instead of writing out all of these zeros and a 5.29 then we can just write that as 5.29 times 10 to the negative 11th power. Now let's look at our other example here which has lots and lots of zeros with it. So we can write those out as 0.000000000000000000000000000500598. So lots and lots of zeros here that we have to use. So let's go ahead and change this, make it a little bit more distinct here and that is some number of grams. So that is some number of grams. Now let's go ahead and look at this. Let's count these decimal places. We've got a lot of ways to move this. So we have to go from here 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 times. So we moved that 24 places and again we moved it to the right. So this very long number that we had here that would be very tedious to write out can also be written as 5.98 times 10 to the negative 24th grams. Much more convenient to be able to write it out this way. So let's summarize a little bit about what we've gone through here and what we've learned. First of all we know that many numbers in science are really too large or small to be conveniently expressed traditionally. Traditional notation as you normally write out a number just does not work for these. We can divide them into two parts, a digit portion and an exponential portion and what we have to recall is that small numbers have negative exponents, large numbers will have positive exponents, and remember that depends on the direction that you moved the decimal point. If you're moving it to the left you're going to have a positive exponent, if you're moving it to the right you're going to have a negative exponent. So that concludes our discussion of scientific notation and we'll be back again next time to look at some more math help for science courses. So until then have a great day everyone and I will see you in class.