Topic B: Magnitude and scientific notation
There are two whole Khan Academy videos on what scientific notation is, why we even worry about it. And it also goes through a few examples. And what I want to do in this video is just use a ck12.org Algebra I book to do some more scientific notation examples. So let's take some things that are written in scientific notation. Just as a reminder, scientific notation is useful because it allows us to write really large, or really small numbers, in ways that are easy for our brains to, one, write down, and two, understand. So let's write down some numbers. So let's say I have 3.102 times 10 to the second. And I want to write it as just a numerical value. It's in scientific notation already. It's written as a product with a power of 10. So how do I write this? It's just a numeral. Well, there's a slow way and the fast way. The slow way is to say, well, this is the same thing as 3.102 times 100, which means if you multiplied 3.102 times 100, it'll be 3, 1, 0, 2, with two 0's behind it. And then we have 1, 2, 3 numbers behind the decimal point, and that'd be the right answer. This is equal to 310.2. Now, a faster way to do this is just to say, well, look, right now I have only the 3 in front of the decimal point. When I take something times 10 to the second power, I'm essentially shifting the decimal point 2 to the right. So 3.102 times 10 to the second power is the same thing as-- if I shift the decimal point 1, and then 2, because this is 10 to the second power-- it's same thing as 310.2. So this might be a faster way of doing it. Every time you multiply it by 10, you shift the decimal to the right by 1. Let's do another example. Let's say I had 7.4 times 10 to the fourth. Well, let's just do this the fast way. Let's shift the decimal 4 to the right. So 7.4 times 10 to the fourth. Times 10 to the 1, you're going to get 74. Then times 10 to the second, you're going to get 740. We're going to have to add a 0 there, because we have to shift the decimal again. 10 to the third, you're going to have 7,400. And then 10 to the fourth, you're going to have 74,000. Notice, I just took this decimal and went 1, 2, 3, 4 spaces. So this is equal to 74,000. And when I had 74, and I had to shift the decimal 1 more to the right, I had to throw in a 0 here. I'm multiplying it by 10. Another way to think about it is, I need 10 spaces between the leading digit and the decimal. So right here, I only have 1 space. I'll need 4 spaces, So, 1, 2, 3, 4. Let's do a few more examples, because I think the more examples, the more you'll get what's going on. So I have 1.75 times 10 to the negative 3. This is in scientific notation, and I want to just write the numerical value of this. So when you take something to the negative times 10 to the negative power, you shift the decimal to the left. So this is 1.75. So if you do it times 10 to the negative 1 power, you'll go 1 to the left. But if you do times 10 to the negative 2 power, you'll go 2 to the left. And you'd have to put a 0 here. And if you do times 10 to the negative 3, you'd go 3 to the left, and you would have to add another 0. So you take this decimal and go 1, 2, 3 to the left. So our answer would be 0.00175 is the same thing as 1.75 times 10 to the negative 3. And another way to check that you got the right answer is if you have a 1 right here, if you count the 1, 1 including the 0's to the right of the decimal should be the same as the negative exponent here. So you have 1, 2, 3 numbers behind the decimal. That's the same thing as to the negative 3 power. You're doing 1,000th, so this is 1,000th right there. Let's do another example. Actually let's mix it up. Let's start with something that's written as a numeral and then write it in scientific notation. So let's say I have 120,000. So that's just its numerical value, and I want to write it in scientific notation. So this I can write as-- I take the leading digit-- 1.2 times 10 to the-- and I just count how many digits there are behind the leading digit. 1, 2, 3, 4, 5. So 1.2 times 10 to the fifth. And if you want to internalize why that makes sense, 10 to the fifth is 10,000. So 1.2-- 10 to the fifth is 100,000. So it's 1.2 times-- 1, 1, 2, 3, 4, 5. You have five 0's. That's 10 to the fifth. So 1.2 times 100,000 is going to be a 120,000. It's going to be 1 and 1/5 times 100,000, so 120. So hopefully that's sinking in. So let's do another one. Let's say the numerical value is 1,765,244. I want to write this in scientific notation, so I take the leading digit, 1, put a decimal sign. Everything else goes behind the decimal. 7, 6, 5, 2, 4, 4. And then you count how many digits there were between the leading digit, and I guess, you could imagine, the first decimal sign. Because you could have numbers that keep going over here. So between the leading digit and the decimal sign. And you have 1, 2, 3, 4, 5, 6 digits. So this is times 10 to the sixth. And 10 to the sixth is just 1 million. So it's 1.765244 times 1 million, which makes sense. Roughly 1.7 times million is roughly 1.7 million. This is a little bit more than 1.7 million, so it makes sense. Let's do another one. How do I write 12 in scientific notation? Same drill. It's equal to 1.2 times-- well, we only have 1 digit between the 1 and the decimal spot, or the decimal point. So it's 1.2 times 10 to the first power, or 1.2 times 10, which is definitely equal to 12. Let's do a couple of examples where we're taking 10 to a negative power. So let's say we had 0.00281, and we want to write this in scientific notation. So what you do, is you just have to think, well, how many digits are there to include the leading numeral in the value? So what I mean there is count, 1, 2, 3. So what we want to do is we move the decimal 1, 2, 3 spaces. So one way you could think about it is, you can multiply. To move the decimal to the right 3 spaces, you would multiply it by 10 to the third. But if you're multiplying something by 10 to the third, you're changing its values. So we also have to multiply by 10 to the negative 3. Only this way will you not change the value, right? If I multiply by 10 to the 3, times 10 to the negative 2-- 3 minus 3 is 0-- this is just like multiplying it by 1. So what is this going to equal? If I take the decimal and I move it 3 spaces to the right, this part right here is going to be equal to 2.81. And then we're left with this one, times 10 to the negative 3. Now, a very quick way to do it is just to say, look, let me count-- including the leading numeral-- how many spaces I have behind the decimal. 1, 2, 3. So it's going to be 2.81 times 10 to the negative 1, 2, 3 power. Let's do one more like that. Let me actually scroll up here. Let's do one more like that. Let's say I have 1, 2, 3, 4, 5, 6-- how many 0's do I have in this problem? Well, I'll just make up something. 0, 2, 7. And you wanted to write that in scientific notation. Well, you count all the digits up to the 2, behind the decimal. So 1, 2, 3, 4, 5, 6, 7, 8. So this is going to be 2.7 times 10 to the negative 8 power. Now let's do another one, where we start with the scientific notation value and we want to go to the numeric value. Just to mix things up. So let's say you have 2.9 times 10 to the negative fifth. So one way to think about is, this leading numeral, plus all 0's to the left of the decimal spot, is going to be five digits. So you have a 2 and a 9, and then you're going to have 4 more 0's. 1, 2, 3, 4. And then you're going to have your decimal. And how did I know 4 0's? Because I'm counting,, this is 1, 2, 3, 4, 5 spaces behind the decimal, including the leading numeral. And so it's 0.000029. And just to verify, do the other technique. How do I write this in scientific notation? I count all of the digits, all of the leading 0's behind the decimal, including the leading non-zero numeral. So I have 1, 2, 3, 4, 5 digits. So it's 10 to the negative 5. And so it'll be 2.9 times 10 to the negative 5. And once again, this isn't just some type of black magic here. This actually makes a lot of sense. If I wanted to get this number to 2.9, what I would have to do is move the decimal over 1, 2, 3, 4, 5 spots, like that. And to get the decimal to move over the right by 5 spots-- let's just say with 0, 0, 0, 0, 2, 9. If I multiply it by 10 to the fifth, I'm also going to have to multiply it by 10 to the negative 5. So I don't want to change the number. This right here is just multiplying something by 1. 10 to the fifth times 10 to the negative 5 is 1. So this right here is essentially going to move the decimal 5 to the right. 1, 2, 3, 4, 5. So this will be 2.5, and then we're going to be left with times 10 to the negative 5. Anyway, hopefully, you found that scientific notation drill useful.