Video Transcript: Scientific Notation

Scientific notation example: 0.0000000003457 

Express 0.0000000003457 in scientific notation. So let's just remind ourselves what it means to  be in scientific notation. Scientific notation will be some number times some power of 10 where this number right here-- let me write it this way. It's going to be greater than or equal to 1, and  it's going to be less than 10. So over here, what we want to put here is what that leading  number is going to be. And in general, you're going to look for the first non-zero digit. And this  is the number that you're going to want to start off with. This is the only number you're going  to want to put ahead of or I guess to the left of the decimal point. So we could write 3.457, and  it's going to be multiplied by 10 to something. Now let's think about what we're going to have  to multiply it by. To go from 3.457 to this very, very small number, from 3.457, to get to this, you have to move the decimal to the left a bunch. You have to add a bunch of zeroes to the left of  the 3. You have to keep moving the decimal over to the left. To do that, we're essentially  making the number much much, much smaller. So we're not going to multiply it by a positive  exponent of 10. We're going to multiply it times a negative exponent of 10. The equivalent is  you're dividing by a positive exponent of 10. And so the best way to think about it, when you  move an exponent one to the left, you're dividing by 10, which is equivalent to multiplying by  10 to the negative 1 power. Let me give you example here. So if I have 1 times 10 is clearly just  equal to 10. 1 times 10 to the negative 1, that's equal to 1 times 1/10, which is equal to 1/10. 1  times-- and let me actually write a decimal, which is equal to 0-- let me actually-- I skipped a  step right there. Let me add 1 times 10 to the 0, so we have something natural. So this is one  times 10 to the first. One times 10 to the 0 is equal to 1 times 1, which is equal to 1. 1 times 10 to the negative 1 is equal to 1/10, which is equal to 0.1. If I do 1 times 10 to the negative 2, 10 to the negative 2 is 1 over 10 squared or 1/100. So this is going to be 1/100, which is 0.01. What's  happening here? When I raise it to a negative 1 power, I've essentially moved the decimal from  to the right of the 1 to the left of the 1. I've moved it from there to there. When I raise it to the  negative 2, I moved it two over to the left. So how many times are we going to have to move it  over to the left to get this number right over here? So let's think about how many zeroes we  have. So we have to move it one time just to get in front of the 3. And then we have to move it  that many more times to get all of the zeroes in there so that we have to move it one time to  get the 3. So if we started here, we're going to move 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 times. So this is  going to be 3.457 times 10 to the negative 10 power. Let me just rewrite it. So 3.457 times 10 to  the negative 10 power. So in general, what you want to do is you want to find the first non-zero  number here. Remember, you want a number here that's between 1 and 10. And it can be equal to 1, but it has to be less than 10. 3.457 definitely fits that bill. It's between 1 and 10. And then  you just want to count the leading zeroes between the decimal and that number and include  the number because that tells you how many times you have to shift the decimal over to  actually get this number up here. And so we have to shift this decimal 10 times to the left to get this thing up here. 

Scientific notation examples 

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.