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# Scientific Notation

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- 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.

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