Scientific notation
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Scientific Notation (old)
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Scientific Notation Examples
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Scientific notation intuition
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Scientific Notation
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Scientific Notation I
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Scientific Notation 3 (new)
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Scientific Notation Example 2
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Scientific notation
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Scientific notation 3
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Multiplying in Scientific Notation
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Multiplying and dividing scientific notation
Scientific Notation Examples More scientific notation examples
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- It always helps me to see a lot of examples of something so I
- figured it wouldn't hurt to do more scientific
- notation examples.
- So I'm just going to write a bunch of numbers and then write
- them in scientific notation.
- And hopefully this'll cover almost every case you'll ever
- see and then at the end of this video, we'll actually do some
- computation with them to just make sure that we can
- do computation with scientific notation.
- Let me just write down a bunch of numbers.
- 0.00852.
- That's my first number.
- My second number is seven trillion, twelve billion.
- I'm just arbitrarily stopping the zeroes.
- The next number is 0.0000000 I'll just draw a couple more.
- If I keep saying zero, you might find that annoying.
- five hundred The next number -- right here, there's a
- decimal right there.
- The next number I'm going to do is the number seven hundred and twenty-three.
- The next number I'll do -- I'm having a lot of seven's here.
- Let's do 0.6.
- And then let's just do one more just for, just to make sure
- we've covered all of our bases.
- Let's say we do eight hundred and twenty-three and then let's throw some -- an
- arbitrary number of zero's there.
- So this first one, right here, what we do if we want to write
- in scientific notation, we want to figure out the largest
- exponent of ten that fits into it.
- So we go to its first non-zero term, which
- is that right there.
- We count how many positions to the right of the decimal point
- So it's going to be equal to this.
- So it's going to be equal to eight -- that's that guy
- right there -- 0.52.
- So everything after that first term is going to
- be behind the decimal.
- So 0.52 times 10 to the number of terms we have.
- One, two, three.
- ten to the minus three.
- Another way to think of it: this is a little bit more.
- This is like eight one / two thousands, right?
- Each of these is thousands.
- We have eight one / two of them.
- Let's do this one.
- Let's see how many zero's we have.
- We have three, six, nine, twelve.
- So we want to do -- again, we start with our largest
- term that we have.
- Our largest non-zero term.
- In this case, it's going to be the term all
- the way to the left.
- That's our seven.
- So it's going to be 7.012.
- It's going to be equal to 7.012 times 10 to the what?
- Well it's going to be times ten to the one with this many zero's.
- So how many things?
- We had a one here.
- Then we had one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve zero's.
- I want to be very clear.
- You're not just counting the zero's.
- You're counting everything after this first
- term right there.
- So it would be equivalent to a one followed by twelve zero's.
- So it's times ten to the twelfth.
- Just like that.
- Not too difficult.
- Let's do this one right here.
- So we go behind our decimal point.
- We find the first non-zero number.
- That's our five.
- It's going to be equal to five.
- There's nothing to the right of it, so it's 5.00 if we wanted
- to add some precision to it.
- But it's five times and then how many numbers to the right, or
- behind to the right of the decimal will do we have?
- We have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, and we
- have to include this one, fourteen.
- five times ten to the minus fourteenth power.
- Now this number, it might be a little overkill to write this
- in scientific notation, but it never hurts to get
- the practice.
- So what's the largest ten that goes into this?
- Well, one hundred will go into this.
- And you could figure out one hundred or ten squared by saying, "OK, this
- is our largest term." And then we have two zero's behind it
- because we can say one hundred will go into seven hundred and twenty-three.
- So this is going to be equal to 7.23 times, we could say times
- one hundred, but we want to stay in scientific notation, so I'll
- write times ten squared.
- Now we have this character right here.
- What's our first non-zero term?
- It's that one right there, so it's going to be six times and
- then how many terms do we have to the right of the decimal?
- We have only one.
- So times ten to the minus one.
- That makes a lot of sense because that's essentially
- equal to six divided by ten because ten to the minus
- 1 is 1/10 which is 0.6.
- One more.
- Let me throw some commas here just to make this a
- little easier to look at.
- So let's take our largest value right there.
- We have our eight.
- This is going to be 8.23 -- we don't have to add the other
- stuff because everything else is a zero -- times ten to the --
- we just count how many terms are after the eight.
- So we have one, two, three, four, five, six, seven, eight, nine, ten.
- 8.23 times 10 to the 10.
- I think you get the idea now.
- It's pretty straightforward.
- And more than just being able to calculate this, which is a
- good skill by itself, I want you to understand why
- this is the case.
- Hopefully that last video explained it.
- And if it doesn't, just multiply this out.
- Literally multiply 8.23 times 10 to the 10 and
- you will get this number.
- Maybe you could try it with something smaller
- than ten to the ten.
- Maybe ten to the fifth.
- And well, you'll get a different number but
- you'll end up with five digits after the eight.
- But anyway, let me do a couple more computation examples.
- Let's say we had the numbers -- let me just make something
- really small -- 0.0000064.
- Let me make a large number.
- Let's say I have that number and I want to multiply it.
- I want to multiply it by -- let's say I have a really large
- number -- three two -- I'm just going to throw a bunch of zero's here.
- I don't know when I'm going to stop.
- Let's say I stop there.
- So this one, you can multiply out.
- But it's a little difficult.
- But let's put it into scientific notation.
- One, it'll be easier to represent these numbers and
- then hopefully you'll see that the multiplication actually
- gets simplified as well.
- So this top guy right here, how can we write him in
- scientific notation?
- It would be 6.4 times 10 to the what?
- one, two, three, four, five, six.
- I have to include the six.
- So times ten to the minus six.
- And what can this one be written as?
- This one is going to be 3.2.
- And then you count how many digits are after the three.
- one, two, three, four, five, six, seven, eight, nine, ten, eleven.
- So 3.2 times 10 to the 11th.
- So if we multiply these two things, this is equivalent to six
- -- let me do it in a different color -- 6.4 times 10 to
- the minus 6 times 3.2 times 10 to the 11th.
- Which we saw in the last video is equivalent to 6.4 times 3.2.
- I'm just changing the order of our multiplication.
- Times ten to the minus six times ten to the eleventh power.
- And now what will this be equal to?
- Well, to do this, I don't want to use a calculator.
- So let's just calculate it.
- So 6.4 times 3.2.
- Let's ignore the decimals for a second.
- We'll worry about that at the end.
- So two times four is eight, two times six is twelve.
- Nowhere to carry the one, so it's just one hundred and twenty-eight.
- Put a zero down there.
- three times four is twelve, carry the one.
- three times six is eighteen.
- You've got a one there, so it's one hundred and ninety-two.
- Right?
- Yeah.
- one hundred and ninety-two.
- You had them up and you get eight, four, one plus nine is ten.
- Carry the one.
- You get two.
- Now, we just have to count the numbers behind
- the decimal point.
- We have one number there, we have another number there.
- We have two numbers behind the decimal point,
- so you count one, two.
- So 6.4 times 3.2 is equal to 20.48 times 10 to the -- we
- have the same base here, so we can just add the exponents.
- So what's minus six plus eleven?
- So that's ten to the fifth power, right?
- Right.
- Minus six and eleven.
- ten to the fifth power.
- And so the next question, you might say, "I'm done.
- I've done the computation." And you have.
- And this is a valid answer.
- But the next question is is this in scientific notation?
- And if you wanted to be a real stickler about it, it's not in
- scientific notation because we have something here that could
- maybe be simplified a little bit.
- We could write this -- let me do it this way.
- Let me divide this by ten.
- So any number we can multiply and divide by ten.
- So we could rewrite it this way.
- We could write one / ten on this side and then we can multiply
- times ten on that side, right?
- That shouldn't change the number.
- You divide by ten and multiply it by ten.
- That's just like multiplying by one or dividing by one.
- So if you divide this side by 10, you get 2.048.
- You multiply that side by ten and you get times ten to
- the -- times ten is just times ten to the first.
- You can just add the exponents.
- Times ten to the sixth.
- And now, if you're a stickler about it, this is good
- scientific notation right there.
- Now, I've done a lot of multiplication.
- Let's do some division.
- Let's divide this guy by that guy.
- So if we have 3.2 times 10 to the eleventh power divided by
- 6.4 times 10 to the minus six, what is this equal to?
- Well, this is equal to 3.2 over 6.4.
- We can just separate them out because it's associative.
- So, it's this times ten to the eleventh over ten to
- the minus six, right?
- If you multiply these two things, you'll
- get that right there.
- So 3.2 over 6.4.
- This is just equal to 0.5, right?
- 32 is half of 64 or 3.2 is half of 6.4, so this
- is 0.5 right there.
- And what is this?
- This is ten to the eleventh over ten to the minus six.
- So when you have something in the denominator, you
- could write it this way.
- This is equivalent to ten to the eleventh over ten to the minus six.
- It's equal to ten to the eleventh times ten to the
- minus six to the minus one.
- Or this is equal to ten to the eleventh times ten to the sixth.
- And what did I do just there?
- This is one over ten to the minus six.
- So one over something is just that something to the
- negative one power.
- And then I multiplied the exponents.
- You can think of it that way and so this would be equal
- the same bases, ten in this case, and you're dividing them,
- you just take the one the numerator and you subtract the
- exponent in the denominator.
- So it's eleven minus minus six, which is eleven plus six,
- which is equal to seventeen.
- So this division problem ended up being equal to
- 0.5 times 10 to the 17th.
- Which is the correct answer, but if you wanted to be a
- stickler and put it into scientific notation, we want
- something maybe greater than one right here.
- So the way we can do that, let's multiply
- it by ten on this side.
- And divide by ten on this side or multiply by one / ten.
- Remember, we're not changing the number if you multiply
- by ten and divide by ten.
- We're just doing it to different parts of the product.
- So this side is going to become five -- I'll do it in pink -- ten
- times 0.5 is 5, times 10 to the 17th divided by 10.
- That's the same thing as ten to the seventeenth times ten
- to the minus one, right?
- That's ten to the minus one.
- So it's equal to ten to the sixteenth power.
- Which is the answer when you divide these two
- guys right there.
- So hopefully these examples have filled in all of the
- gaps or the uncertain scenarios dealing with
- scientific notation.
- If I haven't covered something, feel free to write a comment on
- this video or pop me an e-mail.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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