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 3 Scientific notation 3
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- On February 2, 2010 the U.S. Treasury estimated the
- national debt at 1.2278 times 10 to the 13th power.
- And just to get a sense of things, 1 times 10 to the
- sixth is a million, 1 times 10 to the ninth is a billion, 1
- times 10 to the 12th is a trillion.
- So we're talking on the order of magnitude
- of 10 trillion dollars.
- So this is about 12 trillion dollars.
- Then they tell us that the U.S. Census Bureau's estimate
- for the U.S. population was about 3.086 times 10 to the
- eighth power.
- So this is a little over 300 million people.
- So that's an interesting number right there, it's the
- population.
- And then they say, using these estimates calculate the
- per-person share of the national debt.
- So essentially, we want to take the entire debt and
- divide by the number of people.
- That'll give us the per-person share of the national debt.
- Use scientific notation to make your calculations and
- express your answer in both
- scientific and decimal notation.
- Which means just as a regular number.
- Round to four decimal places while making calculations.
- So we want the per-person debt.
- So we want to take the total debt and divide by
- the number of people.
- So the total debt is 1.2278 times 10 to the 13th.
- And we want to divide that by the total number of people,
- which is 3.086 times 10 to the eighth.
- And we could separate this into two division problems. We
- could say that this is equal to the division right here,
- 1.2278 divided by 3.086.
- And then times 10 to the 13th divided by 10 to the eighth.
- Now, what's 10 to the 13th divided by 10 to the eighth?
- Let me do it over here.
- The way I think about it, this is the exact same thing as 10
- to the 13th times 10 to the negative eight.
- This is an eight right here.
- If you have a 10 to the eighth in the denominator, that's
- like multiplying by 10 to the negative eight.
- So you have 13, you the same base 10, so 10 to the 13th
- times 10 to the negative eight is going to be 10 to
- the 13 minus 8.
- Which is 10 to the fifth.
- Or another way to think about it: If you have the base in
- the denominator, you subtract the exponents.
- So it's 13 minus 8.
- 10 to the fifth.
- So it's this blue expression times 10 to the fifth.
- And let's get a calculator out to calculate this right here.
- And they say round everything to four decimal places, so
- I'll keep that in mind.
- Let me turn my calculator on.
- 1.2278 divided by 3.086 is equal to 0.3979.
- Because we want to round right there.
- Let me remember that.
- Or let me just put it on the side so I can
- still look at it.
- So this this little dividing decimals
- problem results in 0.3979.
- And of course, times 10 to the fifth dollars per person.
- Once again, you might be tempted to say, hey this is in
- scientific notation.
- I have some number times a power of ten.
- But notice, this number is not greater than or equal to 1.
- Remember, this number, if you want to be formal about
- scientific notation, has to be greater than or equal to 1, or
- less than 10.
- So what we can do here is we can multiply.
- If we don't want to change the number, we can multiply this
- number by 10 and divide this number by 10.
- Or another way you can think about it is, this whole thing
- can be rewritten as 0.3979 times 10
- times 10 to the fourth.
- What I did was just now was I broke up the 10 of the fifth
- into a 10 and a 10 to the fourth.
- And I did that because I want to multiply this by 10 so I
- can get a 3 out front instead of a 0.3.
- So let's do that.
- So essentially, I took a 10 out of the 10 to the fifth.
- I divided it by 10, I multiplied this other guy by
- 10, not changing the whole number.
- So then this right here will become 3.979 and then times 10
- to the fourth.
- So that's how much debt there is per-person
- in scientific notation.
- Now, in the problem they also wanted us to express it in
- decimal notation.
- Which is just kind of standard writing it as a number with
- our standard numeric decimal system.
- So what is 3.979 times 10 to the fourth?
- Let's think about it.
- Well let me just do it this way.
- Let's just move the decimal space.
- If we multiply it by 10, we're going to get 39.79.
- If we multiply it by 10 squared, we're
- going to get 397.9.
- If we multiply it by 10 to the third, we're
- going to get 3,979.
- If we multiply it by 10 to the fourth, we're going to get one
- more zero right there.
- So we're essentially going to move the
- decimal four to the right.
- So I could write it like this.
- This is equal to $39,970.
- So if you think about the national debt per person.
- Every man, woman, and child in the United States essentially
- owes $39,970.
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