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Scientific notation example: 0.0000000003457

Can you imagine if you had to do calculations with very, very small numbers? How would you handle all those zeros to the right of the decimal? Thank goodness for scientific notation! Created by Sal Khan and Monterey Institute for Technology and Education.

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  • female robot ada style avatar for user Brittany Harper
    is there a concept for this?
    (79 votes)
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  • male robot hal style avatar for user Denis Orlov
    Is there any short variant for writing 100000000000000000005000000000000000 in scientific notation?
    (11 votes)
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    • aqualine ultimate style avatar for user Johnathan
      Counting Zeroes... clunk
      Loading... clonk

      100,000,000,000,000,000,005,000,000,000,000,000
      Reporter:
      Here lies 100 decillion 5 quadrillion in its natural habitat. How do we simplify this dinosaur? Like in word form, one could split the numbers apart, so in scientific notation, we could also split the numbers apart. Here we have 100 decillion 5 quadrillion with its liver a-chopped and innards askew.
      (1•10^35) + (5•10^15)
      Something of that sort, should be fine and dandy. Hope this helps! :-)
      (77 votes)
  • spunky sam blue style avatar for user faryal ahmed
    Hi, I have a question, I was doing practice on Khan Academy site. There was a question 52 thousandths, which I have to turn in scientific notation, I answered 5.2x10^4 because I know 52000 has 3 zeros and I also add 2, so it gave me 10^4 but I was shocked it is incorrect, but why? Can anyone tell me?
    (20 votes)
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    • orange juice squid orange style avatar for user Lovish Garg
      Well the answer will be 5.2*10^-2. Because you have written thousandths not thousands. Both are very different. thousands are on the left side of the decimals but thousandths will be on the right side of the decimal. By the was 52 thousandths will be 0.052
      (5 votes)
  • stelly yellow style avatar for user epaintsil
    what is the easiest way to solve scientific notation.
    (5 votes)
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    • blobby blue style avatar for user acupofcchino
      Hi Epaintsil,
      There isn’t really a ‘easier’ way, so I’ll try to explain as best as I can. So a random number with TONS of zeros, 62427000000, how can we possibly simplify this? I can’t just write that TERRIBLY long number forever!

      So scientific notation is like: x * 10^y, where x is a number between 1 and 10, including the 1 and 10.

      In our case, our number, n, can be changed onto 6.2427, because that’s the only way we can fit it into our 1-10.

      So now we have x in our equation, what’s y?

      How I find this, is I count the digits that are to the right of the decimal point, which is 4, plus the amount of 0s, which leads to 10 digits in total.

      Y is 10, and X is 6.2427. Putting that into our equation ( x * 10^y, remember?), gives us 6.2427 * 10^10. And that’s the answer!

      If you have any questions, or have figured a mistake, please reply. Otherwise, upvote and have a great day! (Not forcing you to upvote though :) )
      (29 votes)
  • mr pants teal style avatar for user EXCUSED
    Anyone wanna ride the struggle bus with me?
    (15 votes)
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  • stelly blue style avatar for user thatsferlin1
    can someone teach me how to do this I dont really understand (simplify what Sal said)
    (6 votes)
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    • starky ultimate style avatar for user Taders09
      basically it is a way of showing a really tiny number with a lot of zeros without writing 0.0000000ect
      so if you have 0.000000000034 and you wanted to write it in scientific notation then you count the zeros and look at the numbers that aren't zeros and then take them and rewrite the problem as 3.4, then when you know how many zeros there are you do 3.4 * 1 to the -10 power
      and if you were to solve that you would get the original number.
      in short scientific notation is a way to simplify numbers.
      Hope this helps!
      (10 votes)
  • mr pants pink style avatar for user mnivema
    There is no end to the math
    (9 votes)
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  • cacteye green style avatar for user Ava Duran
    Can someone tell me when I would use this in the real world?
    (4 votes)
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    • aqualine ultimate style avatar for user evie
      Okay, sure. Let's say you become a cool scientist and you have to measure cells, bacteria, and protists, etc. If you're measuring atoms, A carbon atom is 0.000000000001 meters. Luckily, there's an easier way to measure that, 10^-12 meters. Scientific notation could make your job that much easier. That's not the only way this makes life easier. Let's say you're into space and you want to measure Saturn's measurements. Luckily, there's already a fact sheet that you can check out here: https://nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html
      As you can see throughout the page, almost every measurement has a scientific notation attached to a common measurement like a kilometre.
      (10 votes)
  • blobby green style avatar for user schltri27
    Is there any other way's to handle 0's than the scientific notation method
    (7 votes)
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  • blobby green style avatar for user Foz
    me monkey brain this hurt head ooga ooga ;(
    (8 votes)
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Video transcript

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.