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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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  • 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?
    (19 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)
  • male robot hal style avatar for user Denis Orlov
    Is there any short variant for writing 100000000000000000005000000000000000 in scientific notation?
    (8 votes)
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    • blobby blue style avatar for user Legend19374646
      Sorry Johnathan, but that is quite wrong. If we have a number greater than 10, we move the decimal point to the left until we have a number between 1 and 10. Then, we count the number of times we moved the decimal and write that as an exponent over a base of 10. Finally, we write our number multiplied by the power of 10. So basically, to write this stupendously big number 100,000,000,000,000,000,005,000,000,000,000,000 in scientific notation, you will have to first get to the quadrillions and then think until the decillions.

      So, for the answer, you will write:
      1.00000000000000000005 * 10^35


      Note: the sign * is the multiplication sign.
      (2 votes)
  • aqualine ultimate style avatar for user The JVP
    Is there any application for scientific notation?
    (2 votes)
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    • male robot hal style avatar for user Kunal Chugh
      Yes, there is! When dealing with big and long numbers, it can really be a pain writing out all of them, this is when scientific notation helps out! For example it will be really hard to write 300000000000, so we write it like 3•10^11
      (12 votes)
  • starky ultimate style avatar for user ⚜
    Is there an algorithm for this?
    (3 votes)
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  • starky tree style avatar for user Cameron Holbrook
    how do you convert an ending equation like 3.05 x 10^5 back to scientific notation though?
    (0 votes)
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    • stelly blue style avatar for user Kim Seidel
      Your number is already in scientific notation.
      Do you mean how do you go back to standard notation? If yes, you just need to shift the decimal point. The 10^5 means you are multiplying by 5 instances of 10. Each 10 shifts the decimal point 1 place to the right (you get a bigger number). Since there are 5 10's, shift the decimal point 5 places to the right and you get 305000 or 305,000

      If the exponent had been negative 5, then you are essentially dividing by 5 tens and the decimal point will shift left (creating a smaller number).

      Hope this helps.
      (8 votes)
  • starky seedling style avatar for user daniel.cox
    Does it always have to be a decimal or can it be like 32 * 10^2 instead of 3.2 * 10^3?
    (2 votes)
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    • duskpin sapling style avatar for user proxima
      The number can never exceed 10, so it would be wrong to use 32 instead of 3.2. That being said, you can still have it as whole numbers that are less than ten. For example, 300 would turn into 3 * 10^2.
      (3 votes)
  • aqualine seed style avatar for user clora solai
    Hey, i have a question. i understood whats the meaning of "thousandths" but there was something like that
    19 hundred-thousandths
    i tried to write it. firstly, i thought 19 hundred can be 190 or 0.19 then i took that 190.. so then, it had to be 0.190 cause of thousandths. when i looked the answer, seriously it was so complicated. english isnt my main language but in this video, its not mention these ^^molds. i want to learn how its changing when we adding something or when we remove some number from that.
    thanks for read.
    (3 votes)
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  • aqualine ultimate style avatar for user Icey Isis
    do you have a video with multiplying scientific notations?
    (2 votes)
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  • blobby green style avatar for user Iann Foster
    how is it neagative ten power
    (2 votes)
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    • aqualine tree style avatar for user Page
      Okay , here is a problem to try and help you,
      0.3643 into scientific notation ,
      0.3643 start at your decimal point and move it one place to the right , then you have 03.643 .
      now you drop the zero because you dont need it ,
      because you dident move it to the left and you went the wrong way on the number line you have to make negative (-). once you have finished moving the decimal point to the RIGHT then you put it into the correct form .
      3.643 times 10^-1 ( negative one because you went wrong way on number line *intentional *
      and one because there is one number behind it .
      Hope this help's !
      (2 votes)
  • hopper cool style avatar for user ⅉ𝐨𝐬𝐞𝐩𝐡ℒ𝐨𝐯𝐞
    Makes sense. I wonder if there is a good way to remember which way is which for negative and positive. Don't you think?
    (3 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.