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### Course: Grade 7 (VA SOL)>Unit 8

Lesson 5: Scientific notation intro

# 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.

## Want to join the conversation?

• is there a concept for this?
• Chemistry and physics use large numbers and this format helps us deal with large and small numbers.
• Is there any short variant for writing 100000000000000000005000000000000000 in scientific notation?
• Counting Zeroes... clunk

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! :-)
• 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?
• 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
• what is the easiest way to solve scientific notation.
• 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 :) )
• can someone teach me how to do this I dont really understand (simplify what Sal said)
• 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!
• Is there any other way's to handle 0's than the scientific notation method
• Sorry but no. If there were, I think there would be more videos about scientific notation.
• Is there anywhere in the real world that we would use scientific notation other then a math worksheet?
• Sure... many sciences deal with very large and very small numbers. Some examples would be: distances between planets or solar systems, the size of small things like atoms, etc. These numbers may have too many digits for calculators to display. Scientific notation is needed in those situations. And, calculators will sometimes give results in scientific notation when you input a calculation that results in a number to large to display.
• Is there any application for scientific notation?
• 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
• So 3.457 x 10^-10= 3.457/10^10
• Yep! We keep it as a multiplication as it makes it easier to do arithmetic with numbers in scientific notation.
• how do you convert an ending equation like 3.05 x 10^5 back to scientific notation though?
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.

## 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.