The speed of light is 299,792,458 m/s … which
is a very long number to write. It takes up a lot of paper, and it takes me
a long time to type it into my calculator. For most calculations, I don’t really need
this degree of precision. It’s rare for me to do a calculation with
more than 3 or 4 significant figures. If I wanted to round this value to 4 significant
figures, it would be 299,800,000 m/s. That’s still quite long to write out. And now it’s really easy to make a mistake
about the number of zeros to write. Scientific Notation is a much quicker way
to write very large numbers. In Scientific Notation, the speed of light
written to 4 significant figures is 2.998 x 10^8 m/s. If I need to write it with 3 significant figures,
it’s 3.00 x 10^8 m/s. This notation is much more compact and easier
to read. We can also use Scientific Notation to abbreviate
very small numbers. For example, the mass of a proton is
0.000000000000000000000000001672626219 kilograms. Imagine if you had to write this out every
time you did a problem involving atomic mass. You’d waste so much paper. It’s so much faster and easier to write
this number in Scientific Notation. With 3 sig figs, it’s 1.67 x 10^-27 kilograms. Notice that for very large numbers, the exponent
on the 10 is positive, and for very small numbers, the exponent on the 10 is negative. How do we convert these very long numbers
into the much shorter Scientific Notation? Let’s start with the speed of light example. The first step is to find where the decimal
point is. When a number is written as an integer like
this – there is an understood decimal at the end. So let’s go ahead and put that in. The next step is to move the decimal point
until there is only one non-zero digit to the left of the decimal. Each time you move a decimal point, you are
changing the number by a factor of 10, so we need to keep track of the number of moves. Let’s count. 1..2..3..4..5..6..7..8 We moved the decimal
to the left 8 times. That’s equivalent to dividing the number
by 10^8. To make sure we don’t actually change the value of the number, we have to
multiply by 10^8. Here’s another way to think of it. Let’s use a smaller number like 45000 so
we can see the pattern more easily. We’ll put in the decimal at the end of 45000. 45000 is the same as 45000 x 10^0. 10^0=1, so you can multiply any number by
10^0 and you won’t change the number. I’m putting that in so you can see it’s
not a big deal to have that x 10 at the end. X 10^0 is a way to say the decimal hasn’t
changed position at all. If we move the decimal one place to the left,
we now have 4500.0 That’s a different number. 4500 is much less than 45000. 10 times less. By moving the decimal point one step to the
left, we divided 45000 by 10. So to cancel the effect of moving the decimal,
we need to multiply by 10 So it’s like 4500.0 x 10^0 x 10^1=4500.0 x 10^1 Let’s move the decimal again. 450.00 x 10^1 x 10^1=450.00 x 10^2 Each time we move the decimal, we need to multiply by another 10. Count up all the 10s and you’ll see that
to write 45000 in Scientific Notation, we moved the decimal point 4 times, so we needed
to multiply by 10, 4 times. We get 4.5 x 10^4 We do something very similar when we use Scientific
Notation for very small numbers. Let’s use the human red blood cell as an
example. It’s 8 micrometers in diameter. That’s 0.000008 m. Once again, we really don’t want to have
to write down all those zeros. Let’s re-write it in Scientific Notation. Step 1 – find the decimal point. This one is here already. Step 2 – move the decimal point until there
is one non-zero digit to the left of the decimal. Don’t forget to count how many steps you
move the decimal point. 1..2..3..4..5..6. Notice we’re moving the decimal point to
the right. Each time we move the decimal point to the
right, that’s the same as multiplying the number by 10. That means we’re going to have to DIVIDE
the final number by 10 for every step that decimal point takes to the right. Step 3 – write the number and multiply by 10 raised to the negative of the number of steps. So in this case, we get 8 x 10^-6. Are you already familiar with negative powers
of 10? Let’s review this idea. When we move the decimal to the right, we
need to undo that by dividing by 10. We can write that as 10^-1, which is the same
as 1/10. If we move the decimal point twice, that means
we need to divide by 10 twice. We can think of that as 1/10 x 1/10, which
is the same as 1/10^2, which is the same as 10^-2. Let’s continue in this way until we get
to 10-6. How about if we want to go in the other direction? What if we started with a number in scientific
notation, and we want to expand it? For example, the approximate age of the Earth
is 4.543 x 10^9 years. To “get rid” of that 10^9, we need to
move the decimal point to the right. Remember, moving the decimal point to the
right is the same as multiplying by 10. So each time you move the decimal to the right,
take away one of those 10s on the right. 45.43 x 10^8 454.3 x 10^7 4543 x 10^6 45430 x 10^5 454300 x 10^4 4543000 x 10^3 45430000 x 10^2 454300000 x 10^1 4543000000. A quick word about vocabulary: We say that in
Scientific Notation, a number is written as a product between a coefficient and a power
of 10. Here you can see on the left is the coefficient. It’s always a number between 1 and 10. The coefficient is multiplied by a power of
10 – and it can either be a positive or a negative power of 10. 10 is the base, and this is the exponent. Recognizing each of these parts is important
when it comes to adding and subtracting numbers in Scientific Notation. Now let’s turn to calculations using Scientific
Notation. First, we’ll look at the rules for adding
and subtracting, and then we’ll look at examples for multiplying and dividing. Add the mass of a proton and the mass of a
neutron. 1.673 x 10^-27 kg + 1.675 x 10^-27 kg. These numbers have the same base and exponent,
so all we need to do is add together the coefficients, so we get 3.348 x 10^-27 kg. The coefficient is still between 1 and 10,
so there is no need to change the exponent on the right. Let’s look at an example where the exponent
would change. Find the mass of 10 protons:
We can add 1.673 x 10-27 10 times – that’s the same as multiplying the coefficient by
10. We get 16.73 x 10-27 We’ll move the decimal
point one step to the left, and then we can rewrite this as 1.673 x 10-26 What do we do when the exponents are different? We’ll need to manipulate the numbers so
they have the same exponents. This involves moving the decimal point. For example, if we do this calculation:
8.43 x 10^3 + 9.765 x 10^6 we need to shift the decimal point on one of these numbers
so the exponents match. Let’s move the decimal point on 9.765 x
10^6 so we have 103. To do that, we move the decimal to the right
3 places. 9.765 x 10^6 is the same as 97.65 x 10^5,
which is the same as 976.5 x 10^4, which is the same as 9765 x 10^3. Now that these two numbers have the same base
and exponents, we can just add their coefficients. 8.43 + 9765=9773.43 and don’t forget the
x 103. Move the decimal point to get back to scientific
notation: 9.77343 x 10^6 Notice that there were only two decimal places in 8.43, so we’ll
round to two decimal places in our final answer: 9.77 x 10^6 If you need a refresher on significant
figures, check out our video on that subject. The same rules apply if you are subtracting. As long as the base and exponent of the two
numbers are the same, you can just do the operation on the coefficients. For example,
3.1415 x 1064 – 2.71828 x 10^4=0.42322 x 10^4 Rewrite in scientific notation by moving
the decimal point one place to the right. 4.2322 x 103 – and we already have the correct
number of sig figs. To multiply two numbers in Scientific Notation,
you multiply their coefficients and add their exponents. Keep the same number of significant digits
as the number with the fewest significant digits. For example:
(5.56 x 10^3 ) x (2.124 x 10^5)=11.80944 x 10^8 Round to 3 sig figs: 11.8
x 10^8, Again if you need a refresher on sig figs, we have a video all about those rules. To divide two numbers in Scientific Notation,
you divide the coefficients and subtract the exponent. Round your final answer to the number of significant
figures in the coefficient with the smallest number of significant figures. (4.5 x 10^7) / (7.82 x 10^9)=0.57544757
x 10^-2 Round to 2 sig figs: 5.8 x 10^-3 If you’re working with Scientific Notation
on a calculator, make sure you know how to enter in these values correctly. It depends on the calculator. There may be a 10x button, or there may be
a button that says E or EE. These all mean the same thing. If you want to enter in 5 x 10^3, you would
type in 5, then the multiply button, then the 10x button, then 3. Or you would type in 5, then the E button,
then 3. Either way, you’ll get 5000.

Using Scientific Notation | Scientific Notation Calculator | Calculations using Scientific Notation

5 thoughts on “Using Scientific Notation | Scientific Notation Calculator | Calculations using Scientific Notation

  • September 14, 2016 at 11:57 pm
    Permalink

    Uma pena que esse canal tenha tido pouca divulgação na versão em Português. Ele é simplesmente incrível, quando criar meu canal irei sempre divulgar vocês.

    Reply
  • September 15, 2016 at 1:05 am
    Permalink

    I never tire of vector math. I'd love to see videos on vectors and matrices.

    Reply
  • September 16, 2016 at 7:08 am
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    Nice straight forward overview guys!

    Reply
  • September 18, 2016 at 6:37 pm
    Permalink

    Pelo amor de Deus Socrática, postem no canal em português! Todos esses vídeos de vcs são de mais! Por favor.

    Reply
  • December 27, 2017 at 5:28 pm
    Permalink

    Thank you so much.

    Reply

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