ma006 ยป

Contents

- 1 important digits
- 2 important digits
- 3 Why Scientific Notation
- 4 positive powers of ten
- 5 positive powers of ten
- 6 Negative Powers and Reciprocals
- 7 negative powers of ten
- 8 negative powers of ten
- 9 scientific notation
- 10 moving the decimal
- 11 moving the decimal
- 12 writing scientific notation
- 13 writing scientific notation
- 14 cells in scientific notation
- 15 cells in scientific notation

This is the number of cells we found from our last problem. Let's think about how big this number is. What digit or digits do you think are the most important in this number? Is it the 1, the 6, the middle 7, the 2, or the last 6? Check the numbers that you think are right. And this is debatable, so don't worry if you don't get it right on your first try.

Here is our number in a place value chart. I know that the one is really important. It's in the ten millions spot. So, this tells us how big the number is. The next important number is the six. 16 million is halfway between ten million and 20 million, so I want to use the one and six to gauge how big my answer is. Notice, too, that if I add the powers of ten to my chart, I can see that the ten millions is So, this number is on the order if 10^7 or ten millionth. The one is definitely the most important number and then six would be a close second.

We can write this number as 16,000,000, or a 16 followed by 6 zeros. When we think about writing this number, this can kind of be a pain. We have a ton of zeros and our number's pretty big. There are other extremely large and small numbers like 2 trillion and 48 billionths. We don't want to have to write all these zeros all the time, so, let's figure out how we can do this differently. Let's learn Scientific Notation. We don't want to have to write these zero's all the time, it can be a pain. So instead, let's find another way to write these numbers. Let's learn scientific notation.

When multiplying by a positive power of right, or stays in place. Think of this as your decimal point. And just to note, zero's actually not a positive power of You should look into that if you've got questions about it.

When multiplying 1 by a power of 10, I can see that the decimal point moves one place to the right. It goes here. When I multiply by another power of 10, the decimal place moves again, to the right, right here. So I can easily tell that the decimal point is always moving right.

Okay. Let's look at negative powers of
ten. Okay. Remember, 0 is not a positive
nor negative but my first negative power
will be 10^-1, then I have 10^-2 or 10^-3.
I know 10^0 is 1 because anything at the 0
power is 1. For negative exponents,
remember, I take the reciprocal of the
number. So, I have 1/10 to the 1 or 1/10.
For a 10^-2, I have 1/10^2, or 1/10*1/10,
which is 1/100. For 10^-3, I have the*
reciporical of 10 because the exponent's
negative, and it becomes a 3. 1/10^3 is
just 1/10*1/10 1/0 or 1/1000. So, I have
the decimal point 1 because the 1 is in
the tenth spot. 1/100, here, it's 0.01
because the 1 is in the hundredths place.
And then, I have one thousandth. I know
it's one thousandth because the 1 is in
the thousandths place.*

When multiplying by a negative power of ten, the decimal point moves left, moves right, or stays in place. Go ahead and answer. And remember, here's our decimal.

Here's my decimal, I know I need to move it to the left side of 1, so I have to move it this way. My decimal definitely moves left, let's make sure it keeps going. Now, that I have it on the other side of 1 or on the left side of 1, I can move the decimal again. When I move the decimal to the left, I create a zero in that place value, so I move from tenths to hundredths. And here, when I move the decimal again, I'm going to go from hundredths to thousandths. When I move the decimal left, I create a zero for a place holder and I move from hundredths to thousandths. Nice work on that quiz.

These are the two bottom numbers from the
last video. They're already written in
scientific notation, let's look at the
form. For a number to be in scientific
notation, the first number has to be a
number greater than or equal to 1, and
less than 10. We'll call that number a.
Next, we multiply that number by a power
of 10, so we have a*10 to some exponent.
Any number written in this form is*
considered to be in scientific notation.
Just be sure that this number is greater
than or equal to 1 and less than 10.
Usually it's in the form of the decimal,
but sometimes it won't have the decimal
after the numbers. It might just be 6*10*
of the -3.

Now that you know positive powers of 10 moves the decimal right and negative powers of 10 move the decimal left, let's see you use this in action. Here's another quiz. Fill in the powers or exponents to make each statement true. You can put your exponent in these boxes. Remember, you want to think about where should this decimal point move. How can I get 63 and turn into 6300? How many spots does it need to move and in which direction?

Alright. If you said positive two for this exponent, great work. When I multiply by a positive power of ten, I move the decimal two places right. For this next one, I notice that my decimal point is here. I need to make it go to the left side of six. I know that my exponent needs to be negative, since I'm moving my decimal left. If I go one spot, the decimal would be here, but I need two more zeros, so I'm going to go one, two and the decimal will end up here. Remember, I add zeros as placeholders whenever I move the decimal. I moved the decimal three places to the left, so my exponent is -3. Okay, for this one, my decimal starts here. I don't have anything after the decimal and I notice that I need to move it to the left. So I'm going to go one, two, three, four. My decimal moved four spots to the left, so I have to have an exponent of -4. In this case, I want my decimal point to end up to the right of six and one more place, here. So I'm going to move it one, two, three, and then I have to add 1 more spot where a zero would be as a placeholder. So I moved it one, two, three, four spots. So, positive four would be this exponent. Remember,, positive exponents move our decimal place to the right. Negative exponents move the decimal place to the left.

For your quiz I want you to write the number of seconds in 40 years, and the length of a red blood cell using scientific notation. You can put your numbers in these boxes.

For the number of seconds in 40 years, I
know I need to start the number with a 1.
So, here's my decimal point. I need to
move this decimal place over here. So, if
I thought about moving this decimal place,
I would need to move it 1, 2, 3, 4, 5, 6,
This should make sense, my exponent is a
positive 9, so I just move this decimal
place to the right 9 spaces. Okay. For the
red blood cell, I know I have 6, so 6
needs to start my answer. I'm moving this
decimal place to the left because I have
zeroes to the left of six. So, I would
move it 1, 2, 3, 4, 5, 6, 6 places left.
So, I have 6*10^-6.*
Okay, scientific notation is also pretty
helpful because when I compare two
numbers, like the dominoes toppled, or for
a million, I can compare it to 1 billion.
I notice that the difference in the powers
is 10^3.
That means there's a thousand times the
number of dominoes. When I look at these
two 2 numbers in scientific notation, I
can tell how many times larger a number is
than another. For example, the number of
seconds in 40 years is a thousand times
the number of dominoes toppled. I know
this because the difference of the powers
is 10^3.
So, the number of seconds in 40 years is a
thousand times the number of dominoes.
Alright, let's get back to our original
problem.

Let's use scientific notation one last time. Here is the number of cells that we had, 16,777,216. 16 million was the number of cells we decided upon. We knew the 1 and 6 were most important. So, let's take this number and write it in scientific notation. Put your answer in these boxes.

We need a number greater than or equal to going to use 1.6. I know large numbers have positive powers of 10, so I know there needs to be a positive number in this box. When I think about changing 1.6 into 16,000,000., I know the decimal place needs to move 7 places to the right. So I have 1.6 10^7.