Download Scientific Notation Notes

August 30, 2012
EQ: How do I convert between standard form and scientific notation?
HW: Practice Sheet
Bellwork: Simplify each expression
1. (5x3)4
2. 5(x3)4
3. 5(x3)4
8
20x
Simplify and leave in standard form
4. -73
5. (-12)2
August 30, 2012
EQ: how do I operate in scientific notation?
HW: Have a good weekend.
Bellwork Quiz:
Simplify each expression and leave in exponential form.
1. (3x2y4)3
2. 5(a4b3)5
15a4b2
3. 5g-2h3
Simplify each expression and leave in standard form.
4. (-5)
5. -24
3
August 30, 2012
EQ: HOw do I operate in scientific notation?
HW: Practice Sheet
Bellwork:
11
2
1. (5.43 x 10 ) x (2.1 x 10 )
2. (3.4 x 103) x (4.5 x 102)
3. (9.1 x 1012) x (5.6 x 105)
4. (3.15 x 1011) ÷ (2.1 x 105)
5. (5.27 x 1011) ÷ (2.1 x 109)
August 30, 2012
Scientific notation is nothing more than a fancy way that scientists and
mathematicians represent very large and very small numbers.
Can you think of some situations that would call for very large or very
small numbers?
Consider the following situations:
· The distance from the earth to the sun in centimeters
· The total number of cells in a square inch of human tissue
· The diameter of a grain of sand
These area all either very large or very small numbers, and if we were
doing a research study on one of the above topics, we wouldn't want to
have to write that number over and over again.
Thus, scientific notation is really just something we came up with
because we are lazy. When you become a famous mathematician, you
can be lazy too!
August 30, 2012
Before we can start writing in scientific notation, though ,we have to go
back and review what know about the decimal number system.
As its name implies, the decimal number system is just a number
system that has 10 as its base. Think back to elementary school. Did
your teacher ever use base ten blocks? Maybe when you were learning
a out place value? or borrowing?
*Note: yes there are number systems out there that are NOT base 10,
but those are better left for another class.
So let's put what we know about the decimal number system together
with what we know about exponents.
1
10 or just 10 is our base
What is 102?
100
What is 103?
1000
What pattern do you notice?
August 30, 2012
If you said we added a zero every time, you're not wrong, but there is a
better way to put it.
101 = 10
102 = 100
What we are actually doing is adding a decimal
place each time we raise the power of 10.
103 = 1000
104 = 10000
The same is even true if we use negative exponents:
10-1 = 1 = 0.10
10
10-2 =
1
100
= 0.01
1
10-3 = 1000 = 0.001
So what? How does this help us write very large and very small
numbers?
August 30, 2012
Consider this: What happens when you multiply any number by 10?
5 x 10 = 50
We just move the decimal 1 place to the right!
What happens when i multiply any number by 102 (or 100)?
5 x 100 = 500
We just move the decimal 2 places to the right. let's try just one more.
3
what happens when I multiply any number by 10 (or 1000)?
5 x 1000 = 5000
Exactly! I just move the decimal 3 places to the right.
So the pattern is that whenever I multiply by a power of 10, I just use the
exponent to tell me how many places to move my decimal.
Do you think the same is true for negative exponents?
August 30, 2012
Of course it is, we just move the decimal in the other direction!
5 x 10-1 = 5 x
1 = 0.5
10
1
5 x 10-2 = 5 x
= 0.05
100
And that is the basis for scientific notation!
Now the fundamentals:
True scientific notation consists of 2 parts:
1. A number between 1 and 10
and
2. A power of 10
Note: We can write large or small numbers as powers of 10, but true
scientific notation has a leading digit between 1 and 10.
3.21 x 107
Good Scientific Notation
32.1 x 106
Bad Scientific Notation
August 30, 2012
To convert from standard form to scientific notation:
Step 1: Place the decimal point so that there is one non-zero digit to the
left of the decimal.
145,000,000
1.45
Step 2: Count the number of places the decimal has been moved.
145,000,000
1.45
In this case, the decimal was moved 8 places.
Step 3: Rewrite as a product of a power of 10. If the original number was
greater than 1, the exponent is positive. If the original was less than 1. the
exponent is negative.
1.45 x 108
Yes, it is that easy.
August 30, 2012
Let's try another one together.
Convert the following number to scientific notation:
0.0000000000054
Step 1: Place the decimal point such that there is one non-zero digit to
the left of the decimal.
5.4
Step 2: Count the number of places we moved the decimal.
0.0000000000054
5.4
We moved the decimal point 12 places.
Step 3: The original number was less than one, so my exponent will be
negative.
5.4 x 10-12
August 30, 2012
You try:
Convert each of the following numbers to scientific notation.
1. 2,340,000,000
2. 0.00000321
3. 5,400
4. 0.000918
5. 15,000,000,000,000
August 30, 2012
To convert from scientific notation to standard form:
1. Move the decimal point to the right for positive exponents.
2. Move the decimal point to the left for negative exponents.
Example:
3.41 x 106
3,410,000
*Note: I am NOT counting zeros! I am counting decimal places!!!!!
Look at another one:
2.179 x 10-4
0.0002179
Easy enough, right?
August 30, 2012
You try some:
1. 5.45 x 109
2. 4.7 x 10
-9
3. 2.114 x 1012
4. 3.12 x 10
5. 7.8 x 108
-4
August 30, 2012
Like everything else in math, we also have to be able to operate using
scientific notation. What if I have two numbers in scientific notation that I
want to multiply?
5
3
(4.5 x 10 ) x (2.2 x 10 )
I could convert them to standard form and then multiply the traditional
way:
450,000 x 2,200 = 990,000,000
And then convert back to scientific notation:
990,000,000 = 9.9 x 108
But that's a lot of converting. There must be an easier way. We are lazy
mathematicians after all.
August 30, 2012
let's look at the problem again:
5
3
(4.5 x 10 ) x (2.2 x 10 )
Are those parentheses really necessary?
The commutative property of multiplication says that I can multiply terms
in any order I want and still get the same answer. So it doesn't really
matter how I group the terms in the problem above. Let's look at it without
the parentheses:
5
3
4.5 x 10 x 2.2 x 10
Now why don't we move things around a little bit:
5
3
4.5 x 2.2 x 10 x 10
Have I changed the problem?
4.5 x 2.2 = 9.9
105 x 103 = 108
So....
5
3
8
(4.5 x 10 ) x (2.2 x 10 ) = 9.9 x 10
August 30, 2012
So what's the algorithm?
Step 1: Multiply the decimal terms.
Step 2. Multiply the exponential terms.
Step 3: Put the two parts together.
Let's look at another one:
8
4
(5.7 x 10 ) x (3.1 x 10 )
Using our algorithm, we get:
(5.7 x 108) x (3.1 x 104) = 17.67 x 1012
Done, right?
Not exactly. Go back to our definition of scientific notation. In order for a
number to be in true scientific notation, the leading decimal has to be a
number between 1 and 10. So we have one more step to do:
17.67 x 1012 = 1.767 x 1013
August 30, 2012
What about division?
Well, since division is really just a special type of multiplication, does it
make sense that division would work the same way?
Step 1: Divide the decimal terms
Step 2: Divide the exponential terms
Step 3: Put the two parts together.
Example:
(9.9 x 107) ÷ (3.6 x 104)
9.9 ÷ 3.6 = 2.75
107 ÷ 104 = 103
So...
(9.9 x 107) ÷ (3.6 x 104) = 2.75 x 103
August 30, 2012
Try another one:
8
(1.38 x 10 )
(4.6 x 103)
First we divide the decimals:
1.38 ÷ 4.6 = 0.30
Then we divide the exponential expressions:
108 ÷ 103 = 105
Then we put them together again:
(1.38 x 108)
(4.6 x 103)
= 0.30 x 105
But that's not official scientific notation is it?
0.30 x 105 = 3.0 x 104
Now we have a solution in true scientific notation
August 30, 2012
You try some:
1. (4.3 x 104) x (2.2 x 103)
2. (6.9 x 108) x (1.3 x 107)
9.
3. (2.9 x 103) x (4.1 x 109)
4. (5.4 x 103) x (3.1 x 102)
5. (3.9 x 1012) x (4.1 x 103)
6. (3.4 x 107) ÷ (1.5 x 103)
7. (7.2 x 1013) ÷ (2.1 x 108)
7
3
8. (8.7 x 10 ) ÷ (1.6 x 10 )
10.
9
(2.18 x 10 )
(5.2 x 106)
5
(1.14 x 10 )
(5.2 x 102)
Use hundredths as your level of
precision for decimals that seem to
go on for a while.
August 30, 2012
So what about adding and subtracting?
Well, just like exponential addition, addition in scientific notation has
some very strict rules it has to follow.
First, we have to look and see if the power of 10 is the same. Consider
the following expression:
5
5
(2.34 x 10 ) + (5.41 x 10 )
Because both numbers have the same power of 10, I can just add the
decimal term and leave the power of 010 the same:
5
5
5
(2.34 x 10 ) + (5.41 x 10 ) = 7.75 x 10
We can see why this works if we convert to standard form:
234,000
+ 541,000
775,000
Since the power of 10 just moves the decimal place, numbers with the
same power of 10 can be added without changing anything.
August 30, 2012
But what if my numbers don't have the same power of 10?
(1.34 x 104) + (4.45 x 105)
There are two things I can do here. The first is to convert each number to
standard form and then add:
13,400 + 445,000 = 458,400
Then I convert back to scientific notation:
5
458,400 = 4.584 x 10
But what if the conversion is not so simple. Consider the following:
(1.35 x 1013) + (2.47 x 1012)
Well, I can just rewrite each number so that it has the same power of 10
and then follow the rule from the previous slide:
(1.35 x 1013) + (2.47 x 1012) = (13.5 x 1012) + (2.47 x 1012)
12
12
12
(13.5 x 10 ) + (2.47 x 10 ) = 15.97 x 10
which is not good scientific notation, so we convert it to:
1.597 x 1013
August 30, 2012
And subtraction works the same way:
(5.4 x 109) - (2.1 x 109)
5.4 - 2.1 = 3.3
So...
(5.4 x 109) - (2.1 x 109) = 3.3 x 109
Not too bad, right?
August 30, 2012
1. (4.3 x 104) + (2.2 x 104)
2. (6.9 x 108) + (1.3 x 107)
3. (2.9 x 1012) + (4.1 x 1012)
4. (5.4 x 105) + (3.1 x 103)
5. (3.9 x 1015) + (4.1 x 1014)
6. (3.4 x 107) - (1.5 x 107)
7. (7.2 x 1013) - (2.1 x 1012)
8. (8.7 x 107) - (1.6 x 107)
9. (8.7 x 107) - (1.6 x 106)
10. (8.7 x 1015) - (1.9 x 1015)