Transcript File

Practical Estimation
Using
Scientific Notation
Created
by
Dick Heckathorn
Can you quickly approximate an
answer to the following problem?
How many seconds are there in
one year?
365 days
24 hours
3600 sec
1 year x
x
x
1 year
1 day
1 hour
Can you quickly determine which
is larger?
3684 x 27.36 x 416.3 ?
or
63.72 x 273 x 4175 ?
Can you quickly approximate an
answer to the following problem?
3746 x 21.4 x 81.74
0.0975
A review of place holders
57,246.9318
5
7
2
4
6
ten-thousands
thousands
hundreds
tens
ones
9
3
1
8
tenths
hundredths
thousandths
ten-thousandths
A review of what happens to a
number when it is multiplied by
ten (10).
2
2 x 10 = 20
2 x 10 x10 = 200
2 x 10 x 10 x 10 = 2000
2 / 10 = 0.2
2 / 10 / 10 = 0.02
2 / 10 / 10 / 10 = 0.002
A review of numbers that are
different by a power of 10.
Number How number is formed
1
10
100
1000
10000
100000
1000000
=
=
=
=
=
=
=
1
1 x 10
1 x 10 x 10
1 x 10 x 10 x 10
1 x 10 x 10 x 10 x 10
1 x 10 x 10 x 10 x 10 x 10
1 x 10 x 10 x 10 x 10 x 10 x 10
A review of how numbers are
changed to power of ten format.
Power
of 10
Number
How number is formed
1
10
100
1,000
10,000
100,000
1,000,000
1 x no tens
1 x 10
1 x 10 x 10
1 x 10 x 10 x 10
1 x 10 x 10 x 10 x 10
1 x 10 x 10 x 10 x 10 x 10
1 x 10 x 10 x 10 x 10 x 10 x 10
100
101
102
103
104
105
106
A review of how numbers are
changed to power of ten format.
Power
of 10 Number
How number is formed
100
10-1
10-2
10-3
10-4
10-5
10-6
1
1 / 10
1 / 10 / 10
1 / 10 / 10 / 10
1 / 10 / 10 / 10 / 10
1 / 10 / 10 / 10 / 10 / 10
1 / 10 / 10 / 10 / 10 / 10 x 10
1
0.1
0.01
0.001
0.0001
0.00001
0.000001
Scientific Notation
Is a system in which the numbers are
expressed as the product of the
COEFFICIENT
which is a number that is equal to or greater
than one but less than ten
and the appropriate
POWER OF TEN
Here is an example.
The Number
63847
63847
Same number in scientific
notation
6.3847 x 104
Coefficient
Power of Ten
Let us now examine the steps to
change a number to scientific
notation format.
Number to be Changed
76348
76348
First move the decimal point
so that one non-zero digit is
to the left of the decimal
point.
7.6348
Next determine what you did
with the decimal point.
76348
7.6348
It was moved 4 places to the
left.
How does this number
compare to the original
number?
76348
7.6348
The number is 10,000 (104)
times smaller.
76348
7.6348
To make the number in
scientific notation the same as
the original number, what must
we do to the powers of ten?
76348
7.6348
We must multiply the number by
ten to the fourth power (4).
7.6348 x 104
Number to be Changed
0.000385
0.0003857
First move the decimal point
so that one non-zero digit is
to the left of the decimal
point.
3.857
Next determine what you did
with the decimal point.
0.0003857
3.857
It was moved 4 places to the
right.
How does this number
compare to the original
number?
0.0003857
3.857
The number is 10,000 (104)
times larger.
0.0003857
3.857
To make the number in
scientific notation the same as
the original number, what must
we do to the powers of ten?
0.0003857
3.857
We thus multiply 7.6348 by 10
to the minus four power (10-4).
3.857 x 10-4
Change the following
number to scientific
notation format.
724000
Check your answer on the
next slide when finished.
724000
The answer is
7.24 x 104
Change the following
number to scientific
notation format.
0.0000517
Check your answer on the
next slide when finished.
0.0000517
The answer is
5.17 x 10-5
Let us now solve a problem
where the numbers are
written in scientific
notation.
The Problem
(3.1 x 102) x (2.0 x 104)
(3.1 x
2
10 )
x (2.0 x
4
10 )
First separate the coefficients
& powers of ten.
(3.1 x 2.0) x (102 x 104)
(3.1 x
2
10 )
x (2.0 x
4
10 )
(3.1 x 2.0) x (102 x 104)
Next show how the powers of
ten are combined.
(3.1 x 2.0) x
(2
+
4)
10
(3.1 x
2
10 )
x (2.0 x
4
10 )
(3.1 x 2.0) x (102 x 104)
(3.1 x 2.0) x 10(2 + 4)
Finally finish the calculation.
6.2 x
6
10
Let us solve another
problem where the
numbers are written in
scientific notation.
The Problem
3
6.4 x 10
-5
3.2 x 10
3
6.4 x 10
-5
3.2 x 10
First separate the coefficients
& powers of ten.
3
6.4
10
x -5
3.2
10
3
6.4 x 10
-5
3.2 x 10
3
6.4
10
x -5
3.2
10
Next show how the powers of
ten are combined.
6.4
3-(-5)
x 10
3.2
3
6.4 x 10
-5
3.2 x 10
3
6.4
10
x -5
3.2
10
6.4
3-(-5)
x 10
3.2
Finally finish the calculation.
2.0 x 10
8
Let us now solve a problem by
first changing the number to
scientific notation format and
then determine the answer.
The Problem
6749 x 263
6749 x 263
First write the number in
scientific notation.
(6.749 x 103) x (2.63 x 102)
6749 x 263
(6.749 x 103) x (2.63 x 102)
Next combine the coefficients and
powers of ten.
(6.749 x 2.63 ) x 103 + 2
6749 x 263
(6.749 x 103) x (2.63 x 102)
(6.749 x 2.63 ) x 103 + 2
Next combine the coefficients and
the powers of ten.
17.75 x 105
6749 x 263
(6.749 x 2.63 ) x 103 + 2
17.75 x 105
Finally write in proper scientific
notation.
1.775 x 106
Set up the following problem
on a piece of paper showing
each step.
0.00463 x 6734 x 27
When finished check your
answer on the next slide.
The answer is:
8.42 x 102
Fermi Questions
By
Dick Heckahtorn
To solve Fermi questions such
as how may blades of grass are
there inside the fence that
surrounds the soccer field, one
needs to learn how to round a
number to its nearest order of
magnitude.
For example:
6.35 x 105
In the desired format is:
106
For example:
2.35 x 105
In the desired format is:
105
But what about 4.35 x
5?
10
4.35 x
5?
10
The answer is:
6
10
How many of you said 105 ?
How can this be?
Since we are dealing only with
orders of magnitude, we need
to determine the half-way point
between one order of
magnitude and the next.
For example between
101 and 102.
Would you agree that the half
way point between
1
2
1.5
10 and 10 is 10 ?
1.5
10 is
I hope you agree because
the midpoint between
1
2
10 and 10 ?
Our task then is quite simple.
What is the value of 101.5 ?
Did you say 50?
If you did, you have responded
with an answer that most
students give.
But 50 is an incorrect answer.
You see, we are not dealing
with an ordinary number scale.
We are dealing with an
exponential scale.
This is true because each order
of magnitude is 10 times larger
than the previous.
Without going into a lot of
mathematical explanation,
perform the following steps on
your calculator.
1. Press the ‘2nd’ key
2. Press the ‘LOG’ key
3. Enter 1.5
4. Press the ‘ENTER’ key
The answer is ……..
Did you get 31.6227766?
You should if you have the
calculator set on float.
Otherwise you will get a
rounded value.
An Example
Examine the following:
100 - order of magnitude = 0
101 - order of magnitude = 1
105 - order of magnitude = 5
10-1 - order of magnitude = -1
-3
10 - order of magnitude = -3
-5
10 - order of magnitude = -5