Transcript Slide 1

SCIENTIFIC NOTATION
• Scientific notation is used to express very
large or very small numbers. A number in
scientific notation is written as the product
of a number (integer or decimal) and a
power of 10. The number has one digit to
the left of the decimal point. The power of
ten indicates how many places the
decimal point was moved.
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• The decimal number 0.00000065 written in
scientific notation would be 6.5x10-7 because the
decimal point was moved 7 places to the right to
form the number 6.5. It is equivalent to
6.5x0.1x0.1x0.1x0.1x0.1x0.1x0.1
• A decimal number smaller than 1 can be converted
to scientific notation by decreasing the power of ten
by one for each place the decimal point is moved to
the right.
• Scientific notation numbers may be written in
different forms. The number 6.5x10-7 could also be
written as 6.5e-7.
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•The number of stars in the Adromeda Galaxy can
be written as:
2.0 x 100,000,000,000
It is that large number, 100,000,000,000 which cause the
problem. But that is just a multiple of ten. In fact it is ten
times itself eleven times:
10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000,000 = 1010
Though we think of zero as having no value, zeroes can make
a number much bigger or smaller. Think about the difference
between 10 dollars and 100 dollars. Any one who has
balanced a checkbook knows that one zero can make a big
difference in the value of the number. In the same way, 0.1
(one-tenth) of the US military budget is much more than 0.01
(one-hundredth) of the budget. (Though either one is probably
more money than most of us will ever see in our checkbooks!)
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• So we would write 200,000,000,000 in
scientific notation as:
• 2.0 x 1011
• This number is read as follows: "two point
zero times ten to the eleventh."
How Does Scientific Notation Work?
As we said above, the exponent refers to the number of zeros that follow the
1. So:
101 = 10;
102 = 100;
103 = 1,000,
and so on.
Similarly, 100 = 1, since the zero exponent means that no zeros follow the 1.
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• Negative exponents indicate negative powers of 10,
which are expressed as fractions with 1 in the
numerator (on top) and the power of 10 in the
denominator (on the bottom).
• So:
• 10-1 = 1/10;
10-2 = 1/100;
10-3 = 1/1,000,
and so on.
• This allows us to express other small numbers this
way. For example:
2.5 x 10-3 = 2.5 x 1/1,000 = 0.0025
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• It is easy to see that all the variations above
are just different ways to represent the same
number:
• 200,000,000,000 = 20 x 1010 (20 x 10,000,000,000)
2.0 x 1011 =(2.0 x 100,000,000,000)
0.2 x 1012 =(.2 x 1,000,000,000,000)
CONSIDER 378,400 or 3.784 x 105?
• The number 378,400 is also small enough to be readable.
There may be two reasons for expressing 378,400 in
scientific notation rather than decimal form.
1) Computation: Scientific Notation makes adding,
subtracting, multiplying and dividing numbers much
simpler.
2) Creating and reading tables.
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Practice with Scientific Notation!
Review of Scientific Notation
• Write out the decimal equivalent (regular form) of the
following numbers that are in scientific notation.
• Section A: Model: 101 = 10
• 1) 102 = ___________
• 2) 104 = ___________
• 3) 107 = ___________
• 4) 10-2 = __________
• 5) 10-5 = __________
• 6) 100 = __________
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• Section B: Model: 2 x 102 = 200
•
•
•
•
•
•
7) 3 x 102
8) 7 x 104
9) 2.4 x 103
10) 6 x 10-3
11) 900 x 10-2
12) 4 x 10-6
= ______________
= ______________
= ______________
= ______________
= ______________
= ______________
• Section C: Now convert from decimal form
into scientific notation. Model: 1,000 = 103
13) 10 = ____
14) 100 = ______
15) 100,000,000 = _____
16) 0.1 = ______
17) 0.0001 = ________
18) 1 = _______
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•Section D: Model: 2,000 = 2 x 103
19) 400 = ____________
20) 60,000 = __________
21) 750,000 = ____________
22) 0.005 = _______________
23) 0.0034 = _____________
24) 0.06457 = _____________
********************
•Section E:
Multiplication (the "easy" operation - remember
that you just need to multiply the main numbers
and add the exponents).
Model: (2 x 102) x (6 x 103) = 12 x 105 = 1.2 x 106
Check next page for exercises!..
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/….
scientific notation
decimal notation
25) (1 x 103) x (3 x 101) = _______________
____________________
26) (3 x 104) x (2 x 103) = ________________
____________________
27) (5 x 10-5) x (11 x 104) = ______________
____________________
28) (2 x 10-4) x (4 x 103) = ________________
____________________
•Section F:
Division
(12 x 103)
Model:
------------- = 2 x (103 x 10-2) = 2 x 101 = 20
(6 x 102)
29) (8 x 106) / (4 x 103)
multiplication problem
final answer (in sci. not.)
= ______________
________________
30) (3.6 x 108) / (1.2 x 104) = ___________
________________
31) (4 x 103) / (8 x 105)
= ______________
________________
32) (9 x 1021) / (3 x 1019)
= ____________
__________________
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•Addition: The first step is to make sure the exponents are the same.
We do this by changing the main number (making it bigger or smaller) so
that the exponent can change (get bigger or smaller). Then we can add the
main numbers and keep the exponents the same.
Model: (3 x 104) + (2 x 103) = (3 x 104) + (0.2 x 104) = 3.2 x 104 = 32,000
First express the problem with the exponents in the same
form, then solve the problem.
same exponent
33) (4 x 103) + (3 x 102) = _________________
34) (9 x 102) + (1 x 104) = _________________
35) (8 x 106) + (3.2 x 107) = ________________
36) (1.32 x 10-3) + (3.44 x 10-4) = ____________
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final answer
____________________
____________________
____________________
____________________
• Subtraction: Just like addition, the first step is to
make the exponents the same. Instead of adding
the main numbers, they are subtracted. Try to
convert so that you will not get a negative answer.
•
Model: (3 x 104) - (2 x 103) = (30 x 103) - (2 x 103) = 28 x 103 = 2.8 x 104
same exponent
37) (2 x 102) - (4 x 101) = _________________
38) (3 x 10-6) - (5 x 10-7) = ________________
39) (9 x 1012) - (8.1 x 109) = _______________
40) (2.2 x 10-4) - (3 x 102) = _______________
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final answer
___________________
___________________
___________________
___________________
More exercise!..
43) What is 1.25 x 10-1 = ?
44) 0.000553 is what in scientific notation?
45) (2 x 103) + (3 x 102) = ?
46) (2 x 103) - (3 x 102) = ?
47) (32 x 104) x (2 x 10-3) = ?
48) (9.0 x 104) / (3.0 x 102) = ?
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ANSWERS:
A)
1)100
5) 0.00001
2) 10,000
6) 1 B)
3) 10,000,000
7) 300
4) 0.01
8) 70,000
9) 2,400
10) 0.006
11) 9
12) 0.000004
14) 102
18) 100
15) 108
16) 10-1
20) 6X104
24) 6.457x10-2
21) 7.5X105
22) 5x10-3
25 b ) 30,000
27 b) 5.5
26a) 6x107
28a) 8x10-1
26b) 60,000,000
28b) 0.8
30) 3x104
31) 5x10-3
32) 3x102
C)
13) 101
17) 10-4
D)
19) 4x102
23) 3.4x10-3
E)
25 a) 3x104
27 a) 5.5x100
F)
29) 2x103
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G)
33) 4.3x103
34) 1.09x104
35) 4x107
H)
37) 1.6x102
38) 2.5x10-6
39) 8.9919x1012
I)
43) 0.125
45) 2.3x103
44) 5.53x10-4
46) 1.7x103
47) 6.4x102
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36) 1.664x10-3
40) -2.9999978x102
48) 3x102
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