MA 1135 Lecture 16 - Scientific Notation
Thursday, April 9, 2015.
Objectives: Review computations with scientific notation.
Occasionally, we run into very large or very small numbers. For example, consider the expression
(1)
5
7 15
= 0
.
00000000000105317224
. . .
That’s an extremely small number. If you were to try that in your calculator, you might get something like
1.053172242E-12 , or something similar, indicating scientific notation,
(2)
5
7 15
= 1
.
05317224
×
10 −
12
There are a number of advantages to using scientific notation. For one, once you get used to it, it’s easier to read than counting all those zeros. Since your calculator can only display 10-12 digits, it can give you more information instead of a bunch of zeros.
Scientific Notation.
The rules for writing numbers in scientific notation are illustrated in the number
(3) 1
.
05317224
×
10 −
12
.
We’ll always have a decimal number greater than or equal to 1 and less than 10. In particular, the decimal number will have exactly one non-zero digit to the left of the decimal point. The general size of the number is expressed as a power of 10, and the exponent, the order of magnitude , is a measure of how big or small the number is.
The power of 10 corresponds to counting zeros. For example,
•
•
•
•
•
10 0
10 1
10 2
10 3 etc.
= 1 - no zeros.
= 10 - one zero.
= 100 - two zeros.
= 1000 - three zeros.
A negative exponent corresponds to how many times we’ll divide by 10, so
(4) 10 −
12 =
1
10 12
=
1
1
,
000
,
000
,
000
,
000
= 0
.
000000000001
When we multiply by a power of 10, the result shows itself in a shift of the decimal point. In a practical way, that’s how we’ll convert into and out of scientific notation. In the number we’re considering, multiplying by
10 −
12 moves the decimal point 12 places to the left.
(5) 1
.
05317224
×
10 −
12
= 0
.
00000000000105317224
Basic Principle 1.
In scientific notation, the power on the 10 corresponds to a shift in the decimal point.
A power of 7 corresponds to a shift of 7 places.
Scientific notation is a straight-forward concept, but it is easy to get mixed up which direction you’re supposed to shift the decimal point. Keep the following concept in mind.
Basic Principle 2.
Remember that a positive power on the 10 represents a large number, and a negative power on the 10 represents a very small number. That should help you decide if you shifted the decimal point in the right direction.
1

MA 1135 Lecture 16 - Scientific Notation 2
Examples converting scientific notation into regular decimal notation.
Example 1.
Express
(6) 8
.
964
×
10 5 in regular decimal notation. The power on the 10 is 5, so we’ll shift the decimal point five places. It’s a positive power, so we’ll shift to the right , which will result in a big number.
(7) 8
.
964
×
10
5
= 896
,
400
.
Example 2.
Express
(8) 3
.
76222
×
10 −
4 in regular decimal notation. The power on the 10 is
−
4, so we’ll shift the decimal point four places. It’s a negative power, so we’ll shift to the left , which will result in a small number.
(9) 3
.
76222
×
10 −
4 = 0
.
000376222
Examples converting decimal to scientific notation.
Example 3.
Express
(10) 7
,
863
,
000 in scientific notation. We want exactly one non-zero digit to the left of the decimal point. That means we want the decimal point to be just to the right of the 7. To do that, we need a shift of six places to the left.
This is a big number, so we should end up with a positive power on the 10.
(11) 7
,
863
,
000 = 7
.
863
×
10
6
.
Example 4.
Express
(12) 23
,
101 in scientific notation. We want exactly one non-zero digit to the left of the decimal point. That means we want the decimal point to be just to the right of the 2. To do that, we need a shift of four places to the left.
This is a big number, so we should end up with a positive power on the 10.
(13) 23
,
101 = 2
.
3101
×
10 4
.
Example 5.
Express
(14) 0
.
000000344 in scientific notation. We want exactly one non-zero digit to the left of the decimal point. That means we want the decimal point to be just to the right of the 3. To do that, we need a shift of seven places to the right. This is a small number, so we should end up with a negative power on the 10.
(15) 0
.
000000344 = 3
.
44
×
10 −
7
.
Multiplying and dividing with scientific notation
When we’re multiplying or dividing in scientific notation, remember that we’re just multiplying and dividing normal numbers, and some of them just happen to be powers of 10. The only concern is that we want to end up in scientific notation.
Example 6.
Multiply
(16) (7
.
23
×
10
4
)
·
(5
.
95
×
10
3
)
, and express the answer in correct scientific notation. Since we want to end up in scientific notation, it’s convenient to keep the decimal numbers and powers of 10 separate.
(17) (7
.
23
×
10
4
)
·
(5
.
95
×
10
3
) = (7
.
23)(5
.
95)
×
10
4
·
10
3
= 43
.
0185
×
10
7
.

MA 1135 Lecture 16 - Scientific Notation 3
Remember that when you multiply exponential expressions with the same base, you can add the exponents.
Note that our answer is not in correct scientific notation, because we don’t have exactly one non-zero digit to the left of the decimal point. We can make an adjustment by converting 43
.
0185 into scientific notation
(18) 43
.
0185
×
10 7 = 4
.
30185
×
10 1
×
10 7 = 4
.
30185
×
10 8
.
Basic Principle 3.
When multiplying or dividing numbers in scientific notation, keep the decimal numbers and powers of 10 separate.
Basic Principle 4.
Your result may not be in correct scientific notation, and you may have to shift the decimal point one place to the left or right.
Basic Principle 5.
When you make the adjustment, if you make the decimal part 10 times smaller, you must make the power 10 times bigger. Or you might have to do the opposite. The main thing is that one part gets bigger, and the other part gets smaller.
Example 7.
Divide
4
.
56
×
10 6
(19)
7
.
54
×
10 8 , and express the answer in correct scientific notation. Since we want to end up in scientific notation, it’s convenient to keep the decimal numbers and powers of 10 separate.
(20)
4
.
56
×
10 6
7
.
54
×
10 8
=
4
7
.
.
56
54
×
10 6
10 8
= 0
.
6047
×
10 −
2
.
Remember that when you divide exponential expressions with the same base, you can subtract the exponents.
Note that our answer is not in correct scientific notation, because we don’t have exactly one non-zero digit to the left of the decimal point. We can make an adjustment by converting 0
.
6047 into scientific notation
(21) 0
.
6047
×
10 −
2
= 6
.
047
×
10 −
1
×
10 −
2
= 6
.
047
×
10 −
3
.
Homework 16
1.
a.
b.
c.
d.
e.
2.
a.
b.
c.
Convert the number in scientific notation into normal decimal notation.
7
.
943
×
10 −
8 .
1
.
23
×
10 7 .
5
.
442
×
10 4 .
9
.
992
×
10 13 .
6
.
012
×
10 −
11 .
Convert the given number into scientific notation.
74
,
324
,
450
,
000
1
,
200
0
.
000523 d.
e.
0
.
0000005691
120
.
03
3.
Multiply or divide, and express answer in correct scientific notation (the decimal part should have exactly one non-zero digit to the left of the decimal point).
a.
(3
.
546
×
10 4 )
·
(2
.
113
×
10 −
2 )

b.
c.
d.
e.
(5
.
743
×
10 42 )
·
(8
.
425
×
10 −
3 )
(5
.
667
×
10 17 )
·
(7
.
333
×
10 14 )
15
7
.
75
×
10
2
.
05
×
10 3
3
.
75
×
10 −
5
6
.
20
×
10
−
7
MA 1135 Lecture 16 - Scientific Notation 4
Answers: 1a) 0
.
00000007943; b) 12
,
300
,
000; c) 54
,
420; d) 99
,
920
,
000
,
000
,
000; e) 0
.
00000000006012
2a) 7
.
432445
×
10 10 ; b) 1
.
200
×
10 3 ; c) 5
.
23
×
10 −
4 ; d) 5
.
691
×
10 −
7 ; e) 1
.
2003
×
10 2 a) (3
.
546
×
10 4 )
·
(2
.
113
×
10 −
2 ) = 7
.
4927
×
10 2 b) (5
.
743
×
10 42 )
·
(8
.
425
×
10 −
3 ) = 48
.
385
×
10 39 = 4
.
8385
×
10 40 c) (5
.
667
×
10 17 )
·
(7
.
333
×
10 14 ) = 41
.
556
×
10 31 = 4
.
1556
×
10 32 d) 7
.
2
75
×
10
15
.
05
×
10 3
= 3
.
7805
×
10 12 e)
6
.
75
×
10 −
5
3
.
20
×
10
−
7
= 0
.
6048
×
10 2 = 6
.
048
×
10 1