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029:006 Scientific Notation
In physics we often deal with very small and very large numbers. For example
the speed of light is approximately 300,000,000 m/s. The size of an atom is
roughly 0.0000000001 m. Rather than writing out all of the zeros, we express
these numbers in scientific notation – a decimal number (the coefficient)
followed by the factor 10 raised to some power (the exponent).
In scientific notation, the speed of light is roughly 3  108 m/s (or more precisely
2.99  108 m/s). Small numbers (numbers less than 1) are handled using
negative exponents. The size of the atom would be written as 1.00  10-10 m.
Big numbers (1 or greater than 1)
100 = 1 (any number raised to the power 0 is just 1)
10 = 101
100 = 10 x 10 = 102 (hundred)
1,000 = 10 x 10 x 10 = 103 (thousand)
1,000,000 = 10x10x10x10x10x10 = 106 (million)
1,000,000,000 = 10x10x10x10x10x10 x10x10x10 = 109 (billion)
1,000,000,000,000 = 1012 (trillion)
2000 = 2 x 1000 = 2 x 103
2500 = 2.5 x 1000 = 2.5 x 103
Small numbers (less than 1)
1
= 10-1
10
2
2
0.02 =
= 2 = 2×10-2
100 10
0.1=
0.01=
1
1
= 2 = 10-2
100 10
0.0 057 = 5.7×10-3
Examples
4,500,000 = 4.5  106, 0.000789 = 7.89  104, 5.0001 x 104 = 0.00050001
Multiplying and dividing numbers in scientific notation
Often we need to multiply and divide large or small numbers. For multiplication of
numbers in scientific notation, first multiply the coefficients, and then add the
exponents. Some examples:
(7 x 103) x (3 x 104) = 21 x 107 = 2.1 x 108.
(2.6 x 106) x (2.0 x 10—4) = 5.2 x 106—4 = 5.2 x 102
(1.5 x 10—5) x (3 x 103) = 4.5 x 10—2
(2 x 10—2) x (5 x 10—8) = 10 x 10—2 +(--8) = 10 x 10—10 = 1.0 x 10—9
When dividing two numbers in scientific notation, first divide the coefficients, and
then the exponent is the difference between exponent of the numerator and the
exponent of the denominator. Some examples:
8×105 8
= ×1053 = 4×102
3
2×10
2
4 107 4
 107 12  0.5  105  5  106
12
8 10
8
3 103 3
1
 103 7   103 7  0.333 104  3.33 103
7
9 10
9
3
2
7 10
7
  102 2  3.5  100  3.5
2
2 10
2
29:006 – Practice Exercises on Scientific Notation
1. Express the following numbers in scientific notation.
(a)
(b)
(c)
(d)
(e)
2,530,000
0.0000072
859
0.001
300,000,000
2. Express the following in standard notation.
(a)
(b)
(c)
(d)
7.35  104
8.6  105
1.87  105
6  108
3. Multiply.
(a)
(b)
(c)
(4 x 106) x (3 x 107)
(2 x 10—7) x (5 x 10—6)
(6 x 1013) x (4 x 10—15)
4. Divide.
(a)
(6 x 102) / (3 x 102)
(b)
(9 x 10—2) / (3 x 10—5)
(c)
(1.6 x 1014) / (4 x 1017)
--------------------------------------------------------------------------------------------Answers:
1. (a) 2.53 x 106, (b) 7.2 x 10–6, (c) 8.59 x102, (d) 1x10-3, (e) 3 x 108
2. (a) 73,500, (b) 0.000086, (c) 187000, (d) 600,000,000
3. (a) 1.2 x 1014,
(b) 1.0 x 10—12, (c) 0.24
4. (a) 2 (b) 3000 (c) 4 x 10—4