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Polynomials 4

Copyright © Cengage Learning. All rights reserved.

Section 4.34.3 Scientific Notation

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Objectives

Convert a number from standard notation to scientific notation.

Convert a number from scientific notation to standard notation.

Use scientific notation to simplify an expression.

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Scientific Notation

We now use exponents to express very large and very small numbers that are written in standard notation in a compact form called scientific notation.

In science, almost all large and small numbers are written in scientific notation.

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Convert a number from standard notation to scientific notation

1.

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Convert a number from standard notation to scientific notation

Scientists often deal with extremely large and extremely small numbers. For example,

• The distance from Earth to the Sun is approximately 150,000,000 kilometers.

• Ultraviolet light emitted from a mercury arc has a wavelength of approximately 0.000025 centimeter.

The large number of zeros in these numbers makes them difficult to read and hard to remember.

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Convert a number from standard notation to scientific notation

Scientific notation provides a compact way of writing large and small numbers.

Scientific Notation

A number is written in scientific notation if it is written as the product of a number between 1 (including 1) and 10 and an integer power of 10.

Each of the following numbers is written in scientific notation.

3.67 106 2.24 10–4 9.875 1022

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Convert a number from standard notation to scientific notation

Every number that is written in scientific notation has the following form:

An integer exponent

A number between 1 and 10

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Example 1

Write 150,000,000 to scientific notation.

Solution:

We note that 1.5 lies between 1 and 10. To obtain 150,000,000, the decimal point in 1.5 must be moved eight places to the right.

Because multiplying a number by 10 moves the decimal point one place to the right, we can accomplish this by multiplying 1.5 by 10 eight times.

1.5 0 0 0 0 0 0 0

150,000,000 written in scientific notation is 1.5 108.

8 places to the right

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Convert a number from scientific notation to standard notation

2.

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Convert a number from scientific notation to standard notation

We can convert a number written in scientific notation to standard notation by reversing the process of converting standard notation to scientific notation.

To convert a number to standard notation, move the decimal point the number of places indicated by the exponent.

If the exponent is positive, this represents a large number and the decimal point will move to the right.

If the exponent is negative, this represents a small number and the decimal point will move to the left.

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Convert a number from scientific notation to standard notation

For example, to write 9.3 107 in standard notation, we move the decimal point seven places to the right.

Since we already have one number to the right of the decimal, we will need to insert 6 zeros for place value.

9.3 107 = 9.3 10,000,000

= 93,000,000 Move the decimal point 7 places to the right.

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Example

Write a. 3.4 105 and b. 2.1 10–4 in standard notation.

Solution:

a. 3.4 105 = 3.4 100,000

= 340,000

b.

Move the decimal point 5 places to the right.

Move the decimal point 4 places to the left.

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Convert a number from scientific notation to standard notation

Each of the following numbers is written in both scientific and standard notation.

In each case, the exponent gives the number of places that the decimal point moves, and the sign of the exponent indicates the direction that it moves.

2.37 106 = 2 3 7 0 0 0 0 . Move the decimal point 6 places to the right

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Convert a number from scientific notation to standard notation

8.375 10–3 = 0 . 0 0 8 3 7 5

9.77 100 = 9.77 No movement of the decimal point

Move the decimal point 3 places to the left

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Use scientific notation to simplify an expression

3.

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Use scientific notation to simplify an expression

Another advantage of scientific notation becomes apparent when we simplify fractions such as

that contain very large or very small numbers. Although we can simplify this fraction by using arithmetic, scientific notation provides an alternative way.

First, we write each number in scientific notation; then we do the arithmetic on the numbers and the exponential expressions separately.

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Use scientific notation to simplify an expression

Finally, we write the result in standard form, if desired.

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Example – Speed Of Light

In a vacuum, light travels 1 meter in approximately 0.000000003 second. How long does it take for light to travel 500 kilometers?

Solution:

Since 1 kilometer = 1,000 meters, the length of time for light to travel 500 kilometers (500 1,000 meters) is given by

(0.000000003)(500)(1,000) = (3 10–9)(5 102)(1 103)

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Example 6 – Solution

= 3(5) 10–9 + 2 + 3

= 15 10–4

= 1.5 101 10–4

= 1.5 10–3

= 0.0015

Light travels 500 kilometers in approximately 0.0015 second.

(or 1.5 millisecond).

cont’d