Technical Science

Scientific Tools and Methods

Scientific Notation

Scientific Notation

In this section you will …– Describe the purpose of Scientific Notation– Convert numbers in standard form to scientific

notation– Convert numbers in scientific notation to

standard form.

Purpose of Scientific Notation Very often scientists are forced to work with

very large or very small numbers.

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The sun is 93 million miles from earth.– 93,000,000 miles

Bacterial cells measure about 1 micrometer.– 1 um = 0.000001 meters

A computer has a 10 gigabyte hard drive.– 10 Gb = 10,000,000,000 bytes

Purpose of Scientific Notation There are some problems with these very

large and very small numbers.

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It’s easy to accidentally forget a zero or add an extra one.

Calculating with these numbers is also difficult. Some have too many digits to be entered on a calculator.

93,000,000 mi 0.000001 m 10,000,000,000 bytes

Purpose of Scientific Notation Scientific notation was developed to solve

the problem of writing very large and small numbers.

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Numbers written in scientific notation have two parts: a stem, which is a number between 1 & 10, and a power of 10.

93,000,000 = 9.3 x 107

Stem Power of 10

Purpose of Scientific Notation Let’s take a closer look at the parts of a

number written in scientific notation to see how it works.

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The stem is always a number between 1 and 10. In this example, 9.3 is the stem and it is between 1 and 10.

93,000,000 = 9.3 x 107

Stem

Purpose of Scientific Notation? Which number below is written in scientific

notation?

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A 1.82 x 10 -6 is the only number that is written strictly in scientific notation because it is the only one with a stem that is between 1 and 10.

– 1.82 x 10-6

– 18.2 x 106

– 0.182 x 106

Purpose of Scientific Notation The power of ten has two parts. There is a

base and there is an exponent

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The base will always be 10. The exponent in this example is 7.

93,000,000 = 9.3 x 107

Base = 10

Exponent = 7

Purpose of Scientific Notation The power of ten part is a short way of

writing all the zeros.

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107 means 10 multiplied by itself 7 times. – 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000

Here are some other examples.– 103 = 10 x 10 x 10 = 1000– 102 = 10 x 10 = 100

93,000,000 = 9.3 x 107

Purpose of Scientific Notation Let’s put it all together and see how

scientific notation works.

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9.3 x 107 = 93,000,000

9.3 x 10,000,000 = 93,000,000

6.83 x 103 = 6,830

6.83 x 1,000 = 6,830

Purpose of Scientific Notation

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Multiplying by a multiple of ten can be done by moving the decimal.

The exponent on the “power of ten” tells how many places to move the decimal.

The “power of ten” part of a number in scientific notation is a multiple of 10.

93,000,000 = 9.3 x 107

107 = 10,000,0009 3,000,000.

7 places

9.3 x 10 7

Purpose of Scientific Notation? Which number below is written in scientific

notation?

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A 8.5 x 10 4 is the only number that is written strictly in scientific notation because the base must be 10.

– 1.82 x 610

– 8.5 x 104

– 3.1 x 1006

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Convert Standard Formto Scientific Notation Writing a number in scientific notation is a

three step process.

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We will start by writing large numbers in scientific notation then we will write small numbers in scientific notation.

You will see a step by step example and then you will have a chance to practice.

Convert Standard Formto Scientific Notation ? Write 8,760,000,000 in scientific notation.

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Step 1: Move the decimal until you have a number between 1 and 10 then drop the extra zeros.

8 7 6 0 0 0 0 0 0 0 ..

8.76 is the Stem

Convert Standard Formto Scientific Notation ? Write 8,760,000,000 in scientific notation.

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Step 2: The number of places you moved the decimal will be the exponent on the power of 10

8 7 6 0 0 0 0 0 0 0 ..

8.76 is the Stem

9 places

10 9 is the Power of 10

Convert Standard Formto Scientific Notation ? Write 8,760,000,000 in scientific notation.

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Step 3: Write the number so it is the stem times the power of ten.

8 7 6 0 0 0 0 0 0 0 ..

8.76 is the Stem9 places

10 9 is the Power of 10

8.76 x 10 9 = 8,760,000,000

Convert Standard Formto Scientific Notation ? Write the following numbers in scientific

notation.

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660,000,000 =

90300 =

397 =

4,500,000 =

8,000,100,000 =

159,400 =

7,300 =

6.6 x 108

9.03 x 104

3.97 x 102

4.5 x 106

8.0001 x 109

1.594 x 105

7.3 x 103

Convert Standard Formto Scientific Notation There are two differences when writing a

small number in scientific notation.

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First, the decimal will be moved in the opposite direction.

Second, the exponent will be written as a negative number.

Convert Standard Formto Scientific Notation ? Write 0.00000000482 in scientific notation.

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Step 1: Move the decimal until you have a number between 1 and 10 then drop the extra zeros.

0 0 0 0 0 0 0 0 0 4 8 2. .

4.82 is the Stem

Convert Standard Formto Scientific Notation ? Write 0.00000000482 in scientific notation.

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0 0 0 0 0 0 0 0 0 4 8 2. .

4.82 is the Stem

Step 2: The number of places you moved the decimal will be the exponent on the power of 10. The exponent will be negative because you started with a number less than 1.

9 places

10 -9 is the Power of 10

Convert Standard Formto Scientific Notation ? Write 0.00000000482 in scientific notation.

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0 0 0 0 0 0 0 0 0 4 8 2. .

4.82 is the Stem

Step 3: Write the number so it is the stem times the power of ten.

9 places

10 -9 is the Power of 10

4.82 x 10 -9 = 0.00000000482

Convert Standard Formto Scientific Notation ? Write the following numbers in scientific

notation.

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0.00543 =

0.00000074 =

0.03397 =

0.00006008 =

0.00000000081 =

0.000105 =

0.0000073 =

5.43 x 10 -3

7.4 x 10 -7

3.397 x 10 -2

6.008 x 10 -5

8.1 x 10 -10

1.05 x 10 -4

7.3 x 10 -6

Convert Scientific Notationto Standard Form Converting scientific notation to a number

in standard form is the same process in reverse.

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The exponent tells you how many places to move the decimal on the stem.– A positive exponent says the number is bigger

than the stem; move the decimal accordingly– A negative exponent says the number is smaller

than the stem; move the decimal accordingly

Convert Scientific Notationto Standard Form ? Write 1.98 x 109 in standard form.

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The exponent tells us to move the decimal 9 places. A positive exponent means the number is bigger

than the stem. To make 1.98 bigger, we must move the decimal to the right.

1 9 8 x 10 9. 0 0 0 0 0 0 0.

1.98 x 109 = 1,980,000,000

9 places

? Write the following numbers in standard form.

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2.9 x 104 =

6.87 x 106 =

1.008 x 109 =

3.04 x 105 =

8.8 x 1010 =

7.52 x 102 =

5.73 x 107 =

29,000

6,870,000

1,008,000,000

304,000

88,000,000,000

752

57,300,000

Convert Scientific Notationto Standard Form

Convert Scientific Notationto Standard Form ? Write 5.37 x 10-9 in standard form.

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The exponent tells us to move the decimal 9 places. A negative exponent means the number is smaller

than the stem. To make 5.37 smaller, we must move the decimal to the left.

5 3 7 x 10 -9.0 0 0 0 0 0 0 0 0.

5.37 x 10-9 = 0.00000000537

9 places

? Write the following numbers in standard form.

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2.9 x 10-4 =

6.87 x 10-6 =

1.008 x 10-9 =

3.04 x 10-5 =

8.8 x 10-10 =

7.52 x 10-2 =

5.73 x 10-7 =

0.00029

0.00000687

0.000000001008

0.0000304

0.00000000088

0.0752

0.000000573

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Scientific Notation