Scientific Notation

Section 7-1 Part 2

Goals

Goal •  To write numbers in

scientific notation and standard form.

•  To compare and order numbers using scientific notation.

Vocabulary •  Scientific Notation

The table shows relationships between several powers of 10.

•  Each time you divide by 10, the exponent in the power decreases by 1 and the decimal point in the value moves one place to the left.

•  Each time you multiply by 10, the exponent in the power increases by 1 and the decimal point in the value moves one place to the right.

Powers of 10

Powers of 10

•  You can find the product of a number and a power of 10 by moving the decimal point of the number. –  If the exponent is positive, move the decimal point to

the right. –  If the exponent is negative, move the decimal point to

the left. •  You may need to write zeros to the right or left of

the number in order to move the decimal point.

A. 14 x 104

Multiply.

14.0 0 0 0 Since the exponent is a positive 4, move the decimal point 4 places to the right.

140,000

B. 3.6 x 10-5

0 0 0 0 3.6 Since the exponent is a negative 5, move the decimal point 5 places to the left.

0.000036

Example: Multiplying by Powers of 10

A. 2.5 x 105

Multiply.

2.5 0 0 0 0 Since the exponent is a positive 5, move the decimal point 5 places to the right.

250,000

B. 10.2 x 10-3

0 10.2 Since the exponent is a negative 3, move the decimal point 3 places to the left.

0.0102

Your Turn:

Definition

•  Scientific Notation - is a way to express numbers that are very large or very small. –  Powers of 10 are used when writing numbers in

scientific notation. –  Numbers written in scientific notation are expressed as

2 factors. •  One factor is a number greater than or equal to 1. •  The other factor is a power of 10.

–  Example: •  1.43 ⨯  1012 •  5.8 ⨯  10-9

The first part is a number that is greater than or equal to 1 and less than 10.

The second part is a power of 10.

Scientific Notation

Why Use Scientific Notation?

•  For very large and very small numbers, these numbers can be converted into scientific notation to express them in a more concise form.

•  Numbers expressed in scientific notation can be used in a computation with far greater ease.

Example: Recognizing Scientific Notation

Is the number written in scientific notation? Explain. 1.  53 ⨯  104

2.  3.42 ⨯  10-7

3.  0.35 ⨯  102

4.  9.6 ⨯  100

No, 53 is not less than 10

Yes

No, 0.35 is not greater than or equal to 1

No, 100 is not in power of 10 form

Your Turn: Is the number written in scientific notation?

Explain. 1.  8.15 ⨯  10-6

2.  12.9 ⨯  108

3.  1.003 ⨯  107

4.  0.0045 ⨯  10-32

Yes

No, 12.9 is greater than 10

Yes

No, 0.0045 is not greater than or equal to 1

Procedure: Writing Numbers in Scientific Notation

1.  Place the decimal point so that there is one non-zero digit to the left of the decimal point.

2.  Count the number of decimal places the decimal point has “moved” from the original number. This will be the exponent on the 10.

3.  If the original number was less than 1, then the exponent is negative. If the original number was greater than 1, then the exponent is positive.

Think: The number is less than 1, so the exponent will be negative.

A. 0.00709 Think: The decimal needs to move 3 places to get a number between 1 and 10.

7.09 x 10-3

Write the number in scientific notation.

So 0.00709 written in scientific notation is 7.09 x 10–3.

Example: Writing Numbers in Scientific Notation

Think: The number is greater than 1, so the exponent will be positive.

B. 23,000,000,000 Think: The decimal needs to move 10 places to get a number between 1 and 10.

2.3 x 1010

Write the number in scientific notation.

So 23,000,000,000 written in scientific notation is 2.3 x 1010.

Example: Writing Numbers in Scientific Notation

Think: The number is less than 1, so the exponent will be negative.

A. 0.000811 Think: The decimal needs to move 4 places to get a number between 1 and 10.

8.11 x 10 -4

Write the number in scientific notation.

So 0.000811 written in scientific notation is 8.11 x 10–4.

Your Turn:

Think: The number is greater than 1, so the exponent will be positive.

B. 480,000,000 Think: The decimal needs to move 8 places to get a number between 1 and 10.

4.8 x 108

Write the number in scientific notation.

So 480,000,000 written in scientific notation is 4.8 x 108.

Your Turn:

Standard form refers to the usual way that numbers are written—not in scientific notation.

Reading Math

Procedure: Writing Numbers in Standard Form

1.  Simply move the decimal point to the right for positive exponent 10.

2.  Move the decimal point to the left for negative exponent 10.

(Use zeros to fill in places.)

1.35000

135,000

Think: Move the decimal right 5 places.

A. 1.35 x 105

1.35 x 10 5

Write the number in standard form.

Example: Writing a Number in Standard Form

0002.7 Think: Move the decimal left 3 places. 2.7 x 10–3

B. 2.7 x 10–3

Write the number in standard form.

0.0027

Example: Writing a Number in Standard Form

2.870000000 Think: Move the decimal right 9 places.

A. 2.87 x 109

2.87 x 10 9

Write the number in standard form.

2,870,000,000

Your Turn:

000001.9 Think: Move the decimal left 5 places.

1.9 x 10 –5

B. 1.9 x 10–5

Write the number in standard form.

0.000019

Your Turn:

A certain cell has a diameter of approximately 4.11 x 10-5 meters. A second cell has a diameter of 1.5 x 10-5 meters. Which cell has a greater diameter?

4.11 x 10-5 1.5 x 10-5

Compare the exponents.

Compare the values between 1 and 10.

The first cell has a greater diameter.

4.11 > 1.5

Notice that 4.11 x 10-5 > 1.5 x 10-5.

Example: Comparing Numbers in Scientific Notation

A star has a diameter of approximately 5.11 x 103 kilometers. A second star has a diameter of 5 x 104 kilometers. Which star has a greater diameter?

5.11 x 103 5 x 104 Compare the exponents.

The second star has a greater diameter.

Notice that 3 < 4. So 5.11 x 103 < 5 x 104

Your Turn:

Order the list of numbers from least to greatest.

Step 1 List the numbers in order by powers of 10.

Step 2 Order the numbers that have the same power of 10

Example: Ordering Numbers in Scientific Notation

Order the list of numbers from least to greatest.

Step 1 List the numbers in order by powers of 10.

Step 2 Order the numbers that have the same power of 10

2 x 10-12, 4 x 10-3, 5.2 x 10-3, 3 x 1014, 4.5 x 1014, 4.5 x 1030

Your Turn:

Joke Time

•  Did you hear about the red ship and the blue ship that collided?

•  Both crews were marooned!

•  What did one shark say to the other while eating a clownfish?

•  This tastes funny!

•  What did the cobbler say when a cat wandered into his shop?

•  Shoe!