8.4 The Scientific Notation

ObjectiveThe student will be able to express numbers in scientific and decimal notation.

Helping us write really tiny or

really big numbers

How wide is our universe?

210,000,000,000,000,000,000,000 miles

(22 zeros)

This number is written in decimal

notation. When numbers get this

large, it is easier to write them in

scientific notation.

Mathematicians are Lazy!!!

They decided that by using powers of 10,

they can create short versions of tiny and

really big numbers.

Scientific NotationScientific Notation

A number is expressed in

scientific notation when it is in

the form

a x 10n

where a is between 1 and 10

and n is an integer (such as -12, -5, -1, 0, 4, 9, 17, etc.)

Rules to Scientific NotationParts: a x 10n

1. Coefficient (the a) – must be a number between 1 and 10

2. Exponent (the n) – a power of 10

3.4 x 106

Easier than writing 3,400,000

When we convert a decimal number to the scientific notation, if we move the decimal point to get the a in direction to the

Left Positive exponentRight Negative exponent

Numbers Greater Than 10 (big)

1. Find the number by moving the decimal point that is between 1 and 10

45,300,000 4.53

2. Write a positive exponent which is equal to the number of places you moved the decimal point to the left.

4.53 x 107

Numbers Less Than 1 (tiny)

1. Find the number by moving the decimal point that is between 1 and 10

0.000291 2.91

2. Write a negative exponent which is equal to the number of places you moved the decimal point to the right.

2.91 x 10-4

Write the width of the universe in scientific notation.

210,000,000,000,000,000,000,000miles

Where is the decimal point now?

After the last zero.

Where would you put the decimal to

make this number be between 1 and10?

Between the 2 and the 1

2.10,000,000,000,000,000,000,000

.How many decimal places did you move the

decimal?23

When the original number is more than 1, the exponent is positive.

The answer in scientific notation is2.1 x 1023

Example 1 Express 0.0000000902 in scientific notation.

Where would the decimal go to make the number be between 1 and 10?

9.02

The decimal was moved how many places?

8

When the original number is less than 1, the exponent is negative.

9.02 x 10-8

Additional Example 1: Writing Numbers in Scientific Notation

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.

7.09 103

Write the number in scientific notation.

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

A. 14 104

Multiply.

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

Additional Example 1: Multiplying by Powers of 10

140,000

B. 3.6 105

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

0.000036

Write 28750.9 in scientific notation.

1. 2.87509 x 10-5

2. 2.87509 x 10-4

3. 2.87509 x 104

4. 2.87509 x 105

Your Turn

1) When convert from scientific notation to decimal notation, if the power in the exponent is negative, then move the decimal point in the coefficient equal to the number of places to the LEFT.

2) When convert from scientific notation to decimal notation, if the poser in the exponent is positive, then move the decimal point in the coefficient equal to the number of places to the RIGHT.

Convert from Scientific Notation to Decimal Notation

Example 2 Express 1.8 x 10-4 in decimal notation.

0.00018

Example 3 Express 4.58 x 106 in decimal notation.

4,580,000

On the graphing calculator, scientific notation is done with the “2nd” and “LOG” buttons.

4.58 x 106 is typed 4.58 “2nd” and “LOG” buttons 6

A. 14 104

Write in Decimal Notation

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

Additional Example 2 & 3: Convert Scientific Notation to Decimal Notation

140,000

B. 3.6 105

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

0.000036

Example 4 Use a calculator to evaluate: 4.5 x 10-5

1.6 x 10-2

Type 4.5 -5 1.6 -2

You must include parentheses if you don’t use those buttons!!

(4.5 x 10 -5) (1.6 x 10 -2)

0.0028125Write in scientific notation.

2.8125 x 10-3

Example 5 Use a calculator to evaluate: 7.2 x 10-5

1.2 x 102

On the calculator, the answer is:6.E -7

The answer in scientific notation is 6 x 10-7

The answer in decimal notation is 0.0000006

Example 6 Use a calculator to evaluate (0.0042)(330,000).

On the calculator, the answer is

1386.

The answer in decimal notation is

1386

The answer in scientific notation is

1.386 x 103

Example 7 Use a calculator to evaluate (3,600,000,000)(23).

On the calculator, the answer is:

8.28 E +10

The answer in scientific notation is

8.28 x 10 10

The answer in decimal notation is

82,800,000,000

Write in PROPER scientific notation.(Notice the coefficient MUST be between 1 and 10) Example 8 Write 234.6 x 109 in scientific notation.

2.346 x 1011

Example 9 Write 0.0642 x 104 in scientific notation

on calculator: 642

6.42 x 10 2

Write (2.8 x 103)(5.1 x 10-7) in scientific notation.

1. 14.28 x 10-4

2. 1.428 x 10-3

3. 14.28 x 1010

4. 1.428 x 1011

Write 531.42 x 105 in scientific notation.

1. .53142 x 102

2. 5.3142 x 103

3. 53.142 x 104

4. 531.42 x 105

5. 53.142 x 106

6. 5.3142 x 107

7. .53142 x 108

Convert the following numbers into correct scientific notation:

35.9 x 103

556.67 x 104

22.7 x 10-3

0.0348 x 10-1

1845 x 105

123.4 x 1023

0.00345 x 107

3.59 x 104

5.5667 x 106

2.27 x 10-2

3.48 x 10-3

1.845 x 108

1.234 x 1025

3.45 x 104

Application Example Representing Large

Numbers 93,000,000 miles from the Earth to the Sun (sunlight takes 8 minutes to reach us)

93,000,000 = 9.3 x 10,000,000= 9.3 x 10 x 10 x 10 x 10 x 10 x 10 x 10= 9.3 x 107 (Decimal point moved 7 digits to the left)

Number between 1 and 10

Appropriate power of ten

Application Example

• Representing Small Numbers

0.000167To obtain a number between 1 and 10 we must move the decimal point to the right.

0.000167 = 1.67 10-4

10-4 = 1/10000 (one ten-thousandth)

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

4.11 10-5 1.5 10-5

Compare the exponents.

Example 10: Comparing Numbers in Scientific Notation

Compare the values between 1 and 10.

The first cell has a greater diameter.

4.11 > 1.5

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

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

5.11 103 5 104 Compare the exponents.

The second star has a greater diameter.

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

Example 11: Comparing Numbers in Scientific Notation

A star has a radius of approximately 6.74 104 kilometers. If the surface area of a sphere can be calculated as SA = 4r2. What is the surface area of the star in scientific notation?

SA = 4(6.74 104)2 = 4(6.74)2 (104)2

=570.86 108Is this your answer in Scientific Notation?

Note all the learned knowledge will be called.

Power of a Product and Power of a Power Property

= 570.86 108 = 5.7086 1010

Example 12: Converting the Product of Numbers into Scientific Notation