BIG NUMBERSand

SMALL NUMBERS

(Scientific Notation)

In science, you sometimes have to deal with very LARGE numbers

How far is the sun from

the earth?

150,000,000,000

meters

Or there may be times that you’ll use very small numbers…

How wide is one atom of gold?

0. 000 000 000 274 meters

Scientific notation is a system used to avoid dealing with all the ZEROS in very big and very small numbers

In scientific notation, numbers have TWO parts

n

{ a number between 1-10

that may be

followed by decimals}

X 10 x

a power

of 10

How do you write numbers in scientific notation?

Problem Write 150,000,000,000 in scientific notation:

Step 1 Move the decimal point from its original position until it is behind first nonzero digit (1) . This gives you a number between 1 and 10.

150 000 000 000. From here

To here

The number becomes 1.5

Step 2

Count the number of places that the decimal point moves to the left; this becomes the positive exponent

150 000 000 000. (The decimal pt. moves 11 places;

the exponent of 10 must be 11)

Answer: 1.5 x 1011

As you can see, there are two ways to write the SAME number

150 000 000 000

(standard notation)

or

1.5 x 1011

(scientific notation)

Problem: Write 0.000 000 000 274 in scientific notation

Step 1 Move the decimal point from its original position until it is behind first nonzero digit (2). This gives you a number between 1 and 10.

0.000 000 000 274

From here

To here

The number becomes 2.74

Step 2

Count the number of places that the decimal point moves to the right ; this becomes the negative exponent

0.000 000 000 274 (The decimal pt. moves 10 places; the

exponent of 10 must be 10)

Answer: 2.74 x 10-10

Again, there are obviously two ways to write this small number.

0.000 000 000 274

(standard notation)

or

2.74 x 10 -10

(scientific notation)

Remember

Big numbers have positive exponents

Small numbers have negative exponents

Now let’s practice !

Write 45 880 000 in scientific notation.

Correct answer is: 4.588 x 107

Let’s try another one…

Write 0.000 005 397 in scientific notation.

Correct answer is: 5.397 x 10-6

You’re on your own…

Write these numbers in scientific notation:

(1) 6,700

(2) 123,000

(3) 0.0089

(4) 9,362,000

(5) 0.000 008 75

One more thing

Be sure you also know how to change a number in scientific notation back to standard form.

For example:

Scientific notation: 8.32 x 10 4

How do you write this number in standard form?

Answer: Standard form: 83 200

How do you change numbers in scientific notation to standard form?

Big numbers

For numbers with positive exponentsFor numbers with positive exponents Move the decimal point from its current

position to the right. The number of decimal places moved must be the same as the exponent. Fill the spaces with zeros.

6.33 x 10 5

From here …. move decimal point 5 places to

the right (you need to write 3 zeros)

Answer: 633 000

Shall we give it a try?

Change 5.02 x 106 to standard form

5.02 x 106

Move the decimal point 6 places to the right

You will need to write in 4 more zeros

Answer: 5020000 or 5 020 000

Small numbers

For numbers with negative exponentsFor numbers with negative exponents

Move the decimal point from its current position to the left. The number of decimal places moved must be the same as the exponent. Fill the spaces with zeros.

7.88 x 10-4

From here …. move decimal point 4

places to the left (you need to write 3 zeros)

Answer: .000788

Let’s try this problem…

Change 9.12 x 10-3 to standard form

9.12 x 10-3

Move the decimal point 3 places to the left

You will need to write 2 more zeros

Answer: .00912 or .009 12

Arrange these numbers from the largest to the smallest.

6.5 x 104 5.8 x 10-3 - 3.4 x 105

9.2 x 10-2 7.01 x 10-1

4.17 x 108 2.2 x 106

Multiplying Numbers in Scientific Notation (4.2 x 103 ) ( 6.01 x 104 )

1) Multiply the first factors : (4.2 x 6.01)

2) Add the powers of 10:

103+4

ANSWER: 25.2 x 107

Dividing Numbers in Scientific Notation

(3.0 x 105 ) / (6.0 x 102 )

1) Divide the first factors:

3.0 / 6.0

2) Subtract the powers of 10

10 5 – 2

ANSWER: 0.5 x 103 = 5.0 x 102

Adding and Subtracting Numbers in Scientific NotationIf you are adding and subtracting numbers in

scientific notation without a calculator: first,adjust the numbers so that the exponents are the same

( 5.4 x 103 ) + ( 8.0 x 102 )

1) Adjust second number

8.0 x 102 = 0.8 x 103

2) Add the two numbers

(5.4 x 103 ) + (0.8 x 103 )

ANSWER: 6.2 x 103