1. 8.4 The Scientific Notation Objective The student will be able to express numbers in scientific and decimal notation.
Helping us write really tiny or
really big numbers

2. 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.

3. Mathematicians are Lazy!!!
They decided that by using powers of 10,
they can create short versions of tiny and
really big numbers.

4. Scientific Notation Scientific 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.)
Scientific Notation

5. Rules to Scientific Notation
Parts: 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 exponent
Right  Negative exponent

6. Numbers Greater Than 10 (big)
Find the number by moving the decimal point that is between 1 and 10
45,300,000  4.53
Write a positive exponent which is equal to the number of places you moved the decimal point to the left.
4.53 x 107

7. Numbers Less Than 1 (tiny)
Find the number by moving the decimal point that is between 1 and 10
 2.91
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

8. 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

9. 2.10,000,000,000,000,000,000,
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 is
2.1 x 1023

10. 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

11. Additional Example 1: Writing Numbers in Scientific Notation
Write the number in scientific notation. A
Think: The decimal needs to move 3 places to get a number between 1 and 10.
7.09  10-3
Think: The number is less than 1, so the exponent will be negative.
So written in scientific notation is 7.09  10–3.

12. Additional Example 1: Multiplying by Powers of 10
Since the exponent is a positive 4, move the decimal point 4 places to the right.
140,000
B. 3.6  10-5
Since the exponent is a negative 5, move the decimal point 5 places to the left.

13. Write 28750.9 in scientific notation.
Your Turn
Write in scientific notation.
x 10-5
x 10-4
x 104
x 105

14. Convert from Scientific Notation to Decimal Notation
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.
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.

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

16. Write in Decimal Notation
Additional Example 2 & 3: Convert Scientific Notation to Decimal Notation
Write in Decimal Notation
A. 14  104
Since the exponent is a positive 4, move the decimal point 4 places to the right.
140,000
B. 3.6  10-5
Since the exponent is a negative 5, move the decimal point 5 places to the left.

17. Example 4 Use a calculator to evaluate: 4.5 x 10-5 1.6 x 10-2
Type
You must include parentheses if you don’t use those buttons!!
(4.5 x ) (1.6 x )
Write in scientific notation.
x 10-3

18. 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

19. 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

20. 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

21. Write in PROPER scientific notation
Write in PROPER scientific notation. (Notice the coefficient MUST be between 1 and 10) Example 8 Write x 109 in scientific notation.
2.346 x 1011
Example 9 Write x 104 in scientific
notation
on calculator: 642
6.42 x 10 2

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

23. Write 531.42 x 105 in scientific notation.

24. Convert the following numbers into correct scientific notation:
35.9 x 103
x 104
22.7 x 10-3
x 10-1
1845 x 105
123.4 x 1023
x 107
3.59 x 104
x 106
2.27 x 10-2
3.48 x 10-3
1.845 x 108
1.234 x 1025
3.45 x 104

25. Application Example Representing Large Numbers
93,000,000 miles from the Earth to the Sun (sunlight takes 8 minutes to reach us)
Representing Large Numbers
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

26. Application Example Representing Small Numbers 0.000167
To obtain a number between 1 and 10 we must move the decimal point to the right.
= 1.67 10-4
10-4 = 1/ (one ten-thousandth)

27. Example 10: Comparing Numbers in Scientific Notation
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.
4.11 > 1.5
Compare the values between 1 and 10.
Notice that 4.11  10-5 > 1.5  10-5.
The first cell has a greater diameter.

28. Example 11: Comparing Numbers in Scientific Notation
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.
Notice that 3 < 4. So 5.11  103 < 5  104
The second star has a greater diameter.

29. Example 12: Converting the Product of Numbers into 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 = 4r2. What is the surface area of the star in scientific notation?
SA = 4(6.74  104)2
= 4(6.74)2  (104)2
=  108
Is this your answer in Scientific Notation?
=  108
=  1010
Note all the learned knowledge will be called.
Power of a Product and Power of a Power Property