Scientific Notation
Writing large and small numbers
It’s easy to make mistakes when computing
with small numbers such as 0.000022 and large
numbers such as 75,000,000. People who deal
regularly with such numbers use scientific
notation.
Scientific notation – a method of writing very large
and very small numbers using powers of 10.
Take for example, an ordinary penny contains about
20,000,000,000,000,000,000,000 atoms. The
average size of an atom is about 0.00000003
centimeters across
Written in scientific notation:
• Number of atoms in a penny;
2.0 x 1022
• Size of each atom:
3.0 x 10-8
The sign on the exponent tells which direction to
move the decimal. A positive exponent means
move the decimal to the right, and a negative
exponent means move the decimal to the left.
Writing Scientific Notation
• Scientific notation is written using the product of
2 factors.
1.0 x 106 = 1,000,000
• First factor is a number ≥ 1 but < 10.
1,000,000 = 1.0
• Second factor is a power of 10 = (106)
Scientific Notation to
Standard Notation
Write each number in standard notation.
2.64 x 107
2.64 x 10,000,000
10 = 10,000,000
26,400,000
Think: Move the decimal right 7 places for positive exponent.
7
1.35 x 10-4
1.35 x 1/10,000
0.000135
10-4 = 1/10,000
Think: Move the decimal left 4 places for negative exponent.
Standard Notation to
Scientific Notation
Write each number in scientific notation.
0.000002
2
Move the decimal to get a number between 1 and 10
2 x 10□
Set up scientific notation. Think: decimal needs to move left 6 places to
change 2 to 0.000002, moving left makes exponent negative.
2 x 10-6
0.0027
2.7
2.7 x 10□
Move the decimal to get a number between 1 and 10
Set up scientific notation. Think: decimal needs to move left 3 places to
change 2.7 to 0.0027, moving left makes exponent negative.
2.7 x 10-3
Scientific Notation
with a Calculator
Technology Lab with TI-83
calculator
Scientist often have to work with very large or
very small numbers. For example, the Andromeda
Galaxy contains over 200,000,000,000 (billion)
stars. Scientific notation is a compact way of
expressing these numbers.
• Show 200,000,000,000 in scientific notation on
your graphing calculator.
• Enter 200,000,000,000 on your calculator. Press
[ENTER]
2E11 on the graphing calculator display means
2 x 1011
Your calculator automatically puts very large
numbers into scientific notation.
You can enter scientific notation directly into the
calculator yourself using the EE function key
(2nd key then comma , key).
• Enter 2 x 1011 by pressing 2 [2nd] EE [,] 11 [enter]
Evaluating Expressions using
Scientific Notation
Evaluate 230,000,000 x 650,000
• First write 230,000,000 in scientific notation = 2.3 x 108
• Second write 650,000 in scientific notation = 6.5 x 105
• Press 2.3 [2nd] EE [,] 8 x 6.5 [2nd] EE [,] 5 [enter]
In scientific notation, 230,000,000 x 650,000 is equal to
1.495 x 1014 or 149,500,000,000,000
(149 trillion, 500 billion).
•
•
•
•
Evaluate (340,000,000,000 ÷ 1,235) x 4,568
Write each number in scientific notation.
340,000,000,000 = 3.4 x 1011
1,235 = 1.235 x 103
4,568 = 4.568 x 103
• Press (3.4 [2nd] EE [,] 11 ÷ 1.235 [2nd] EE [,] 3 ) x 4.568
[2nd] EE [,] 3 [enter]
• (340,000,000,000 ÷ 1,235) x 4,568 =
1.257587045 x 1012
(because this calculator displays results to only 10 decimal places, this in
not exact. The exact answer is 1.2575870445344110 x 10 12)
Problem Solving
• In space, light travels about 9,460,000,000,000
kilometers per year. This distance is known as a lightyear. The star Altair is 16.3 light-years from earth.
Write this distance in scientific notation.
• Suppose the universe is made up of 125,000,000,000
galaxies, each containing 200,000,000,000 stars. Use
this data to find the total number of stars in the
entire universe. Express your answer in scientific
notation and in standard form.
Answer
(check out what these would look like in standard notation)
1.54E14 kilometers from earth
2.5E22 star in the universe
154,000,000,000,000
154 trillion kilometers
25,000,000,000,000,000,000,000
25 sextillion stars in the universe
Extremely small numbers are written in the
same way but use (-) negative exponents.
• For example, our ordinary penny contains about 20
sextillion atoms, written:
2.0E22 or 20,000,000,000,000,000,000 atoms
• The average size of the atom we discussed earlier
is about 3 hundred millionths centimeters across,
written:
3.0E-8 or 0.00000003 centimeters
Helpful Hint
The sign of the exponent tells which direction
to move the decimal.
A positive exponent means to move the
decimal to the right.
A negative exponent means move the decimal
to the left.
Math Fact !!!!!
• The mass (weight) of the earth is about
5,980,000,000,000,000,000,000,000 kilograms.
(5 septillion, 980 sextillion)
• This is written in scientific notation as
5.98 x 1024
• How would you enter this into your calculator?
Ever wonder what 1 million pennies looks like?
Here’s your answer.