Scientific Notation
Extremely Large and Small Numbers
Large and small numbers are prevalent in
Standard Form or
our world:
Decimal Form
In 2011 the Use National Debt was:
$15,143,572,733,215
The mass of a dust particle is approximately:
0.000000000753 kg
Scientists have created an easier way to express these
extreme numbers.
Introduction to Scientific Notation:
Powers of Ten
Decimal form Powers of 10
1,000,000,000
109
1,000,000
106
1,000
103
10
101
1
100
0.1
10-1
0.001
10-3
0.000001
10-6
Name
One billion
One million
One thousand
Ten
One
One tenth
One thousandth
One millionth
Scientific Notation is based on Powers of Ten.
Definition of Scientific Notation
Numbers in Scientific Notation are written as:
c x 10n
where c is a decimal number greater than or equal to 1
and less than 10 (Typically c is rounded to 2 or 3
decimal places) and n is an integer. Below are two
5
examples:
1.2 10
3.5 10
4
Example: Which expression is written in correct Scientific
3.9 is the only number greater than or equal to 1 and less
Notation?
than 10 AND the exponent is an integer
 a  13.2 105  b  7.035 1010.2  c  3.9 1012  d  0.45 1020
Converting Standard Form to
Scientific Notation
By converting the power of 10 to Standard Form Notation,
the true value of the Scientific Notation number can be
obtained.
Example: Convert the following from Standard Form to
Scientific Notation . Then try to find a shortcut.
1.2 10  1.2 100000   120000
5
The positive 5 makes the number bigger, by
moving the decimal 5 places to the right.
1.2 0 0 0 0.
1
2
3
4
5
3.5 10  3.5  0.0001  0.00035
4
The negative 4 makes the number smaller,
by moving the decimal 4 places to the left.
0 .0 0 0 3.5
4
3
2
1
Examples
Rewrite each Scientific Notation in Standard Form without
a calculator.
a. 1.345 10  134,500,000
8
1.3 4 5 0 0 0 0 0.
1
4
3
2
7
6
5
8
9
b. 4.103 10 
0 .0 0 0 0 0 0 0 0 4.103
9
8
7
6
5
3
4
2
The – 9 moves the
0.000000004103 decimal 9 places to
the left (smaller).
1
5
c. 7.8 10  0.000078
0 .0 0 0 0 7.8
5
4
3
2
1
d. 3.67 10  3,670,000
6
3.6 7 0 0 0 0.
1
2
3
4
5
6
The + 8 moves the
decimal 8 places to
the right (bigger).
The – 5 moves the
decimal 5 places to
the left (smaller).
The + 6 moves the
decimal 6 places to
the right (bigger).
Converting Scientific Notation to
Standard Form
To convert Standard Form to Scientific Notation, find the
largest power of ten such that when it divides the
number, the result is between 1 and 10.
10
Example: Convert the following from Scientific Notation to
Standard Form. Then try to find a shortcut.
13
15,143,572,733, 215
10,000,000,000,000
 1.514
Round to 3
decimal
places
The number is made smaller by moving the
decimal 13 places (power of ten needs is positive)
10-10
0.000000000753
0.0000000001
 7.53
The number is made larger by moving the decimal
10 places (power of ten needs is negative)
1.514 10
13
.
15143572733215.
13 12 11 10 9 8 7 6 5 4 3 2 1
7.53 10
10
0.000000000753
.
1 2 3 4 5 6 7 8 9 10
Examples
Rewrite each Standard Form in Scientific Notation without
a calculator.
a. 245000  2.45 10
5
2 .4 5 0 0 0.
5
4
3
2
1
b. 1.78  1.78
Already in Scientific Notation, it is a number
greater than or equal to 1 and less than 10
c. 0.000722  7.22 10
0 .0 0 0 7. 2 2
1
2
3
Move the decimal + 5 places
to the left (positive since the
number is made smaller).
4
4 Move the decimal - 4 places
to the right (negative since
the number is made bigger).
d. 17, 439,502 1.744 10 places to the left (positive
7 Move the decimal + 7
1.7 4 3 9 5 0 2.
7
6
5
4
3
2
1
since the number is made
smaller).
Arithmetic with Scientific Notation
In order to multiply or divide numbers in Scientific Notation,
use the properties of exponents.
Examples: Simplify the following expressions
a. 1.4 10  7.6 10 
7
4
3
8
1.4  7.6  10 10   10.64 10  1.064 10
4
3
Regroup
b. 1.2 10
Simplify
   4.8 10 
1.2  4.8 10 10   0.25 10
1
4
1
Regroup
4
3
Simplify
Scientific
Notation
 2.5 10
2
Scientific
Notation
Scientific Notation and the Calculator
Most scientific calculator write in large and small numbers
in Scientific Notation. In TI calculators the letter E stands
for x10, for example
1.5 E12 = 1.5x10
12
Make sure to rewrite what the calculator displays into
correct scientific notation.
Examples: Use the calculator to write the following
expressions in Scientific Notation.
a. 78,000  2, 400,000,000
Calculator: 1.872 E14  1.872 1014
b. 0.000000748  500
Calculator: 1.496 E  9  1.496 109