Download Scientific Notation

Student Academic Learning Services
Page 1 of 3
Scientific Notation
What is Scientific Notation?
Scientific notation is simply a different way of representing numbers. It is especially useful
in representing very large or very small numbers. For example, Avogadro’s number, an
important physical constant, would be written as 602200000000000000000000 in decimal
notation, but is written as 6.022 × 1023 in scientific notation.
Numbers in Scientific Notation
Any number in scientific notation consists of three components; the coefficient, the base, and
the exponent. The coefficient is always a number that is less than ten and greater than or
equal to one. The base is always ten. Finally, the exponent can be any integer, positive or
negative, and including zero. Below are two examples of numbers in Scientific Notation,
what each part corresponds to, and how it would look in normal decimal form.
3.0 x 105
1.4 x10-8
3.0 is the coefficient
1.4 is the coefficient
10 is the base
10 is the base
5 is the exponent
-8 is the exponent
= 300000
= 0.000000014
Converting from Decimal to Scientific Notation
There are three steps involved in converting a number from decimal notation to scientific
notation:
Step 1. Move the decimal so that it is just to the right of the first digit, taking note of how
many places you moved the decimal, and whether you moved it to the left or the right.
Step 2. Put a “×10” next to your new coefficient
Step 3. The exponent of the ten will be the number of places you moved the decimal in the
first step. If you moved to the left, then the exponent will be positive, and if you moved to
the right, then the exponent will be negative.
These steps are shown below:
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/20/2011
Student Academic Learning Services
Step 1
34500.
Page 2 of 3
0.0004872
Counting from the decimal we have four Counting from the decimal we have
digits to the left that get skipped over.
four digits to the right that are skipped
over.
Step 2
Step 3
3.45 × 10
4.872 × 10
Putting in the base 10
Putting in the base 10
3.45 × 104
4.872 × 10-4
It is a positive exponent because the
decimal moved to the left.
It is a negative exponent because the
decimal moved to the right.
Converting from Scientific to Decimal Notation:
To convert back to decimal notation, you simply move the decimal place over the number of
spaces given by the exponent of ten. If the exponent is positive then you move to the right
and if the exponent is negative then you move to the left. It is no different than simply
evaluating the expression (i.e. using BEDMAS).
5.26 × 10-4 = 0.000526
9.85 × 108 = 985000000
Multiplying and Dividing with Scientific Notation:
One of the advantages of working with numbers in scientific notation is that it makes them
very easy to multiply and divide. All you have to do when multiplying and dividing numbers
in scientific notation is multiply (or divide) the coefficients and then add (or subtract) the
exponents. Check out the example to the right.
Multiplying
Dividing
(3.4 × 104) × (2.0 ×105)
(3.6 × 105) ÷ (1.2 × 103)
= 3.4 × 2.0 × 104 × 105
4+5
= 6.8 × 10
= 6.8 × 109
=
=
. ×
. ×
.
×
.
= 3 × 102
Using the exponent rules you can solve for the new exponent on the base 10.
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/20/2011
Student Academic Learning Services
Page 3 of 3
Adding and Subtracting with Scientific Notation:
Adding and subtracting with scientific notation is not easy. In fact, the best way to approach
it is to convert both numbers to decimal notation and then add then numbers normally. After
the adding is done, you can convert the sum into scientific notation again.
5.36 × 103 +1.24 × 10-2
Adding and Subtracting are very similar
= 5360 + 0.0124
Convert into decimal notation
= 5360.0124
Complete the operation
= 5.3600124 × 103
Convert back into scientific notation
Additional Examples:
Note that it is important to put the answers in proper scientific notation. You may need to
change your answer after you have worked out the solution as can be seen in the following
two examples.
(4.2 × 102) × (3.5 × 105)
(4.8 × 103) / (5.3 × 10-2)
= 4.2 × 3.5 × 102 × 105
= (4.8 / 5.3) × (103 / 10-2)
= 14.7 × 107
Incorrect!
= 0.906 × 105
Incorrect!
= 1.47 × 108
Correct!!
= 9.06 × 104
Correct!!
Try these ones
Evaluate and present your answer in scientific notation:
1) (1.44 × 1014) ¸ (2.4 × 1015)
2) (6.6 × 1027) × (5 × 10-4) ¸ (1.1 × 1020)
3) (4.591 × 105) + (3.2 × 104)
4) (5.95 × 103) - (7 × 101)
Answers to exercises:
1) 6 × 10-2
2) 3.0 × 104
3) 4.911 × 105
4) 5.88 × 103
www.durhamcollege.ca/sals
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 12/20/2011