Scientific Notation
Chemistry deals with very large and very small numbers. Consider this
calculation:
(0.000000000000000000000000000000663 x 30,000,000,000) ÷ 0.00000009116
Hopefully you can see, how awkward it is. Try keeping track of all those zeros!
In scientific notation, this problem is:
(6.63 x 10̅31 x 3.0 x 1010) ÷ 9.116 x 108
It is now much more compact, it better represents significant figures, and it is
easier to manipulate mathematically. The trade-off, of course, is that you have
to be able to read scientific notation.
This lesson shows you (1) how to write numbers in scientific notation and (2)
how to convert to and from scientific notation. As you work, keep in mind that a
number like 9.116 x 108 is ONE number (0.00000009116) represented as a number
9.116 and an exponent (108).
Format for Scientific Notation
1. Used to represent positive numbers only.
2. Every positive number X can be written as:
(1 < N < 10) x 10 some positive or negative integer
Where N represents the numerals of X with the decimal point after the first
nonzero digit.
3. A decimal point is in standard position if it is behind the first non-zero
digit. Let X be any number and let N be that number with the decimal point moved
to standard position. Then:
* If 0 < X < 1 then X = N x 10 negative number
* If 1 < X < 10 then X = N x 100
* If X > 10 then X = N x 10 positive number
4. Some examples of number three:
* 0.00087 becomes 8.7 x 10-4
* 9.8 becomes 9.8 x 100 (the 100 is seldom written)
* 23,000,000 becomes 2.3 x 107
5. Some more examples of number three:
* 0.000000809 becomes 8.09 x 10-7
* 4.56 becomes 4.56 x 100
* 250,000,000,000 becomes 2.50 x 1011
Note that standard position for the decimal place is always just to the right of
the first non-zero digit in the number. Also, it is the first non-zero digit
counting from the left of the number. Another way to remember standard position
is that it will always produce a number between 1 and 10. For example 45.91 x
107 is not in correct scientific notation. However, the ChemTeam wishes to
stress that it is a correct number, it is just not in scientific notation.
As a rule of thumb, you generally need not convert numbers where the absolute
value exponent will be 3 or less. However, exceptions do exist and this is only
practice in the ChemTeam's classroom. Your instructor may have a different
standard for you to obey.
Example #1 - Convert 29,190,000,000 to scientific notation.
The answer will be written assuming four significant figures. However, if you
are not sure of significant figures, don't worry - you'll get to it.
The solving process is simply one of factoring this number, but in a particular
way. For example, 5,838,000,000 times 5 is not the correct way, even though this
is a correct factoring of the original number.
First Explanation
Step 1 - start at the decimal point of the original number and count the number
of decimal places you move, stopping to the right of the first non-zero digit.
Remember that's the first non-zero digit counting from the left.
Step 2 - The number of places you moved (10 in this example) will be the
exponent. If you moved to the left, it's a positive value. If you moved to the
right, it's negative.
The answer is 2.919 x 1010.
Second Explanation
Step 1 - Write all the significant digits down with the decimal point just to
the right of the first significant digit. Like this: 2.919. Reminder: be aware
that this process should ALWAYS result in a value between 1 and 10.
Step 2 - Now count how many decimal places you would move from 2.919 to recover
the original number of 29,190,000,000. The answer in this case would be 10
places to the RIGHT. That is the number 10,000,000,000. Written in exponential
notation, it would be 1010.
To emphasize the factoring idea, we would have this:
2.919 x 10,000,000,000 = 29,190,000,000
Step 3 - Write 2.919 times the other number, BUT, write the other number as a
power of 10. The number of decimal places you counted gives the power of ten. In
this example, that power would be 10 also. The correct answer to this step is:
2.919 x 1010
Please note the the value of the exponent is positive, because you counted to
the RIGHT in step 2.
It may help to think of scientific notation as simply factoring a number, only
you are following rules which dictate how to write the two factors. The first
factor is always between one and ten, while the second factor is always some
power of 10.
Example 2 - Write 0.00000000459 in scientific notation.
Step 1 - Write all the significant digits down with the decimal point just to
the right of the first significant digit. Like this: 4.59. Please be aware that
this process should ALWAYS result in a value between 1 and 10.
Step 2 - Now count how many decimal places you would move from 4.59 to recover
the original number of 0.00000000459. The answer in this case would be 9 places
to the LEFT. That is the number 0.000000001. Be aware that this number in
exponential notation is 10̅9.
To emphasize the factoring idea, we would have this:
4.59 x 0.000000001 = 0.00000000459
Step 3 - Write 4.59 times the other number, BUT, write the other number as a
power of 10. The number of decimal places you counted gives the power of ten. In
this example, that power would be 9. The correct answer to this step is: 4.59 x
10-9
Please note the the value of the exponent is negative, because you counted to
the LEFT in step 2.
Keep in mind two important ideas when converting to scientific notation: how
many decimal places did you move and in what direction. Both of these affect the
power of ten. Also keep in mind that your answer in scientific notation will
always equal the original value. Suprising as it may seem, students in the
ChemTeam classroom make this elementary mistake.
Now, convert both of these to scientific notation, then click the value to see
the answer and an explanation.
35,800,000,000,000
0.00000000821
Suppose the number to be converted looks something like scientific notation, but
it really isn't. For example, look carefully at the example below. Notice that
the number 428.5 is not a number between 1 and 10. Although writing a number in
this fashion is perfectly OK, it is not in standard scientific notation. What
would it look like when converted to standard scientific notation?
Example #3 - Convert 428.5 x 109 to scientific notation.
Step 1 - convert the 428.5 to scientific notation. (The lesson up to this point
has been covering how to do just this step). Answer = 4.285 x 102.
Step 2 - write out the new number. Answer = 4.285 x 102 x 109.
Step 3 - combine the exponents according to the usual rules for exponents.
Answer = 4.285 x 1011.
You don't know the rules for exponents. Click for a brief note on the rules.
Example #4 - convert 208.8 x 10-11 to scientific notation.
Step 1 - convert the 208.8 to scientific notation. Answer = 2.088 x 102.
Step 2 - write out the new number. Answer = 2.088 x 102 x 10-11.
Step 3 - combine the exponents according to the usual rules for exponents.
Answer = 2.088 x 10-9.
Now, convert both of these to scientific notation, then click the value to see
the answer and an explanation.
0.000531 x 1014
0.00000306 x 10-17
1. When converting a number greater than one (the 428.5 and the 208.8 in the
previous examples), the resulting exponent will become more positive (11 is more
positive than 9 while -9 is more positive than -11).
2. When converting a number less than one (the 0.000531 and the 0.00000306 in
the previous examples), the resulting exponent will always be more negative (10
is more negative than 14 and -23 is more negative than -17).
Another way to put it:
If the decimal point is moved to the left, the exponent goes up in value
(becomes more positive).
If the decimal point is moved to the right, the exponent goes down in value
(becomes more negative).
Practice Problems
Convert to scientific notation:
1) 28,000,000
2) 305,000
3) 0.000000463
4) 0.000201
5) 3,010,000
6) 0.000000000000057
7) 20,100
8) 0.00025
9) 65,000,000,000,000,000
10) 8.54 x 1012
11) 2101 x 10-16
12) 305.1 x 107
13) 0.0000594 x 10-16
14) 0.00000827 x 1019
15) 386 x 10-22
16) 2511 x 1012
17) 0.000482 x 10-12
18) 0.0000321 x 1012
19) 288 x 105
20) 4.05 x 1011
