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Chemistry deals with very large and very small numbers. Consider this calculation with the numbers in standard notation:

(0.000000000000000000000000000000663 x 31,000,000,000) ÷ 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 10¯^{31}x 3.1 x 10^{10}) ÷ 9.116 x 10¯^{8}

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 as well as convert between scientific notation and standard notation (also called floating-point 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 10¯^{8} is ONE number (0.00000009116) represented as a number 9.116 and an exponent (10¯^{8}).

1. Scientific notation is used to represent positive numbers only. On the Internet, I have come across occasional mentions that scientific notation can be used for negative numbers so, if your teachers claims this is so, jst go with it and not protest that some guy on the Internet (me!) diagrees.

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. For example, 934,000 would, in scientific notation, wind up with the decmal point between the 9 and the 3.

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 10
^{0} - 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 10
^{0}(the 10^{0}is seldom written) - 23,000,000 becomes 2.3 x 10
^{7}

5. Some more examples of number three:

- 0.000000809 becomes 8.09 x 10¯
^{7} - 4.56 becomes 4.56 x 10
^{0} - 250,000,000,000 becomes 2.5 x 10
^{11}

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 10¯^{7} is not in correct scientific notation. (However, the ChemTeam wishes to stress that 45.91 x 10¯^{7} 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 of the exponent will be 3 or less. However, exceptions do exist (and are usually context-based, so no rules on this; your experience will come into play). Your instructor may guidelines for you to obey that are different from the rule of thumb I just described.

**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 10^{10}

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 10^{10}.

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 10^{10}

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 10¯^{9}.

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.

**Example #3:** 35,800,000,000,000

**Example #4:** 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 #5:** Convert 428.5 x 10^{9} 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 10^{2}.

Step 2 - write out the new number. Answer = 4.285 x 10^{2} x 10^{9}.

Step 3 - combine the exponents according to the usual rules for exponents. Answer = 4.285 x 10^{11}.

You don't know the rules for exponents? Click for a brief note on the rules.

**Example #6:** Convert 208.8 x 10¯^{11} to scientific notation.

Step 1 - convert the 208.8 to scientific notation. Answer = 2.088 x 10^{2}.

Step 2 - write out the new number. Answer = 2.088 x 10^{2} 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.

**Example #7:** 0.000531 x 10^{14}

**Example #8:** 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).

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 10^{12}

11) 2101 x 10¯^{16}

12) 305.1 x 10^{7}

13) 0.0000594 x 10¯^{16}

14) 0.00000827 x 10^{19}

15) 386 x 10¯^{22}

16) 2511 x 10^{12}

17) 0.000482 x 10¯^{12}

18) 0.0000321 x 10^{12}

19) 288 x 10^{5}

20) 4.05 x 10^{11}