Scientific Notation

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 10¯31 x 3.0 x 1010) ÷ 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.

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).

Format for Scientific Notation

1. Used to represent positive numbers only.

2. Every positive number X can be written as:

(1 < N < 10) x 10some 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 10negative number

·  If 1 < X < 10 then X = N x 100

·  If X > 10 then X = N x 10positive 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 10¯7 is not in correct scientific notation. However, the Chemists 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 Chemists'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 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 Chemists classroom make this elementary mistake.

Now, convert both of these to scientific notation, and then click the value to see the answer and an explanation.

35,800,000,000,000

The answer is 3.58 x 1013

1. Write all significant figures as a number between 1 and 10. Do this by putting a decimal point just to the right of the first non-zero digit.

2. Count the number of decimal places needed to get back to the original starting number, while noting the direction you must move to do that.

3. Write step one times step two, but write step two in exponential notation. The number of decimal places moved gives the exponent value and the direction moved gives the sign (left = positive; right = negative).

Note that the exponent indicates HOW MANY places the decimal point was moved and the sign of the exponent indicates the DIRECTION of the movement.

There were then two graphics showing with arrows the movement of the decimal point.

In this problem, the decimal point was moved 13 places to the LEFT (positive exponent).

0.00000000821

The answer is 8.21 x 10¯9

1. Write all significant figures as a number between 1 and 10. Do this by putting a decimal point just to the right of the first non-zero digit.

2. Count the number of decimal places needed to get back to the original starting number, while noting the direction you must move to do that.

3. Write step one times step two, but write step two in exponential notation. The number of decimal places moved gives the exponent value and the direction moved gives the sign (left = positive; right = negative).

Note that the exponent indicates HOW MANY places the decimal point was moved and the sign of the exponent indicates the DIRECTION of the movement.

In this problem, the decimal point was moved 9 places to the RIGHT (negative exponent).

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.

Exponent Rules

1) When exponents are multiplied, you add them.

103 x 102 = 105

2) When exponents are divided, you subtract them.

103 ÷ 102 = 101

3) When parenthesis are involved, as in the example, you multiply.

(103)2 = 106

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

Convert 0.000531 x 1014 to scientific notation.

Step 1 - convert the 0.000531 to scientific notation. Answer = 5.31 x 10¯4.

Step 2 - write out the new number. Answer = 5.31 x 10¯4 x 1014.

Step 3 - combine the exponents according to the usual rules for exponents. Answer = 5.31 x 1010.

0.00000306 x 10¯17

Convert 0.00000306 x 10¯17 to scientific notation.

Step 1 - convert the 0.00000306 to scientific notation. Answer = 3.06 x 10¯6.

Step 2 - write out the new number. Answer = 3.06 x 10¯6 x 10¯17.

Step 3 - combine the exponents according to the usual rules for exponents. Answer = 3.06 x 10¯23.

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

The answers:

1) 2.8 x 107

2) 3.05 x 105

3) 4.63 x 10¯7

4) 2.01 x 10¯4

5) 3.01 x 106

6) 5.7 x 10¯14

7) 2.01 x 104

8) 2.5 x 10¯4

9) 6.5 x 1016

10) already in scientific notation

11) 2.101 x 10¯13

12) 3.051 x 109

13) 5.94 x 10¯21

14) 8.27 x 1013

15) 3.86 x 10¯20

16) 2.511 x 1015

17) 4.82 x 10¯16

18) 3.21 x 107

19) 2.88 x 107

20) already in scientific notation

Math with Scientific Notation

Addition and Subtraction

Speaking realistically, the problems discussed below can all be done on a calculator. However, you need to know how to enter values into the calculator, read your calculator screen, and round off to the proper number of significant figures. Your calculator will not do these things for you.

All exponents MUST BE THE SAME before you can add and subtract numbers in scientific notation. The actual addition or subtraction will take place with the numerical portion, NOT the exponent.

The student might wish to re-read the above two sentences with emphasis on the emphasized portions.

It might be advisable to point out again - DO NOT, under any circumstances, add the exponents.

Example #1: 1.00 x 103 + 1.00 x 102

A good rule to follow is to express all numbers in the problem in the highest power of ten.

Convert 1.00 x 102 to 0.10 x 103, then add:

1.00 x 103

+ 0.10 x 103

= 1.10 x 103

Example #2: The significant figure issue is sometimes obscured when numbers are in scientific notation. For example, add the following four numbers:

(4.56 x 106) + (2.98 x 105) + (3.65 x 104) + (7.21 x 103)

When the four numbers are written in the highest power, we get:

4.56 x 106

0.298 x 106

0.0365 x 106

+ 0.00721 x 106

= 4.90171 x 106

The answer upon adding must be rounded to 2 significant figures to the right of the decimal point, thus giving 4.90 x 106 as the correct answer.

Generally speaking, you can simply enter the numbers into the calculator and let the calculator keep track of where the decimal portion is. However, you must then round off the answer to the correct number of significant figures.

Lastly, be warned about using the calculator. Students often push buttons without understanding the math behind what they are doing. Then, when the teacher questions their work, they say "Well, that's what the calculator said!" As if the calculator is to blame for the wrong answer. Remember, it is your brain that must be in charge and it is you that will get the points deducted for poor work, not the calculator.