Scientific Notation

Purpose of Scientific Notation

• We often deal with very large and very small numbers in chemistry.– For example

• There 126600000000000000000000000 individual water molecules in a gallon of water.

• A single molecule of water has a diameter of approximately 0.0000000003 meters

• It is easier to deal with these numbers in scientific notation– For example

• There are 1.266 x 1026 individual water molecules in a gallon of water.

• A single molecule of water has a diameter of approximately 3 x 10-10 meters.

Scientific Notation

• With scientific notation, you have a number and an exponential term.– Let’s take 3000.

• 3000 can be thought of as 3 x 1000• 1000 is equivalent to 10x10x10, which is 103.• This gives us 3 x 103

– Lets’ take 0.003• 0.003 can be though of as 3 / 1000• 1000 is equivalent to 10x10x10, which is 103.• Substituting, we get 3 / 103

• This is the same as 3 x 10-3

Note: when dealing with a number raised to a power in the denominator of a fraction, you can bring it up and multiply it by the numerator, but you must change the sign of the exponent from positive to negative or negative to positive.

Some more examples

• Let’s do a couple more examples:– 5820000000

• This is equivalent to 5.82 x 1000000000• 1000000000 is 10x10x10x10x10x10x10x10x10 or 109

• So, 5820000000 is 5.82 x 109

– 0.0000746• This is equivalent to 7.46 / 100000• 100000 is 10x10x10x10x10 or 105

• So, 0.0000746 is 7.46 / 105 or 7.46 x 10-5

The easy way to get into and out of scientific notation is to move the decimal.

• Let’s revisit an earlier number 5820000000.– To get into scientific notation we need a single non-zero digit to the

left of the decimal. To achieve this, we move the decimal 9 spots to the left:• 5820000000. to give us 5.82.• Since we moved 9 spots over we multiply by 109

• This gives us 5.82 x 109

• If we move the decimal the other way, we get a negative exponent.– 0.0000746 gives us 7.46 x 10-5 since we shifted the decimal 5 spots to

the right.

Pause and Practice

• What are each of the following in scientific notation:– 780500– 0.0068– 0.0000000185– 960000047

• What are each of the following in standard notation:– 6.74 x 10-6

– 1.74 x 109

– 3.85 x 1012

– 4.69 x 10-2

Pause and Practice Answers

• What are each of the following in scientific notation:– 780500 = 7.805 x 105

– 0.0068 = 6.8 x 10-3

– 0.0000000185 = 1.85 x 10-8

– 960000047 = 9.60000047 x 108

• What are each of the following in standard notation:– 6.74 x 10-6 = 0.00000674– 1.74 x 109 = 1740000000– 3.85 x 1012 = 3850000000000– 4.69 x 10-2 = 0.0469

Calculations with Scientific Notation on the Calculator

• Find one of the following buttons on your scientific calculator:• EE or EXP - one of these is preferred• x 10n - not preferred but acceptable if EE or EXP are not present

• Using EE or EXP– Perform the calculation 1 x 104 / 2 x 106.

• Enter – Result will be 0.005 or 5 x 10-3

• Using x 10n

– Perform the calculation 1 x 104 / 2 x 106.• Enter

– Result will be 0.005 or 5 x 10-3

– Note: As you can see, using the x10n button requires the use of parentheses. If you do not use parentheses, you will not get the correct answer.

Misconception Alert!

• Most mistakes with scientific notation are one of the following:– Students like to use x10 and EE. Do not use both. Type in 2 EE 3 or 2 x

103 but do not do both!– Students do not use parentheses with x 10n

• 1 x 104 / 2 x 106 will give 5 x 109 if parentheses are not used.– Some students learned to input scientific notation by typing in x 10 ^.

There is no reason to do this if you are using a scientific calculator since it has an EE, EXP or x10n button.

Pause and Practice

• Complete the following calculations:

Pause and Practice Answers

• Complete the following calculations:

Now try the exercises