As my readers know, I am in pursuit of a better understanding of chemistry in order to be intelligent in conversations about this science with my granddaughter who is in the Cambridge AICE Diploma program at her high school. I started to learn about scientific measurements as part of my pursuit and realized that I did not understand something as simple as scientific notation. It looked complex, but it had to be easy. It is. This is what I learned and what you or your children should know about scientific notation and using notation in mathematical calculations of addition, subtraction, multiplication and division. The basics!

**Scientific notation:** Chemistry requires writing numbers that are very long to show how large or how small a item is. So many zeros! That is when scientific notation becomes important and makes life a lot easier. Calculators understand scientific notations which always have this format: a x 10b (*a* times 10 raised to the power of *b*) and -a x 10b

b is an integer that explains how many times the decimal point was moved in the notation

a is the coefficient and is any real number (a/k/a significand or mantissa)

– proceeds a if the number is negative

**Creating a scientific notation:**

1) Convert 3,760,000,000.0 into notation form. Large numbers require a left move. Move the mighty decimal point to the left until you have a number between 1 and 10; or in this case 3.76.

2) Add a power of ten exponent to show how many places you moved that decimal point. In this case, we moved it 9 places to the left, or 3.76 x 109. This shows that the number 3.76 should be multiplied by 109 to get back to the long number format. This looks like the following: 3.76 x 109 = 3.76 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10, or 3,760,000,000

Example for a teensy weensy number that is less than 1:

1) Our example is 0.0000054. Small numbers require a right move. Move that little decimal point to the right this time until you have a number that is between 1 and 10. In this case, we get 5.4

3) How many places to the right did you move the decimal point? Six. So, your scientific notation is going to be 10-6. Note that this exponent number is minus 6 and not positive 6! So we have 5.4 x 10-6.

**Some other examples from wikipedia**:

2×100 is 2. We did not move the decimal point at all.

3×102 is 300.0 because we moved the decimal point to the left 2 places.

6.72×109 is 6,720,000,000. We moved the decimal point to the left 9 times.

6.72×10-9 is 0.00000000672. We moved the point to the right 9 times.

-5.3×104 is negative 53,000 or -53,000 with the point moved 4 places to the left.

**Can we do math with scientific notation numbers?** Yes!

**How to do the math when the exponents are identical:**

**Addition with same exponents:** 5.4×105 + 5.3x 105

Add the 5.4 and the 5.3 to get 10.7, or 10.7×105

The exponent remains unchanged.

Note: Another way you could do this is through distributive properties. This would then be 105(5.4) + 105(5.3) = 105(5.4+5.3) = 105(10.7) = 10.7×105

Convert to scientific notation. You want only one digit before your decimal point so the base number is rewritten as 1.07 x 106

**Subtraction with same exponents:** 5.4×105 – 5.3x 105

Subtract 5.3 from 5.4 to get .1, or .1×105

Note: Another way you could do this is through distributive properties. This would then be 105(5.4) – 105(5.3) = 105(5.4+5.3) = 105(.1) = .1×105

You want only one digit before your decimal point so the base number is rewritten as 1 x 104

because you subtracted one from your exponent for correct placement of the decimal.

**Multiplication with the same exponents**: 5.4×105 times 5.3x 105

(5.4×105)(5.3x 105) is 5.4 times 5.3 = 28.62

Your exponents are still 5, so you get 28.62×105

**Division with same exponents:** 5.4×105 divided by 3×1011

(5.4×105)/(3x 1011) is 5.4 divided by 3 = 1.8

When dividing, subtract the exponents, or 11-5 is 6 or 106

The answer is 1.8×106

**How to do the math when the exponents are different:**

**Addition with different exponents:** 5.4×105 + 5.3x 107

First we will change the exponents so they are the same (7) which means we move the decimal point in this example to the left. Now we have .054x 107.

We then rewrite the equation to a distributive property form:

107( .054) + 107( 5.3), or 107( .054+5.3) = 107(5.354).

In scientific notation the answer is now 5.354×107

**Subtraction with different exponents:** 5.4×105 + 5.3x 107

First we will change the exponents so they are the same (7) which means we move the decimal point in this example to the left. Now we have .054x 107.

We then rewrite the equation to a distributive property form:

107( .054) – 107( 5.3), or 107( .054-5.3) = 107(5.246).

In scientific notation the answer is now 5.246×107

**Multiplication with different exponents:** 5.4×105 x 5.3x 10-13

First we subtract the exponents. This gives us an exponent of -8.

Then we multiple the base integers of 5.4 and 5.3 and get 10.7 so our distributive form is 10-8 (5.4) times 10-8 (5.3) or 10-8 (5.4×5.3) or 10-8 (10.7)

Remembering that we want only one digit to the left of the decimal point we reformat to scientific notation and get 1.07 x 101-8 or 1.07 x 10-7

**Division with different exponents:** 5.4×105 – 5.3x 107

First we subtract the exponents. This gives us an exponent of 2.

Then we divide the base integers of 5.4 and 5.3 and get .1 so our distributive form is 102 (5.4) / 102 (5.3) or 102 (5.4/5.3) or 102 (.1)

Remembering that we want one digit to the left of the decimal point we reformat to scientific notation and get .1 x 102 or 1.0 x 101

**Multiplication of positive and negative exponents:** 5.4×105 times 5.3x 10-9 (5.4×105)(5.3x 10-9) is 5.4 times 5.3 = 28.62

Your exponents are still 5 and -9 which you simply add to get -2 or 28.62 x 10-4

But in scientific notation you want only one digit before your decimal point so this is rewritten as 2.862 x 10-3

Wow. That got longer than I anticipated, but now I better understand scientific notation and how to use it in math calculations. Do you?

**Sources:**

Fact Monster

Wikipedia

Math Warehouse

Brain Pop (has a calculator for converting exponential forms)

About.com Chemistry

http://jc-schools.net/