SCIENTIFIC SCIENTIFIC NOTATIONNOTATION

What is it?

And

How it works?

Much of the data collected and used in Physics is either very large or very small.

When we talk about data, we are talking about the measurements or numbers used to represent what we are looking for.

Let’s look for example at the distance from the Sun to Mars.

The mean distance from the Sun to Mars is:

227 billion, 800 million meters

227 800 000 000 m

OR

How about the mass of an electron:

0.000 000 000 000 000 000 000 000 000 000 911 kg

“SCIENTIFIC NOTATION”

Because of this problem, Scientist have developed a type of short hand to work with these numbers.

2.278 x 1011 m

Do you remember the the distance from the Sun to Mars ?

Written in scientific notation, it would look like this.

227 800 000 000 m

9.11 x 10-31 kg

How about the mass of an electron ?

Written in scientific notation, it would look like this.

0.000 000 000 000 000 000 000 000 000 000 911 kg

Scientific Notation Rules:

•When moving the decimal to the left, the exponent will increase.

•When moving the decimal to the right, the exponent will decrease.

•Only one digit should be to the left of the decimal.

Convert from Scientific Notation to Real Number:

5.14 x 105 = 514000•Scientific notation consists of a coefficient (here 5.14) multiplied by 10 raised to an exponent (here 5).

•To convert to a real number, start with the coefficient and multiply by 5 tens like this: 5.14 x 10 x 10 x 10 x 10 x 10 = 514000 . Multiplying by tens is easy: one simply moves the decimal point in the base (5.14) 5 places to the right, adding extra zeroes as needed.

•Convert from Real Number to Scientific Notation:

0.000 345 = 3.45 x 10-4 •Here we wish to write the number 0.000345 as a coefficient times 10 raised to an exponent.

•To convert to scientific notation, start by moving the decimal place in the number until you have a number equal to or greater 1 and less than 10; here it is 3.45.

•We move the decimal 4 places to the right, so the

exponent decreases to -4.

•Examples: Express in Scientific Notation

1. 5800 5.8 x 103

2. 450 000 4.5 x 105

3. 86 000 000 000 8.6 x 1010

4. 0.000 508 5.08 x 10-4

5. 0.000 360 3.60 x 10-4

6. 0.004 4 x 10-3

•Examples: Express in Real Numbers

1. 6.3 x 103 6300

2. 9.723 x 109 9 723 000 000

3. 5.8 x 101 58

4. 4.75 x 10-4 0.000 475

5. 3.56 x 10-7 0.000 000 356

6. 6.3 x 10-1 0.63

Calculating with Scientific Notation

Not only does scientific notation give us a way of writing very large and very small numbers, it allows us to easily do calculations as well.  Calculators are very helpful tools, but unless you can do these calculations without them, you can never check to see if your answers make sense.   Any calculation should be checked using your logic, so don't just assume an answer is correct. 

Rule for Multiplication

When you multiply numbers with scientific notation, multiply the coefficients together and add the exponents.  The base will remain 10.

Example:

Multiply  (3 x 107) x (6 x 105)

First rewrite the problem as:    (3 x 6) x (107 x 105)

Then multiply the coefficients and add the exponents:    18 x 1012

Then change to correct scientific notation: 1.8 x 1013

Remember that correct scientific notation has a coefficient that is less than 10, but greater than or equal to one to the left of the decimal.

Rule for Division

When dividing with scientific notation, divide the coefficients and subtract the exponents.  The base will remain 10.

Example:

Divide 3.0 x 108 by 6.0 x 104

Rewrite the problem as:       3.0 x 108                                                 6.0 x 104    

Divide the coefficients and subtract the exponents to get:      0.5 x 104

Then change to correct scientific notation: 5 x 103

Remember that correct scientific notation has a coefficient that is less than 10, but greater than or equal to one to the left of the decimal.

Rule for Addition and Subtraction

When adding or subtracting in scientific notation, you must express the numbers as the same power of 10.  The exponents must match before you do any math. This will often involve changing the decimal place of the coefficient.

Example:

Add 3.5 x 104 and 5.5 x 104

Rewrite the problem as: (3.5 + 5.5) x 104

Add the coefficients and leave the base and exponent the same:  3.5 + 5.5 = 9 x 104

Example:

Add 2.75 x 104 and 5.5 x 103

First we must pick one of the factors and move the decimal to make the exponents match. Let’s change 5.5 x 103 to 0.55 x 104    

Rewrite the problems as: (2.75 + 0.55) x 104

Add the coefficients and leave the base and exponent the same:  2.75 + 0.55 = 3.3 x 104

Example:

Subtract (4.8 x 105) - (9.7 x 104)

Pick one of the factors and move the decimal to make the exponents match. Let’s change 9.7 x 104 to 0.97 x 105

Rewrite the problems as: (4.8 – 0.97) x 105

Subtract the coefficients and leave the base and exponent the same:  4.8 - 0.97 = 3.83 x 105

•Examples:

1. (3.2 x 103) + (4.8 x 103) = 8 x 103

2. (3.2 x 103) - (4.8 x 10-3) = 3.1999952 x 103

3. (5 x 103) X (12 x 104) = 6 x 108

4. (6.6 x 103) ÷ (2 x 104) = 3.3 x 10-1