SCIENTIFIC NOTATION 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 m OR

How about the mass of an electron: kg

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

2.278 x m Do you remember the the distance from the Sun to Mars ? Written in scientific notation, it would look like this m

9.11 x kg How about the mass of an electron ? Written in scientific notation, it would look like this 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 10 5 = 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 = 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: = 3.45 x Here we wish to write the number 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 We move the decimal 4 places to the right, so the exponent decreases to -4.

Examples: Express in Scientific Notation x x x x x x 10 -3

Examples: Express in Real Numbers x x x x x x

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 10 7 ) x (6 x 10 5 ) First rewrite the problem as: (3 x 6) x (10 7 x 10 5 ) Then multiply the coefficients and add the exponents: 18 x Then change to correct scientific notation: 1.8 x 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 10 8 by 6.0 x 10 4 Rewrite the problem as: 3.0 x x 10 4 Divide the coefficients and subtract the exponents to get: 0.5 x 10 4 Then change to correct scientific notation: 5 x 10 3 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 10 4 and 5.5 x 10 4 Rewrite the problem as: ( ) x 10 4 Add the coefficients and leave the base and exponent the same: = 9 x 10 4

Example: Add 2.75 x 10 4 and 5.5 x 10 3 First we must pick one of the factors and move the decimal to make the exponents match. Let’s change 5.5 x 10 3 to 0.55 x 10 4 Rewrite the problems as: ( ) x 10 4 Add the coefficients and leave the base and exponent the same: = 3.3 x 10 4

Example: Subtract (4.8 x 10 5 ) - (9.7 x 10 4 ) Pick one of the factors and move the decimal to make the exponents match. Let’s change 9.7 x 10 4 to 0.97 x 10 5 Rewrite the problems as: (4.8 – 0.97) x 10 5 Subtract the coefficients and leave the base and exponent the same: = 3.83 x 10 5

Examples: 1.(3.2 x 10 3 ) + (4.8 x 10 3 ) = 8 x (3.2 x 10 3 ) - (4.8 x ) = x (5 x 10 3 ) X (12 x 10 4 ) = 6 x (6.6 x 10 3 ) ÷ (2 x 10 4 ) = 3.3 x 10 -1