Scientific Notation Rules

Scientific notation is simply a method for expressing, and working with, very large or very small numbers. It is a short hand method for writing numbers, and an easy method for calculations. Numbers in scientific notation are made up of three parts: the coefficient, the base and the exponent. Observe the example below: 5.67 x 105 This is the scientific notation for the standard number, 567 000. Now look at the number again, with the three parts labeled.5.67 x 105 coefficient base exponent In order for a number to be in correct scientific notation, the following conditions must be true: 1. The coefficient must be greater than or equal to 1 and less than 10.2. The base must be 10.3. The exponent must show the number of decimal places that the decimal needs to be moved to change the number to standard notation. A negative exponent means that the decimal is moved to the left when changing to standard notation. Changing numbers from scientific notation to standard notation Ex.1 Change 6.03 x 107 to standard notation remember, 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 so, 6.03 x 107 = 6.03 x 10 000 000 = 60 300 000 answer = 60 300 000 Instead of finding the value of the base, we can simply move the decimal seven places to the right because the exponent is 7. So, 6.03 x 107 = 60 300 000 Now let us try one with a negative exponent Ex.2 Change 5.3 x 10-4 to standard notation The exponent tells us to move the decimal four places to the left. so, 5.3 x 10-4 = 0.00053 Changing numbers from standard notation to scientific notation Ex.1 Change 56 760 000 000 to scientific notation Remember, the decimal is at the end of the final zero. The decimal must be moved behind the five to ensure that the coefficient is less than 10, but greater than or equal to one. The coefficient will then read 5.676 The decimal will move 10 places to the left, making the exponent equal to 10. Answer equals 5.676 x 1010 Now we try a number that is very small. Ex.2 Change 0.000000902 to scientific notation The decimal must be moved behind the 9 to ensure a proper coefficient. The coefficient will be 9.02 The decimal moves seven spaces to the right, making the exponent -7 Answer equals 9.02 x 10-7 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. This page will explain the rules for calculating with scientific notation. Rule for Multiplication - When you multiply numbers with scientific notation, multiply the coefficients together and add the exponents. The base will remain 10. Ex 1 - Multiply (3.45 x 107) x (6.25 x 105) ,[object Object]

Then multiply the coefficients and add the exponents: 21.5625 x 1012

Then change to correct scientific notation and round to correct significant digits: 2.16 x 1013NOTE - we add one to the exponent because we moved the decimal one place to the left. Remember that correct scientific notation has a coefficient that is less than 10, but greater than or equal to one. Ex. 2 - Multiply (2.33 x 10-6) x (8.19 x 103) ,[object Object]

Then multiply the coefficients and add the exponents: 19.0827 x 10-3

Scientific Notation Rules

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Scientific Notation Rules