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Student Academic Learning Services Page 1 of 3 Scientific Notation What is Scientific Notation? Scientific notation is simply a different way of representing numbers. It is especially useful in representing very large or very small numbers. For example, Avogadro’s number, an important physical constant, would be written as 602200000000000000000000 in decimal notation, but is written as 6.022 × 1023 in scientific notation. Numbers in Scientific Notation Any number in scientific notation consists of three components; the coefficient, the base, and the exponent. The coefficient is always a number that is less than ten and greater than or equal to one. The base is always ten. Finally, the exponent can be any integer, positive or negative, and including zero. Below are two examples of numbers in Scientific Notation, what each part corresponds to, and how it would look in normal decimal form. 3.0 x 105 1.4 x10-8 3.0 is the coefficient 1.4 is the coefficient 10 is the base 10 is the base 5 is the exponent -8 is the exponent = 300000 = 0.000000014 Converting from Decimal to Scientific Notation There are three steps involved in converting a number from decimal notation to scientific notation: Step 1. Move the decimal so that it is just to the right of the first digit, taking note of how many places you moved the decimal, and whether you moved it to the left or the right. Step 2. Put a “×10” next to your new coefficient Step 3. The exponent of the ten will be the number of places you moved the decimal in the first step. If you moved to the left, then the exponent will be positive, and if you moved to the right, then the exponent will be negative. These steps are shown below: www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/20/2011 Student Academic Learning Services Step 1 34500. Page 2 of 3 0.0004872 Counting from the decimal we have four Counting from the decimal we have digits to the left that get skipped over. four digits to the right that are skipped over. Step 2 Step 3 3.45 × 10 4.872 × 10 Putting in the base 10 Putting in the base 10 3.45 × 104 4.872 × 10-4 It is a positive exponent because the decimal moved to the left. It is a negative exponent because the decimal moved to the right. Converting from Scientific to Decimal Notation: To convert back to decimal notation, you simply move the decimal place over the number of spaces given by the exponent of ten. If the exponent is positive then you move to the right and if the exponent is negative then you move to the left. It is no different than simply evaluating the expression (i.e. using BEDMAS). 5.26 × 10-4 = 0.000526 9.85 × 108 = 985000000 Multiplying and Dividing with Scientific Notation: One of the advantages of working with numbers in scientific notation is that it makes them very easy to multiply and divide. All you have to do when multiplying and dividing numbers in scientific notation is multiply (or divide) the coefficients and then add (or subtract) the exponents. Check out the example to the right. Multiplying Dividing (3.4 × 104) × (2.0 ×105) (3.6 × 105) ÷ (1.2 × 103) = 3.4 × 2.0 × 104 × 105 4+5 = 6.8 × 10 = 6.8 × 109 = = . × . × . × . = 3 × 102 Using the exponent rules you can solve for the new exponent on the base 10. www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/20/2011 Student Academic Learning Services Page 3 of 3 Adding and Subtracting with Scientific Notation: Adding and subtracting with scientific notation is not easy. In fact, the best way to approach it is to convert both numbers to decimal notation and then add then numbers normally. After the adding is done, you can convert the sum into scientific notation again. 5.36 × 103 +1.24 × 10-2 Adding and Subtracting are very similar = 5360 + 0.0124 Convert into decimal notation = 5360.0124 Complete the operation = 5.3600124 × 103 Convert back into scientific notation Additional Examples: Note that it is important to put the answers in proper scientific notation. You may need to change your answer after you have worked out the solution as can be seen in the following two examples. (4.2 × 102) × (3.5 × 105) (4.8 × 103) / (5.3 × 10-2) = 4.2 × 3.5 × 102 × 105 = (4.8 / 5.3) × (103 / 10-2) = 14.7 × 107 Incorrect! = 0.906 × 105 Incorrect! = 1.47 × 108 Correct!! = 9.06 × 104 Correct!! Try these ones Evaluate and present your answer in scientific notation: 1) (1.44 × 1014) ¸ (2.4 × 1015) 2) (6.6 × 1027) × (5 × 10-4) ¸ (1.1 × 1020) 3) (4.591 × 105) + (3.2 × 104) 4) (5.95 × 103) - (7 × 101) Answers to exercises: 1) 6 × 10-2 2) 3.0 × 104 3) 4.911 × 105 4) 5.88 × 103 www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 12/20/2011