Skills to Develop To express a large number or a small number in scientific notation. To express a number in scientific notation to standard form. The instructions for making a pot of coffee specified 3 scoops (rather than 12,000 grounds) because any measurement is expressed more efficiently with units that are appropriate in size. In science, however, we often must deal with quantities that are extremely small or incredibly large. For example, you may have 5,000,000,000,000 red blood cells in a liter of blood, and the diameter of an iron atom is 0.000000014 inches. Numbers with many zeros can be cumbersome to work with, so scientists use scientific notation. Scientific notation is a system for expressing very large or very small numbers in a compact manner. It uses the idea that such numbers can be rewritten as a simple number multiplied by 10 raised to a certain exponent, or power. Let us look first at very large numbers. Suppose a spacecraft is 1,500,000 miles from Mars. The number 1,500,000 can be thought of as follows: That is, 1,500,000 is the same as 1.5 times 1 million, and 1 million is 10 × 10 × 10 × 10 × 10 × 10, or 10 6 (which is read as “ten to the sixth power”). Therefore, 1,500,000 can be rewritten as 1.5 times 10 6 , or 1.5 × 10 6 . The distance of the spacecraft from Mars can therefore be expressed as 1.5 × 10 6 miles. Recall that: 10 0 = 1 10 1 = 10 10 2 = 100 10 3 = 1,000 10 4 = 10,000 and so forth The standard convention for expressing numbers in scientific notation is to write a single nonzero first digit, a decimal point, and the rest of the digits, excluding any trailing zeros (see rules for significant figures in the next section for more details on what to exclude). This number is followed by a multiplication sign and then by 10 raised to the power necessary to reproduce the original number. For example, although 1,500,000 can also be written as 15. × 10 5 (which would be 15. × 100,000), the convention is to have only one digit before the decimal point. How do we know to what power 10 is raised? The power is the number of places you have to move the decimal point to the left to place it after the first digit , so that the number being multiplied is between 1 and 10 : Example \(\PageIndex{1}\): Scientific Notation Express each number in scientific notation. 67,000,000,000 1,689 12.6 Answer a Moving the decimal point 10 places to the left gives 6.7 × 10 10 . Answer b The decimal point is assumed to be at the end of the number, so moving it three places to the left gives 1.689 × 10 3 . Answer c In this case, we need to move the decimal point only one place to the left, which yields 1.26 × 10 1 . Exercise \(\PageIndex{1}\) Express each number in scientific notation. 1,492 102,000,000 101,325 Answer 1. 1.492 × 10 3 2. 1.02 × 10 8 3. 1.01325 × 10 5 To change scientific notation to standard notation , we reverse the process, moving the decimal point to the right. Add zeros to the end of the number being converted, if necessary, to produce a number of the proper magnitude. Lastly, we drop 10 and its power. Example \(\PageIndex{2}\) Express each number in standard, or conventional notation. 5.27 × 10 4 1.0008 × 10 6 Answer a Move the decimal four places to the right and add zeros to give 52,700. Answer b Moving the decimal six places to the right gives 1,000,800. Exercise \(\PageIndex{2}\) Express each number in standard, or conventional notation. a. 6.98×10 8 b. 1.005×10 2 Answer: a. 698,000,000 b. 100.5 We can also use scientific notation to express numbers whose magnitudes are less than 1. For example, the quantity 0.006 centimeters can be expressed as follows: That is, 0.006 centimeters is the same as 6 divided by one thousand, which is the same as 6 divided by 10 x 10 x 10 or 6 times 10 –3 (which is read as "ten to the negative third power"). Therefore, 0.006 centimeters can be rewritten as 6 times 10 –3 , or 6 × 10 –3 centimeters. Recall that: 10 −1 = 1/10 10 −2 = 1/100 10 −3 = 1/1,000 10 −4 = 1/10,000 10 −5 = 1/100,000 and so forth We use a negative number as the power to indicate the number of places we have to move the decimal point to the right to make it follow the first nonzero digit so that the number is between 1 and 10. This is illustrated as follows: Note: In writing scientific notations, the convention is to have only one digit before the decimal point. Numbers that are greater than one have a positive power in scientific notation. If the decimal point is moved to the left n places, the power (n) of 10 is positive. Numbers that are less than one have a negative power in scientific notation. If the decimal point is moved to the right n places, the power (n) of 10 is negative. Example \(\PageIndex{3}\) Express each number in scientific notation. 0.000006567 −0.0004004 0.000000000000123 Answer a Move the decimal point six places to the right to get 6.567 × 10 −6 . Answer b Move the decimal point four places to the right to get −4.004 × 10 −4 . The negative sign on the number itself does not affect how we apply the rules of scientific notation. Answer c Move the decimal point 13 places to the right to get 1.23 × 10 −13 . Exercise \(\PageIndex{3}\) Express each number in scientific notation. 0.000355 0.314159 −0.051204 Answer a Moving the decimal point four places to the right gives 3.55 × 10 −4 . Answer b Moving the decimal point one place to the right gives 3.14159 × 10 −1 . Answer c Moving the decimal point one place to the right gives −5.1204 × 10 −2 . As with numbers with positive powers of 10, when changing from scientific notation to standard or conventional format, we reverse the process. Note Changing a number in scientific notation to standard form: If the scientific notation has a positive power, the standard number is greater than one. Example: 8 x 10 4 = 80,000 If the scientific notation has a negative power, then the standard number is less than one. Example: 8 x 10 -2 = 0.08 Example \(\PageIndex{4}\) Change the number in scientific notation to standard form. 6.22 × 10 −2 9.9 × 10 −9 Answer a 0.0622 Answer b 0.0000000099 Exercise \(\PageIndex{4}\) Change the number in scientific notation to standard form. 9.98 × 10 −5 5.109 × 10 −8 Answer a 0.0000998 Answer b 0.00000005109 Although calculators can show 8 to 10 digits in their display windows, that is not always enough when working with very large or very small numbers. For this reason, many calculators are designed to handle scientific notation. The method for entering scientific notation differs for each calculator model, so take the time to learn how to do it properly on your calculator, asking your instructor for assistance if necessary . If you do not learn to enter scientific notation into your calculator properly, you will not get the correct final answer when performing a calculation. Concept Review Exercises Why it is easier to use scientific notation to express very large or very small numbers? What is the relationship between how many places a decimal point moves and the power of 10 used in changing a conventional number into scientific notation? Answers Scientific notation is more convenient than listing a large number of zeros. The number of places the decimal point moves equals the power of 10. The power of 10 is positive if the decimal point moves to the left and negative if the decimal point moves to the right. Key Takeaway Large or small numbers are expressed in scientific notation, which use powers of 10. Exercises Why is scientific notation useful in expressing numbers? What is the relationship between the power and the number of places a decimal point is moved when going from standard to scientific notation? Express each number in scientific notation. 0.00064 5,230,000 −56,200 0.000000000220 1.0 Express each number in scientific notation. 678 −1,061 0.000560 0.0000003003 100,000,000 Express each number in standard form. 6.72 × 10 4 2.088 × 10 −4 −3 × 10 6 9.98 × 10 −7 Express each number in standard form. 9.05 × 10 5 1.0 × 10 −3 6.022 × 10 23 8.834 × 10 −12 Complete the following table: Incorrect Scientific Notation Correct Scientific Notation 54.7 × 104 NaN 0.0066 × 103 NaN 3,078 × 100 NaN Complete the following table: Incorrect Scientific Notation Correct Scientific Notation 234.0 × 101 NaN 36 × 10−4 NaN 0.993 × 105 NaN Answers Scientific notation is more convenient than listing a large number of zeros. 2. The power is the number of places a decimal point is moved when going from standard to scientific notation. It is positive if the decimal point is moved to the left; negative if moved to the right. 6.4 × 10 −4 5.23 × 10 6 −5.62 × 10 4 2.20 × 10 −10 1.0 × 10 0 4. 6.78 × 10 2 −1.061 × 10 3 5.60 × 10 −4 3.003 × 10 −7 1 × 10 8 67,200 0.0002088 −3,000,000 0.000000998 6. Express each number in standard form. 905,000 0.0010 602,200,000,000,000,000,000,000 0.000000000008834 Incorrect Scientific Notation Correct Scientific Notation 54.7 × 104 5.47 × 105 0.0066 × 103 6.6 × 100 3,078 × 100 3.078 × 103 8. Incorrect Scientific Notation Correct Scientific Notation 234.0 × 101 2.340 × 103 36 × 10−4 3.6 × 10−3 0.993 × 105 9.93 × 104