Scientific Notation – Tutorial
Converting From Decimal Notation to Scientific Notation
Science deals with both very large and very small numbers. For example, the diameter of the Earth is about 13,000,000 meters. The radius of a hydrogen atom is 0.00000000012 meters. Consequently,scientists use a "shorthand" way (scientific or exponential notation) to write very large or very small numbers. Scientific notation expresses numbers as powers of ten. For example:
1 = 100 0.1 = 1/10 = 10-1 10 = 101 0.01 = 1/100 = 10-2 100 = 102 0.001 = 1/1000 = 10-3 1000 = 103 0.0001 = 1/10,000 = 10-4 10,000 = 104 100,000 = 105 1,000,000 = 106
In general, any number X can be written as the product of another number N and a power of ten. It's important to remember that 1 < N <10. In other words, N MUST be at least 1 but less than 10.
The general format for a number written in scientific notation will be:
N x 10power
Examples: 20 = 2 x 10 = 2 x 101 3500 = 3.5 x 1000 = 3.5 x 103 0.0055 = 5.5 x 0.001 = 5.5 x 10-3
Converting a Number into Proper Scientific Notation
Find all of the significant figures in the number. Re-write those digits as a number with 1 digit in front of the decimal point and the rest of the digits after the decimal point (i.e. as a number greater than or equal to 1 but less than 10)
Look at the new number you have written. Count the number of places you must move the decimal point in order to get back to where the decimal point was originally located.
If you have to move the decimal point to the right to get the original number, then write the exponent as a positive number. If you have to move the decimal point to the left to get the original number, then write the exponent as a negative number.
Examples:
Write 22,650,000 in proper scientific notation:
Write all significant figures as a number > 1 but <10: 22,650,000 = 2.265 x 10? To get back to the original number the decimal place must be moved 7 places to the right so the exponent will be positive 7. 22,650,000 = 2.265 x 107.
Write 0.0004050 in proper scientific notation:
Write all significant figures as a number > 1 but < 10: 0.0004050 = 4.050 x 10? To get back to the original number, the decimal place must be moved 4 places to the left so the exponent will be negative 4. 0.0004050 = 4.050 x 10-4
Practice Problems
Express the following numbers using proper scientific notation.
1. 13,000,000 2. 7500.3 3. 209,000 4. 0.00970 5. 0.0000605 6. 0.00300 Answers to Practice Problems:
1. 1.3 x 107 2. 7.5003 x 103 3. 2.09 x 105 4. 9.70 x 10-3 Notice that the trailing zero is kept because it is a signficant figure. 5. 6.05 x 10-5 6. 3.00 x 10-3 Notice that both trailing zeros are kept because they are significant figures. Converting from Scientific Notation to Decimal Notation.
In order to convert a number written in scientific notation to one written in standard or decinal notation, follow these steps.
1. Write the number down without the "x 10power" part. 2. Use the sign and numerical value of the exponent (power) to determine the direction and number of places to move the decimal place.
Move the decimal point to the right if the exponent is positive. Move the decimal point to the left if the exponent is negative.
Remember, numbers with an exponent of 0 are between 1 and 10. Numbers with a positive exponent are greater than or equal to 10 while those with a negative exponent are between zero and 1.
Examples:
Convert 6.53 x 104 into decimal (standard) notation.
Write the number without the "x 104" and add some extra zeros after in order to move the decimal point. 6.53 x 104 becomes 6.5300 Since the exponent is positive, move the decimal 4 places to the right. 6.53 x 104 becomes 65300 Notice that the decimal place doesn't actually appear in this case; it is understood to be at the end of the number. In science, however, placing a decimal point after the last zero in a number greater than or equal to 10 indicates that the zeros are significant figures. If I wrote the number as 65300., then I would have changed the number of signficant figures from 3 in the original number to 5.
Convert 2.50 x 10-3 into decimal (standard) notation.
Write the number without the "x 10-3" part and put some extra zeros in front of the number. 2.50 x 10-3 becomes 0002.50 Since the exponent is negative, move the decimal 3 places to the left. 2.50 x 10-3 becomes 0.00250 Remember that the number should have the same number of significant figures as the original number. Practice Problems
Convert the following numbers from scientific notation to standard (decimal) notation.
1. 3.0900 x 103 2. 6.55 x 10-5 3. 2.455 x 102 4. 1.9 x 10-4 5. 8.008 x 102 6. 2.05 x 10-3 Answers to Practice Problems
1. 3090.0 Notice that the number written in scientific notation included a positive exponent. Therefore, the decimal was moved to the right and a number greater than 10 was obtained. Also notice that the original number had 5 significant figures so my answer must also have 5 signficant figures. 2. 0.0000655 Notice that the number written in scientific notation included a negative exponent. Therefore, the decimal was moved to the left and a number between 0 and 1 was obtained. 3. 245.5 4. 0.00019 5. 800.8 6. 0.00205