Scientific notation is used as a type of shorthand to express very large or very small numbers. It is a way to write numbers so that they do not take up as much space on the page. The format of a number written in scientific notation has two parts. The first part is a number greater than or equal to 1 and less than 10 (for example, 2.35). The second part is a power of 10 (for example, 10^6). The number 2,350,000 is expressed in scientific notation as 2.35 × 10^6. It is important that the decimal point is always placed to the right of the first digit. Notice that very large numbers always have a positive power of 10 and very small numbers always have a negative power of 10.
Example: The velocity of the speed of light is over 186,000,000 mph. This can be expressed as 1.86 × 10^8 mph in scientific notation. The mass of an electron is approximately 0.000,000,000,000,000,000,000,000,000,911 grams. This can be expressed in scientific notation as 9.11 × 10^-28 grams.
Converting Numbers from Standard Notation to Scientific Notation
Example: Convert 1,244,000,000,000 to scientific notation as follows. First, note that the decimal point is to the right of the last zero. (Even though it is not usually written, it is assumed to be there.)
1,244,000,000,000 = 1,244,000,000,000
To change to the format of scientific notation, the decimal point must be moved to the position between the first and second digits, which in this case is between the 1 and the 2. Since the decimal point must be moved 12 places to the left to get there, the power of 10 will be 12. Remember that large numbers always have a positive exponent. Therefore, 1,244,000,000,000 = 1.244 × 10^12 when written in scientific notation.
Example: Convert 0.000000457 from standard notation to scientific notation. To change to the format of scientific notation, the decimal point must be moved to the position between the first and second numbers, which in this case is between the 4 and the 5. Since the decimal point must be moved 7 places to the right to get there, the power of 10 will be −7. Remember that small numbers (those less than one) will have a negative exponent. Therefore, 0.000000457 = 4.57 × 10^-7 when written in scientific notation.
Converting Numbers from Scientific Notation to Standard Notation
Example: Convert 3.68 × 10^7 from scientific notation to standard notation, as follows. To convert from scientific notation to standard notation, move the decimal place 7 places to the right. 3.68 × 10^7 = 36800000 = 36,800,000. Another way to think about the conversion is 3.68 × 10^7 = 3.68 × 10,000,000 = 36,800,000.
Example: Convert 7.1543 × 10^-10 from scientific notation to standard notation. Move the decimal place 10 places to the left: 7.1543 × 10^-10 =.00000000071543. Another way to think about the conversion is 7.1543 × 10^-10 = 7.1543 × .0000000001 = .00000000071543
When converting, remember that large numbers always have positive powers of ten and small numbers always have negative powers of ten. Refer to Figure 1-11 to determine which direction to move the decimal point.
Addition, Subtraction, Multiplication, and Division of Scientific Numbers
To add, subtract, multiply, or divide numbers in scientific notation, change the scientific notation number back to standard notation. Then add, subtract, multiply or divide the standard notation numbers. After the computation, change the final standard notation number back to scientific notation.