Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Approximations of π wikipedia , lookup
Principia Mathematica wikipedia , lookup
Bra–ket notation wikipedia , lookup
Abuse of notation wikipedia , lookup
Large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
Elementary mathematics wikipedia , lookup
History of mathematical notation wikipedia , lookup
Musical notation wikipedia , lookup
Scientific Notation Scientific Notation In the sciences, one frequently needs to use numbers that have a lot of digits. For example: Distance to the Sun: 150000000000 km Speed of Light: 300000000 m/s Mass of a Proton: 0.00000000000000000000000000167 kg Seconds in One Year: 31536000 s It would be very cumbersome if every time a very large or a very small number was expressed, all of its digits had to be written out. Scientific notation is one method used to compactly express numbers that have numerous digits. Scientific notation allows the previous examples to be expressed as: Distance to the Sun: 1.5 × 1011 km Speed of Light: 3 × 108 m/s Mass of a Proton: 1.67 × 10-27 kg Seconds in One Year: 3.1536 × 107 s The key to understanding scientific notation is to realize that a number expressed in scientific notation is always in the form a × 10n where 1 ≤ |a|< 10 and n is an integer Standard Notation Scientific Notation 123 1.23 × 102 20693 2.0693 × 104 0.08273 8.273 × 10-2 5.6 5.6 × 100 -0.792 -7.92 × 10-1 971.56 9.7156 × 102 a × 10n To convert numbers from standard notation to scientific notation: 1) Determine the appropriate value of a. 2) Determine the value of n. Example: Convert 382 to scientific notation: 1) What are the possible values of a? Possible values: 382., 38.2, 3.82, 0.382, etc The only potential value of a that fits the criterion of 1 ≤ |a|< 10 is 3.82. a = 3.82 2) What is the value of n? n is the exponent on the base 10. Its value affects the location of the decimal. To determine n, ask yourself, “How many times does a have to be multiplied or divided by 10 to reconstruct the original number expressed in standard from?”. Whatever that number is, it is the absolute value of n. In our example, 3.82 needs to be multiplied by 10 two times to get back to 382. n=2 Finally, if you had to multiply, the value of n will remain positive; if you had to divide, the value of n must be made negative. In our example we had to multiply so n = 2 remains n = 2 and the final answer is: 382 = 3.82 × 102 Examples Standard Form a tentative n Multiply or Divide? n Scientific Notation 8700 8.7 3 Multiply 3 8.7 × 103 0.56 5.6 1 Divide -1 5.6 × 10-1 -0.00492 -4.92 3 Divide -3 -4.92 × 10-3 579000 5.79 5 Multiply 5 5.79 × 105 105.9 1.059 2 Multiply 2 1.059 × 102