﻿ Scientific Notation
 Examples of Large and Small Numbers Using Scientific Notation Example   Einstein's famous equation says that         E = mc2 where c is the speed of light          c = 300,000,000 m/sec How much energy is emitted if 2,000 kg of matter is destroyed? Solution  (Without Scientific Notation) We compute:           E  =  2,000(300,000,000)2  =  180,000,000,000,000,000,000 This is a very awkward number.   Solution  (With Scientific Notation) We write         c  =  300,000,000  =  3.0 x 108        Count the number of digits to the left of the three and         m  =  2.0 x 103 So that         E  =  mc2  =  (2.0 x 103) (3.0 x 108)2           = (2.0 x 103) (9.0 x (108)2)        Distributing the exponent         = (2.0 x 103) (9.0 x 1016)           Using the Power Rule         = 18.0 x 1019                             Using the Addition Rule         = 1.8 x 10 x 1019                       Putting 18.0 into Scientific Notation         = 1.8 x 1020                               Using the Addition Rule   In General we count the number of digits to the right of the first digit to determine the exponent. Exercise The number of molecules in a mole is         602,000,000,000,000,000,000,000 Write this number in scientific notation.           Example:   The mass of a hydrogen atom is         .00000000000000000000000167 kg. what is the mass of 75,000 atoms? Solution:   To convert a decimal to scientific notation we count how many moves we must make to bring the decimal to the right of the first nonzero digit.  In our case we need to move the decimal over to the right 24 places hence we can write:         mass of one atom  =  1.67 x 10-24   We have that         Total mass  =  Mass of one atom  x  number of atoms or         =  (1.67 x 10-24) (7.5 x 104)  =  12.525 x 10-20  Using the Addition Rule         = 1.3 x 10 x 10-20   =  1.3 x 10-19                    Addition Rule again   Exercise: A computer can  perform an addition calculation in 3.1 x 10-7 seconds.  How long will it take to perform 4 x 106  (4 million) addition calculations?         Example  How many seconds are there in 70 years? Solution                         365.25 days       24 hrs        60 min        60 sec         70yrs                                                                                                                                                                                  1 yr               1 day          1 hr            1 min         =  2,209,032,000 seconds         =  2.2 x 109  seconds. Exercise One Angstrom = 0.0000001 cm.  One light year is 5.86 x 1012 miles.  How many angstroms are there in a light year? Note:  There are 5,280 feet in a mile and one centimeter is 0.39 inches.   Exercise What percentage of the US population plays professional baseball?  Assume that there are 30 teams and each team has 40 players.  Also the population of the US is 300,000,000.   Exercise The distance from the earth to the sun is 93,000,000 miles.  The speed of light is 3.0 x 108 meters per second.  How many minutes does it take for a solar flare to reach the earth?   Exercise Estimate how many particles of sand there are on the California coast.   Back to Math 152A Home Page Back to the Math Department Home