2.14   Scientific Notation

from Basic Algebra: One Step at a Time © 2002

P. 227-234

Dr. Robert J. Rapalje

Seminole State College of Florida

 

ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE END OF THIS PAGE

 

To see selected solutions in Living C O L O R  click here!

 

When working with very large (astronomical!) numbers or very small (microscopic!) numbers, it is often convenient to write the numbers in scientific notation.  In scientific notation, the number must be expressed as a number between 1 and 9.99... times a power of 10.  Numbers with magnitude 10 or greater will be expressed with a positive power of 10, while numbers with magnitude smaller than 1 must be expressed with a negative power of 10.

Before using the calculator to compute with scientific notation, first it may be helpful to review the concept and make sure you understand how to convert from standard decimal form to scientific notation and back, using the calculator in your head!  Your calculator may or may not convert from one to the other (this is something you can easily do in your head!), but it will certainly perform calculations and sometimes give you answers in scientific notation.  First, the “non-calculator” explanation, for those times in which you do not have a calculator in hand and for those times, like certain exams, in which calculators are not allowed!  By the way, converting from standard decimal form to scientific notation (and vice-versa) is so simple you hardly need a calculator to do it!  Just count the number of places you must move the decimal to obtain a number between 1 and 10.   Numbers larger than (astronomical!) 10 have a positive power of 10.  Small decimal numbers (microbiology!) must have a negative power of 10.  Study the following examples.

EXAMPLES:

STANDARD DECIMAL FORM                                         SCIENTIFIC NOTATION

 1.  300                                    = 3.0 × 100                              = 3.0  × 102

 2.  3000                                  = 3.0 × 1000                            = 3.0  × 103

 3.  3250                                  = 3.25 × 1000                          = 3.25 × 103

 4.  32,500,000                        move decimal 7 places left              = 3.25 × 107

 5.  0.03                                   = 3.0 × 1/100                           = 3.0  × 10-2

 6.  0.0003                               = 3.0 × 1/10000                       = 3.0  × 10-4

 7.  0.0000000325                        move  8 places right                      = 3.25 × 10-8

 8.  0.00000000032                      move 10 places right                     = 3.2  × 10-10

 

EXERCISES:  Express in scientific notation.

 1.     450 = ________________________                  2.     7500 = _________________________

 3.     12,000,000 = __________________                  4.     720,000,000,000 = ________________

 5.     0.00325 = _____________________                 6.     0.000000246 = ___________________

 7.     0.00000000325 = _______________                 8.     0.00000000436 = _________________

 9.     480,000,000 = __________________               10.     93,200,000,000 = ________________

11.     0.0000876 = ___________________               12.     0.0000122  = ____________________

Continuing with the non-calculator explanation, sometimes scientific notation can be a shortcut in calculation, especially if you do not have a calculator.  However, calculators are really great later!

EXAMPLE 9.     2,000,000 ×  3,000,000,000

Solution:              First, change to scientific notation.   

     2  × 106 ×  3  × 109                        Multiply 2 times 3 and add the exponents 6 plus 9.

     6  × 1015

 

EXAMPLE 10.     4,000,000 ×  3,000,000,000

Solution:              First, change to scientific notation.   

     4  × 106 ×  3  × 109                        Multiply 4 times 3 and add the exponents 6 plus 9.

     12  × 1015                                    Change 12 to scientific notation:   1.2 × 101

     1.2 × 101 × 1015

        1.2 × 1016

 

EXAMPLE 11.     0.0000032 ×  70,000,000,000

Solution:              First, change to scientific notation.   

     3.2 × 10-6 × 7  × 1010         Multiply 3.2 times 7 and add the exponents -6 plus 10.

     22.4  × 104                                  Change 22.4 to scientific notation:   2.24 × 101

     2.24 × 101 × 104

        2.24 × 105 or 224,000

 

EXAMPLE 12.    

Solution:                First, change to scientific notation.

          Divide 6 by 3, and subtract the exponents 6 minus (-5).

 

        2 × 10 11

 

EXAMPLE 13.    

Solution:                First, change to scientific notation.

          Divide 6 by 4, and subtract the exponents -5 minus 7.

         1.5 × 10 -12

 

 

EXAMPLE 14.    

 

Solution:              First, change to scientific notation.   

          Divide 3 by 6, and subtract the exponents 6 minus (-3).

 

       0.5 × 10 9              Change 0.5 to scientific notation:   5 × 10 -1

 

       5 × 10 -1 × 10 9    

 

       5 × 10 8   

Perhaps you noticed that all the numbers in the preceding examples were carefully “arranged” (mathematicians like the word “contrived”!) so everything came out even, and the arithmetic was easy even without a calculator.  Of course, in the real world, life does NOT usually come out even, and neither does the math.  Not to worry, however!!  With most any calculator, you can easily handle such calculations, whether the numbers come out even or not.  Calculators will really be nice, but first here are a few exercises to try without calculators.

 

EXERCISES.             Perform the calculations without using a calculator by converting to scientific notation.

 

13.  20,000 ×  3,000,000                                                       14.     25,000,000 ×  300,000,000         

 

 

 

15.   400,000,000  ×  0.000008                                             16.     0.0000006  ×  0.0004

 

 

 

17.                                          18.                                        19.     

 

 

 

 

20.                                               21.                                       22. 

 

 

 

 

USING YOUR CALCULATOR

Your calculator may or may not be able to convert from standard decimal form to scientific notation and vice versa.  For some calculators (certain Casios and the TI-85/86), you use the [MODE] button, and follow the instructions in your manual.  For other calculators (including the TI-30 and other Texas Instrument models) there are separate functions called [SCI] and [FLO].  FLO means floating point, which is a decimal format.  The details of how different calculators convert from scientific notation to standard decimal form vary greatly.  However, when it comes to using them to calculate problems in this section, they are remarkably similar. 

EXAMPLE 15.    Use your calculator to compute   6,000  ×  7,000.   Express it in scientific notation.

Solution:         Just enter [6000] [x] [7000] [= or ENTER].  The calculator gives you 42000000.  Now you can either change it to scientific notation yourself or use the calculator.  To do it yourself, just move the decimal to the left to write 4.2.  Since you moved the decimal 7 places to the left, this means the answer is           4.2 × 107.

TI30 Note:

If your calculator says 420000000, look for [2nd] [Sci].  The calculator will convert automatically to scientific notation and give you 4.2  7.  Of course this does not mean 4.2 raised to the 7th power.  You have to write it in correct form:  4.2 ×107.

 

 

TI85/86 Note:

If your calculator says 42000000, find the [MORE] button at the top of the middle column of keys.  Above the [MORE] button is [MODE].  Press [2nd] [MODE], and a rather impressive MODE menu opens up on the calculator.  In the top row of this menu, it says [Normal], [Sci] (for Scientific Mode), and [Eng] (for Engineering Mode).  Press the [Right arrow] key, and the cursor moves from [Normal] to [Sci].

Press [ENTER] to lock it in; press [EXIT] to get out of this menu; press [ENTER], and the calculator converts the last calculation to scientific notation.

Now, as a follow up, calculate 3 x 4 [ENTER].  The calculator gives you 1.2 x 101.    The answer you were looking for was 12--obviously, the calculator is still in scientific notation mode.  To change it back, press [2nd]   [MODE]  [Left arrow] [ENTER]  [EXIT]  [ENTER].  The calculator returns to Normal Mode, and it should give you the answer of 12.

 

 

TI85/86 Summary of Keystrokes

Given a number on the calculator in standard decimal form to convert to scientific notation,

press [2nd]  [MODE]  [Right arrow]  [ENTER]  [EXIT]  [ENTER]

 

To convert back to standard decimal, press

[2nd]  [MODE]  [Left arrow]  [ENTER]  [EXIT]  [ENTER]

 

EXAMPLE 16.    Use your calculator to compute   6,000,000  ×  7,000,000.

      Express it in scientific notation.

Solution:        Just enter [6000000] [×] [7000000] [= or ENTER].  The calculator automatically gives you scientific notation, since it can’t doesn’t have enough digits to display the standard decimal form.  Depending upon the type of calculator, you will automatically get  something like  4.2 13 , 4.2 E 13,  or 4.2  13.  What this means, and the way you must write the answer is 4.2 × 1013.

 

EXERCISES.      Use your calculator to calculate.  Express standard decimal or scientific notation.

23.  20,000 ×  3,000,000                                 24.  25,000,000 ×  300,000,000         

 

 

25.   400,000,000  ×  0.000008                       26.  0.0000006  ×  0.0004

 

 

27.                                   28.                            29.  

 

 

30.                                      31.                            32. 

 

 


 

Sometimes problem is given in scientific notation.  If this is the case, then look for the [EE]  or the [EXP] button on your calculator.  This EE (or EXP) function makes scientific notation very easy to enter.  If you have an EE or EXP key on your calculator, to enter 7.2 × 1012, simply enter [7.2] [EE (or EXP)]  [12]  [ENTER].  It doesn't get any easier than that! CAUTION: DO NOT ENTER [7.2]  [×]  [EE], etc. 

 EXAMPLE 17.    Use your calculator to find the value of 

Solution:       Enter [7.5] [EE or EXP] [6] [¸] [1.5] [EE or EXP] [+/- or (-)] [2] [ENTER].

(Note:  To enter -2 exponent, on some calculators you must enter [2], then [+/-].)

The calculator should give you  5 × 108. 

You can check this without using a calculator by dividing 7.5 by 1.5 which is 5.

Then subtract exponents 6 - (-2) to get 8, which is the power of 10.

 

EXERCISES.      Use your calculator to find the value of each of the following.

 33.     =                                                34.     =

   [7.2]   [EE]  [12]  [x]  [6.3]  [EE]  [+/-]  [8]  [=]

 

 

35.     =                                                36.    =

 

 

37.      =                                                      38.      =

 

 

39.     =                                                     40.      =    

 

 

41.      [HINT: 1012 may be entered as 1 X 1012 ]

 

 

42.                                                                 43.      

 

 

 

COMBINED OPERATIONS

Remember, when numerators and denominators have more than one term or factor, you must use  parentheses around the entire numerator and the entire denominator.  (See Section 1.04, p. 26.)

 

EXAMPLE 18.     Calculate: 

Solution:         [( ]  [7.2] [EE]  [12] [x]  [6.3]  [EE]  [+/- or (-)]  [8]  [ )]  [¸]

 [( ]  [3.5] [EE]  [+/- or (-)]  [4] [x]  [8.1]  [EE]   [14]  [ )]  [=]

The answer should be 1.6 x 10 -6 . 

   

44. =                                     45.=

 

 

  

 

46.=

 

 

ANSWERS 2.14

 p. 228-234:

              1. 4.5×102; 2. 7.5×103;  3. 1.2×107; 4. 7.2 × 1011; 5. 3.25×10-3; 6. 2.46×10-7;  7. 3.25×10-9;              

              8. 4.36×10-9;  9. 4.8×108;  10. 9.32×1010;  11. 8.76×10-5;  12. 1.22×10-5;  13. 6×1010; 

            14. 7.5×1015;  15. 3.2×103or 3200  16. 2.4×10-10;  17. 3×10-12;   18. 4×109;  19. 1.2×104; 

            20. 2 × 1010  21. 2.5×109;  22. 7.5×101 or 75;  23. 6×1010;  24. 7.5×1015;  25. 3.2×103or 3200 

            26. 2.4×10-10;  27. 3×10-12;   28. 4×109;  29. 1.2×104;  30. 2× 1010  31. 2.5×109; 

            32.7.5×101or 75; 33. 4.54×105; 34. 2.84×1011; 35. 4.88×1022; 36. 4.88×10-20; 37. 2.06×1016;

            38. 7.78×10-23; 39. 8.89×1025; 40. 8.89×101 or 88.89; 41. 1.25×1015; 42. 2.5×10-9;

            43. 1.67×103 or 1666.67;   44. 1.86×10-28;   45. 1.86×1028;   46. 1.86×10-12 .

 

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Dr. Robert J. Rapalje Altamonte Springs Campus
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