The exponential notation (sometimes called the "scientific" notation) greatly simplifies calculations, especially with very large and very small numbers. It uses positive and negative exponents to write multiples and submultiples of 10:

1000        = 10 x 10 x 10 = 103
 100        = 10 x 10 = 102
  10        = 10 = 101
   1        = 100
   0.1      = 1/10 = 10-1
   0.01	    = 1/100 = 10-2
   0.001    = 1/1000 = 10-3>

and so on.

Note that for numbers greater than one, "l0^p" represents the number "l" followed by p zeros. For negative exponents, the notation 10^-p represents a fractional number, with the digit "1" occupying the pth decimal place to the right of the decimal point.

This idea may he extended to any number, for any number may be written as a multiple of a power of ten. For example, the number 4000 may he written as 4 x 10³. The number 0.038 may be written as 3.8 x 10^-2. These numbers have two parts, the numeric part (3.8) and the exponential part (10^-2).

Interpretation of these numbers is easy if you note that they may be converted back to ordinary notation by simply shifting the decimal point of the numeric part a number of places equal to the value of the exponent. To remove the exponential part of 3.8 x 10^-2, simply move the decimal point of 3.8 two places to the left and it becomes 0.038 again.

Numbers may be written in many equivalent exponential forms. Simply observe the rule:

When the exponent is increased by amount p, the decimal of the numeric part must be shifted p places to the left to compensate. When the exponent is decreased by amount p, the decimal of the numeric part must be shifted p places to the right.

This rule need not be committed to memory, for the mathematical "common sense" you have acquired in other courses should allow you to "derive" the rule as needed, and the reinforcement of frequent use will soon make it a habit.

A conventional style for expressing results makes maximum use of the space-saving economy of this notation. Write all results so that there's only one significant digit to the left of the decimal point in the numeric part. This is called "standard form".

When standard form is used there will be no trailing zeros to the right of the last significant digit. So there's never any doubt how many figures are significant.

Numbers expressed in exponential notation may be easily multiplied or divided by operating on the numeric and exponential parts separately. We utilize the fact that multiplication is commutative. One realistic, non-trivial example should demonstrate the process:

(3×10^-8)(2.4×10⁵)(6×10²)/4×10⁹ = 3(2.4)6/4 × (10^-8 × 10⁵ × 10²) / 4×10⁹ =

10.8×10^(-8+5+2-9) = 10.8×10^-10

Or perhaps this layout is clearer:

       -8          5        2                     -8    5    2
(3 x 10  )(2.4 x 10 )(6 x 10 )    3 x 2.4 x 6   10  x 10 x 10 
—————————————————————————————— = ——————————— x  —————————————— =
                 9                                      9
          (4 x 10 )                    4              10 


             (-8+5+2-9)            -10
    10.8 x 10           = 10.8 x 10

Notice how exponents add when quantities are multiplied, but when one divides by an exponential quantity, its exponent is subtracted. This recalls mathematical theorems you learned in algebra. Care must be taken to observe the rules for adding and subtracting signed numbers; this is the most frequent cause of blunders.

Numbers in exponential notation may be easily raised to powers by applying the usual rules.

(AB)^p = A^pB^p and (X^p)^q = X^pq

Example: (12 x 10³)² = 12² x 10⁶ = 144 x 10⁶ = 1.44 x 10⁸

The rules given for multiplication and division of numbers in exponential notation must not be applied to addition or subtraction. It is necessary to convert both numbers to the same power of 10 before adding them.

Example:

(3 x 10⁵) + (2 x 10⁷) = (3 x 10⁵) + (200 x 10⁵) = 203 x 10⁵

EXERCISES ON EXPONENTIAL NOTATION

1.      (10²)(10⁹) =

2.      (10⁵)(10⁷) =

3.      (10⁵)(10^-7) =

4.      (10^-7)(10⁵) =

5.      (10^-7)(10^-5) =

6.      (10⁷) / (10⁵) =

7.      (10⁵) / (10⁷) =

8.      (10^-5) / (10⁷) =

9.      (10⁷) / (10^-5) =

10.      (10^-5) / (10^-7) =

11.      (10⁸) / (10¹²) =

12.      (10⁶)(10^-3) / (10²) =

13.      (10^-6)(10^-4) / (10^-4) =

14.      (2 x 10³)(5 x 10^-7) / (3 x 10^-2) =

15.      (2 x 10³ + 5 x 10⁴) / (2 x 10^-2) =

ANSWERS

     11                  12                  -2
1. 10               2. 10               3. 10

     -2                  -12                 2
4. 10               5. 10               6. 10 

     -2                  -12                 12
7. 10               8. 10               9. 10 

      2                   -4                  +1
10. 10              11. 10              12. 10
                                                                 
      -6                         -2                5
13. 10              14. 3.33 x 10       15. 26 x 10

© 1999, 2004, by Donald E. Simanek.