Page 28 - Theory and Problems of BEGINNING CHEMISTRY

CHAP. 2] MATHEMATICAL METHODS IN CHEMISTRY 17

EXAMPLE 2.17. Write 455 000 in standard exponential form.
Ans. 455 000 = 4.55 × 10 × 10 × 10 × 10 × 10 = 4.55 × 10 5

The number of 10s is the number of places in 455 000 that the decimal point must be moved to the left to get one
(nonzero) digit to the left of the decimal point.

Using an Electronic Calculator
If you use an electronic calculator with exponential capability, note that there is a special key (labeled EE
3
or EXP ) on the calculator which means “times 10 to the power.” If you wish to enter 5 × 10 , push 5 , then
the special key, then 3 .Do not push 5 , then the multiply key, then 1 , then 0 , then the special key, then 3 .
If you do so, your value will be 10 times too large. See the Appendix.

EXAMPLE 2.18. List the keystrokes on an electronic calculator which are necessary to do the following calculation:
3 2
6.5 × 10 + 4 × 10 =
Ans.
6 . 5 EXP 3 + 4 EXP 2 =

On the electronic calculator, to change the sign of a number, you use the +/− key, not the − (minus)
key. The +/− key can be used to change the sign of a coefficient or an exponent, depending on when it is
pressed. If it is pressed after the EE or EXP key, it works on the exponent rather than the coefficient.

EXAMPLE 2.19. List the keystrokes on an electronic calculator which are necessary to do the following calculation:
−5 6
(7.7 × 10 )/(−4.5 × 10 ) =
Ans. 7 . 7 EXP 5 +/− ÷ 4 . 5 +/− EXP 6 =

The answer is −1.7 × 10 −11 .

2.5. SIGNIFICANT DIGITS
No matter how accurate the measuring device you use, you can make measurements only to a certain
degree of accuracy. For example, would you attempt to measure the length of your shoe (a distance) with
an automobile odometer (mileage indicator)? The mileage indicator has tenths of miles (or kilometers) as its
smallest scale division, and you can estimate to the nearest hundredth of a mile, or something like 50 ft, but
that would be useless to measure the length of a shoe. No matter how you tried, or how many measurements
you made with the odometer, you could not measure such a small distance. In contrast, could you measure the
distance from the Empire State Building in New York City to the Washington Monument with a 10-cm ruler?
You might at first think that it would take a long time, but that it would be possible. However, it would take so
many separate measurements, each having some inaccuracy in it, that the final result would be about as bad as
measuring the shoe size with the odometer. The conclusion that you should draw from this discussion is that
you should use the proper measuring device for each measurement, and that no matter how hard you try, each
measuring device has a certain limit to its accuracy, and you cannot measure more accurately with it. In general,
you should estimate each measurement to one-tenth the smallest scale division of the instrument that you are
using.
Precision is the closeness of a set of measurements to each other; accuracy is the closeness of the average
of a set of measurements to the true value. Scientists report the precision for their measurements by using
a certain number of digits. They report all the digits they know for certain plus one extra digit which is an
estimate. Significant figures or significant digits are digits used to report the precision of a measurement. (Note
the difference between the use of the word significant here and in everyday use, where it indicates “meaningful.”)
For example, consider the rectangular block pictured in Fig. 2-1. The ruler at the top of the block is divided
into centimeters. You can estimate the length of a block to the nearest tenth of a centimeter (millimeter),