Chem 110           When to Use Scientific Notation & How to Round Numbers Off

When to Use Scientific Notation

A number should be expressed in scientific notation (with only one nonzero digit to the left of the decimal) under these conditions:
1.  A number with ambiguous zeroes (tailing zeroes in a number without a decimal)

To remove the ambiguity, it MUST be expressed in sci. not.
e.g.  It is not clear whether 35000 has 2, 3, 4 or 5 sig.fig.  It is not clear whether the three “tailing zeroes” are significant or not.  If you mean 35000 to have 3 sig. fig., then it MUST be expressed as 3.50 x 10 ⁴ .

2.  A number that is very small (as a rule of thumb, less than 0.01).

It is tedious and riskier to copy numbers with a string of avoidable zeroes.
e.g.   0.000 000 83 SHOULD be expressed as 8.3 x 10 ^{- 7}

3.  A number that is in exponential form for any reason

e.g.   324.3 x 10 ^{- 8}  SHOULD be expressed as 3.243 x 10 ^{– 6}
Some students indiscriminately express all their numbers in scientific notation because they are too lazy to figure out when it is necessary.  Although it is not “wrong” to do so, you should learn when it is appropriate.  For example, it would not be appropriate to tell someone to weigh out “2.5 x 10 grams of salt” when “25 grams of salt” would do equally well.

20.0 x 5.0               =  _________________     (42 x 10 ³) x 2    =  _________________
0.004/800              =  _________________      22 x 2.0                =  _________________

Write down your answers before looking at the correct answers.  Be sure to ask if you don't understand why your answers don't match mine!  CLICK HERE TO SEE THE ANSWERS.

Rounding Off Numbers

In correcting a number to express the proper number of sig. fig., we often have to drop off unwanted digits.  The rules for rounding off numbers are explained in your textbook.  Here is a summary:

Rules for rounding off numbers:

If the digit immediately to the right of the last sig. fig. is equal* or greater than 5, you round up.
If the digit immediately to the right of the last sig. fig. is less than 5, you round down.
    35.76 in 3 sig. fig. is 35.8 (round up because it is more than half-way between 35.7 and 35.8)
    35.74 in 3 sig. fig. is 35.7 (round down because it is less than half-way between 35.7 and 35.8)

If the digit immediately to the right of the last sig. fig. is equal to 5 followed by other nonzero digits, you round up.
35.752 in 3 sig. fig. is 35.8 (round up because it is more than half-way between 35.7 and 35.8)

If the digit immediately to the right of the last sig. fig. is equal to 5 and not followed by nonzero digits,
the number is exactly half-way between 35.7 and 35.8, we often follow a simpler rule and round up.
*Properly done, we should round up only if the digit immediately preceding the 5 is odd:
35.65 in 3 sig. fig. is 35.6 because the digit preceding the 5 is 6 which is even and not odd.
35.35 in 3 sig. fig. is 35.4 because the digit preceding the 5 is 3 which is odd.

After rounding off, if the resulting number has ambiguous zeroes, it should be recorded in scientific notation to avoid ambiguity.
   34821.0 in 2 sig. fig. is 35000 where the three zeroes may or may not be significant.
   The correct answer is 3.5 x 10 ⁴  .

72.49  in 3 sig. fig. is   ___________    299000 in 2 sig. fig. is    ___________
45.52  in 3 sig. fig. is   ___________    92,528 in 4 sig. fig. is     ___________
120.05  in 4 sig. fig. is ___________    13,052 in 3 sig. fig. is     ___________
239.5 in 3 sig. fig. is    ___________    2.448 x 104 in 3 sig. fig. is ___________
28.149 in 3 sig. fig. is  ___________    32000.000 in 3 sig. fig. is   ___________
63500 in 2 sig. fig. is   ___________    89999 in 3 sig. fig. is          ___________

Write down your answers before looking at the correct answers.  Be sure to ask if you don't understand why your answers don't match mine!  CLICK HERE TO SEE THE ANSWERS.

Extra practice by doing Silberberg, 4^th Ed, p.35 #50 thru 55 (Answers of questions with blue-colored numbers are at back of book.)

Answers to Use of Sci.Not.:
20.0 x 5.0 = 100 = 1.0x10²
0.004/800  = 0.000005 = 5x10^-6
(2 x 10³)(3 x 10⁴) = 84 x 10³ = 8 x 10⁴
22 x 2.0 = 44 rather than 4.4x10

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Answers to Rounding Off:
72.5,   45.5,   120.1,   120. or 1.20x10²,    28.1,     6.4x10⁴,
 3.0x10⁵,   9.253x10⁴,   1.31x10⁴,   2.45x10⁴,   3.20x10⁴,   9.00x10⁴

You can get more practice by doing #1.21 thru 1.28, p.24 (General Chemistry: The Essential Concepts by Chang, 3^rd ed). Answers of even-numbered problems are at back of book.

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