Scientific Notation and Significant Figures

Scientific Notation

N x 10ⁿ

• N is a number, either an integer or decimal, between 1 and 10.
• n is a positive or negative integer.

• standard: 1.23x10⁶
• non-standard: 123x10⁴

Positive exponents

1. a number greater than 1
   exponent of 10 is a positive whole number
2. value of the exponent
   number of places the decimal point must be moved so that the notation is in standard form
3. 36,600 x 10⁰
   For each place the decimal point is moved to the left, add 1 to the original exponent

3.66 x 10⁴

Negative exponents

1. a number less than 1
   exponent of 10 is a positive whole number
2. value of the exponent
   number of places the decimal point must be moved so that the notation is in standard form
3. 0.00563 x 10⁰
   For each place the decimal point is moved to the right, subtract 1 from the original exponent

5.63 x 10^-3

Exponential notation: Multiplication

1. Multiply digit terms in the normal fashion.
2. Obtain the exponent in the product by adding the exponents of the factors multiplied.
3. If necessary, adjust the exponent to leave just one digit to the left of the decimal point.

(1.25x10⁵) x (4.0x10^-2) = (1.25x4.0) x 10^5+(-2) = 5.0x10³

Exponential notation: Division

1. Divide the digit terms in the normal fashion.
2. Obtain the exponent in the quotient by subtracting the exponent of the divisor from the exponent of the dividend.
3. If necessary, adjust the exponent to leave just one digit to the left of the decimal point.

(7.5x10⁶) / (3.0x10^-2) = (7.5/3.0) x 10^6-(-2) = 2.5x10⁸
 dividend     divisor

Expressing the Uncertainty (Reproducibility) in Measured Quantities Using Significant Figures

14.62 mL: implied precision +/- 0.01 mL

Guidelines for counting significant figures

1. Numbers Always Considered Significant
   1. all non-zero digits
   2. zeros between non-zero digits
   3. in numbers containing a decimal point, all zeros written to the right of the rightmost non-zero digit

   300.16, 1.0200, and 1,000.0 all contain 5 significant figures.

2. Numbers that are NEVER Significant
   Zeros written to the left of the leftmost non-zero digit (these merely indicate the placement of the decimal point)

   0.00416 and 0.00000100 both contain three significant figures

3. Trailing Zeros in Numbers Containing No Decimal Point
   • Zeros trailing to the right of the rightmost non-zero digit may or may not be significant
     For example, the number 100 may have one sig. fig. (100), two sig. figs. (100), or three sig. figs. (100)
   • Remove ambiguity by expressing the number using scientific notation
     100 expressed as:
     • 1 sig. fig. (1x10²)
     • 2 sig. fig. (1.0x10²)
     • 3 sig. fig. (1.00x10²)
4. Exact Numbers
   1. Numbers derived from definition or through counting
   2. Numbers considered to be "infinitely precise" (not subject to errors in measurement)
   Exact numbers have no effect on the precision expressed in a numerical calculation
   

   12 inches = 1 ft	1 liter is 1,000,000 mL
   1 hr. = 3600 sec.	42 students enrolled in a class

Significant Figures: Multiplication and Division

	0.942 atm x 23.482 L
n =	------------------------------------------- =	0.864826127 mol =	0.865 mol
	0.08205 L atm / mol K x 311.73 K		round to 3 sig. fig.

Significant Figures: Addition and Subtraction

		implied precision	correct precision
25.6854 g		25.6854 +/- 0.0001 g
+0.17     g		+0.17     +/- 0.01     g	round off
25.8554 g		25.8554 +/- 0.0001 g -- >	25.86 g +/- 0.01 g

Additional exercises and resources:

• Significant Figures and Exponential Notation from the ChemTeam site (John Park, Diamond Bar, CA)
• On-line exercises:
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