## Scientific Notation

All numbers, regardless of magnitude, can be expressed in the form:N x 10where^{n}

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

- standard: 1.23x10
^{6} - non-standard: 123x10
^{4}

### Positive exponents

36,600- a number greater than 1

exponent of 10 is a positive whole number - value of the exponent

number of places the decimal point must be moved so that the notation is in standard form - 36,600 x 10
^{0}

For each place the decimal point is moved to the left, add 1 to the original exponent

3.66 x 10^{4}

### Negative exponents

0.00563- a number less than 1

exponent of 10 is a positive whole number - value of the exponent

number of places the decimal point must be moved so that the notation is in standard form - 0.00563 x 10
^{0}

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

When multiplying numbers written in exponential notation:- Multiply digit terms in the normal fashion.
- Obtain the exponent in the product by adding the exponents of the factors multiplied.
- If necessary, adjust the exponent to leave just one digit to the left of the decimal point.

(1.25x10^{5}) x (4.0x10^{-2}) = (1.25x4.0) x 10^{5+(-2)}= 5.0x10^{3}

### Exponential notation: Division

When dividing numbers written in exponential notation:- Divide the digit terms in the normal fashion.
- Obtain the exponent in the quotient by subtracting the exponent of the divisor from the exponent of the dividend.
- If necessary, adjust the exponent to leave just one digit to the left of the decimal point.

(7.5x10^{6}) / (3.0x10^{-2}) = (7.5/3.0) x 10^{6-(-2)}= 2.5x10^{8}

dividend divisor

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

14.62 mL: implied precision +/- 0.01 mLIn this measured quantity, the significant figures are those digits known precisely (namely 1, 4, and 6; these digits are known with a high degree of confidence) plus the last digit (2) which is estimated or is approximate

### Guidelines for counting significant figures

**Numbers Always Considered Significant**- all non-zero digits
- zeros between non-zero digits
- 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.**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**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. (1x10
^{2}) - 2 sig. fig. (1.0x10
^{2}) - 3 sig. fig. (1.00x10
^{2})

- 1 sig. fig. (1x10

- Zeros trailing to the right of the rightmost non-zero digit may or may not be significant
**Exact Numbers**- Numbers derived from definition or through counting
- Numbers considered to be "infinitely precise" (not subject to errors in measurement)

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

The result of these operations will contain the same**number of significant figures**as the quantity in the calculation having the

**fewest number**of significant figures.

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

The result must be expressed with the same number of decimal places (i.e., the same absolute uncertainty) as the quantity carrying the least number of**decimal places**(i.e., the least precisely determined quantity)

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

References:

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