L2: Uncertainty in Measurement: Significant
Digits

__Exact Numbers - Counted
Quantities__

Two Parts:

- Magnitude

- Identity

Example: 3 cows, 7
horses

Can be fractions (I
have a dozen eggs and eat 3, how many dozen do I
have left)?

Inexact Numbers - Measured
Quantities

Two Parts:

-Magnitude

-Scale

"Value" - depends on scale and is expressed by
the number of Significant Digits

Report all certain values

Report first uncertain value

Uncertain Value is a "guess" between
smallest unit of scale

Successive Measurements will vary by uncertain
value

__Accuracy__ - How close a measured value is to the true value.

__Precision -__ How close successive measured values are to each
other. The smallest the unit of a
scale the more precise successive measurements.

__Significant Digits -__ First
uncertain and all certain digits of a measured number.

__Representing Significant
Digits__

1. Non Zeros are always significant

2. Leading Zeros are never significant

3. Captive Zeros are always significant

4. Trailing Zeros are only significant if a number has a
decimal point

Round Off (Truncation) Rules

1. Round Up if the second
uncertain digit is greater than or equal to 5.

2. Round down if the second
uncertain digit is less than 5.

Significant Digits in
Calculations

1. Addition and Subtraction

-Result is limited to precision of least precise
measurement (determined by the largest uncertain
digit)

You can not do this by sight
but actual need to do the math (see examples).

101-99 = 2

(A number with 2 sig figs
subtracted from a number with 3 sig figs gives a difference with one sig fig).

2. Multiplication and
Division

-Result is limited to the number of significant figures of
the value with the least number
of significant figures.

101 X 99 = 9999=9,900

3. Compound Calculations

Use all digits during intermittent steps and only truncate
the final answer. You have to manually
determine the number of significant digits of intermediate addition or
subtraction steps (but use all digits in subsequent calculations).

Example:

_{}

The answer has three sig figs
because of the rules for addition in the denominator gives that value 3 sig
figs.

Example: (correct solution)

(1.89+1.55+1.0) x 7.55=

4.44 x7.55=33.522=34

Note that although the
intermittent addition step resulted with an answer with 2 sig figs we used all
three in the subsequent step, but then used the knowledge to truncate the final
answer to 2 significant digits.

Wrong way (truncate early)

(1.89+1.55+1.0) x 7.55=

4.44 x7.55 = 4.4x7.55=33
(which is the wrong answer as we made the mistake of truncating the summation
value of the intermediate step.