L2: Uncertainty in Measurement: Significant Digits
Exact Numbers - Counted Quantities
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
"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).
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=
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.