SCIENTIFIC MEASUREMENTS
A measurement can be defined as number with attached units.
The numerical value of a measurement should reflect the sensitivity of the instrument used to make the measurement.
Consider a bathroom scale, which measurement is reasonable?
165.674 lbs or 166 lbs
An ‘exact measurement’ does not really exist because every instrument has some degree of uncertainty. An instrument reads only a finite number of digits that have meaning.
Every measurement has some degree of uncertainty in the last decimal place. The last digit read with an instrument, with analog readout, is estimated.
Ruler A 4.2 +/- 0.1 cm
Ruler B4.25 +/- 0.05 cm
RULES FOR DETERMINING SIGNIFICANT DIGITS
1. Nonzero digits are always significant.
2. Leading zeros that appear at the start of a number are never significant because they act only to fix the position of the decimal point in a number less than 1.
3. Confined zeros that appear between nonzero numbers are always significant.
4. Trailing zeros at the end of a number are significant only if the number contains a decimal point or contains an over-bar.
Consider the following list of numbers and ask yourself how many of the digits are actually needed to retain accuracy (number of significant figures).
950100
9501
950.1
95.01
9.501
0.9501
0.0000009501
Many will be duped into thinking that the number of significant figures was; 6, 4, 4, 4, 4 and 10. There are four significant digits in each of the above numeric expressions. The only thing that differs
in each of the above expressions is the position of the decimal.
Now let us write these values in scientific notation without loss of accuracy.
9.501 x 105
9.501 x 103
9.501 x 102
9.501 x 101
9.501 x 100
9.501 x 10-1
9.501 x 10-7
Here it becomes more apparent that the zeros in the first number and last were merely establishing the position of the decimal.
MASS VERSUS WEIGHT
Mass is a measure of the amount of matter in an abject.
Weight is the force exerted by gravity on an object.
A scale uses a spring to measure weight. A balance measures mass by comparing the force acting equally on both pans of a balance.
Rules for Rounding Off
1. If the first non-significant digit is less than 5, drop it and the last significant digit remains the same. Thus, 47.21 (rounded to 3 sig. figs) is:
2. If the first non-significant digit is more than 5 or is 5 followed by numbers other than zeros, drop the non-significant digit(s) in increase the last significant digit by 1. Hence, 47.26 and 47.252 are both equal to 47.3 ( when rounded to 3 sig. figs)
3. If the first non-significant digit is 5 and and is followed by zeros, drop the 5 and.
A) increase the last significant digit by one if it is odd, or
b) leave the significant digit the same if it is even
Thus, rounded to 3 sig-figs, 47.250 and 47.350 become:
4. Non-significant digits to the left of the decimal point are not discarded, but are replace by zeros.
Thus 1781 becomes 1780 and not 178 when rounded to three significant digits.
Rule for Addition and Subtraction
The answer must not contain a smaller place (that is, decimal, units, tens, and so on) than the number with the smallest place.
25.1
+22.11
47.21 rounds to
47.2
Rule for Multiplication and Division
In multiplication and division, the answer must not contain any more significant digits than the least number of significant digits in the numbers used in the multiplication or division.
Rounds to 11.8
A Special Rule: Exact Numbers
Exact numbers are precisely know and can have as many significant digits as a calculation requires, so they are not used to determine the number of significant digits for an answer. For example there are exactly 12 inches in 1 foot.
Rounds to 2.248 ft
Exponential notation
Form of mathematical expression in which a number is expressed as the product of two numbers, one a decimal and the other a power of 10.
1000 = 1 x103
100 = 1 x 102
10 = 1 x 101
1 = 1 x 100
0.1 = 1 x 10-1
0.01 = 1 x 10-2
0.001 = 1 x 10-3
The Earth
1 x 107 meters in diameter.
Red Blood Cells
1 x 10-6 meters in
diameter.
Scientific notation
Form of exponential notation is which the decimal part must have exactly one nonzero digit to the left of the decimal point; it widely used by scientists.
There are 26,800,000,000,000,000,000,000 helium atoms in 1 liter of helium gas at standard conditions.
2.68 x 1022 helium atoms
Not
26.8 x 1021
The Percent Concept
Exponential Notation
Form of mathematical expression in which a number is expressed as the product of two numbers, one a decimal and the other a power of 10.
1000 = 1 x103
100 = 1 x 102
10 = 1 x 101
1 = 1 x 100
0.1 = 1 x 10-1
0.01 = 1 x 10-2
0.001 = 1 x 10-3
Q. So just how easy is the math that we do in CHEM 1005?
A. The math we do here gets no easier than multiplying fractions.
Equals one.
Unit Equations and Unit Factors
A unit equation is a simple statement of two equivalent quantities.
1 dollar = 10 dimes
1 dime = 10 pennies
Unit Factors
and 
Unit Analysis Problem Solving
1. Write down the units asked for.
2. Write down the given value.
3. Apply a unit factor to convert the units in the given value to the units in the answer.
A can of pop contains 12.0 fluid ounces of liquid. What is the volume of the can in quarts?
(1 quart = 32 ounces)
