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Introduction to Math and Measurement
 

Displaying (and Processing) Data
 

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Study of Motion
 

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Motion in Two Dimensions
 

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Study of Thermodynamics
 

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Student Information
 

An Introduction to Math and Measurement

"The fundamentals: 'scientific skills, tools, and processes used by
Physicists and Astronomers' ".

Table of Contents and Links by topic below.

Intrinsic and Extrinsic Properties

The Metric System:

Fundamental Units of Measure:

Derived Units of Measure:

Working with Metric Measurements:

Using Scientific Notation:

Displaying and processing data:

Categories of Observations:

Accuracy and Precision:

Significant Digits:
 

Measurements in science and engineering are made in either the Metric or the British (English) system. The engineering people in the United States still often use the British system.

Regardless of what system is used measurements fall into one of two categories. These are called the intrinsic and extrinsic properties of measurements:

In physics and chemistry an intrinsic property (or intensive property) of a system is a physical property of the system which does not depend on the system size or the amount of material in the system. By contrast, an extrinsic property (or extensive property) of a system does depend on the system size or the amount of material in the system.

Examples of intrinsic properties are temperature, pressure and density.

Examples of extrinsic properties are mass, volume and energy.

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1. Measurement in the science of Physics and Astronomy, as in all of the sciences, today, are made in the metric system. It is used as the basis for all measurements.

a) The metric system: A system of measurement that uses "standardized" definitions of the base unit for each category of measurement

The metric system is measuring system defined in terms of the decimal system. Each category of measurement, such as length, has a base unit of measurement. (i.e. The base unit for length is the meter, the base unit for mass is the gram, the base unit for volume is the liter, and the base unit for time is the second.)

For measurement values larger or smaller than each category's base unit of measure the metric system uses prefixes. The more common prefixes are:
 
 

Prefix Symbol Value (x base unit) Value (x base unit)
Tera T 1 x 10^12 1,000,000,000,000
Giga G 1 x 10^9 1,000,000,000
Mega M 1 x 10^6 1,000,000
Kilo K 1 x 10^3 1,000
hecto h 1 x 10^2 100
deka da 1 x 10^1 10
base unit (symbol for unit used) 1 x 10^0 1
deci d 1 x 10^-1 0.1
centi c 1 x 10^-2 0.01
milli m 1 x 10^-3 0.001
micro m 1 x 10^-6 0.000001
nano n 1 x 10^-9 0.000000001
pico p 1 x 10^-12 0.000000000001
femto f 1 x 10^-15 0.000000000000001

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b) Fundamental units of measure: The fundamental units of measure are the measurements from which all other measurements are derived. Or to put it another way, all other units of measure are defined in terms of one or more of these measurements. Here is a list of the fundamental units of measure.
 
MEASUREMENT UNIT ABBREVIATION
Length meter m
Time second s
Mass kilogram kg
Electric Current ampere A
Temperature Kelvin K
Amount of Substance mole mol
Luminous Intensity candela cd

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c) Derived units of measure: The derived units of measure are all of the units of measure other than the fundamental units of measure. Derived units of measure are themselves derived from one or more fundamental units of measure. Some examples of derived are described here in the following table.
 
Measurement Unit of Measure symbol Unit(s) it is derived from Example
Area meter^2 m^2 meter 16 m^2
Volume meter^3 m^3 meter 8 m^3
Speed (velocity) meter / second m / s meter & second 5 m / s
Acceleration meter / second^2 m / s / s meter & second 2 m / s / s
Force newton N Kg, meter, & second 25 N
Density kg / meter^3 kg / m^3 Kg & meter 1.5 Kg / m^3

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d) Working with metric measurements:  In any measurement system it is often necessary to be able to work with more than one kind of measurement unit while doing math calculations. Units of measurements are handled and manipulated in calculations by essentially the same rules used for handling variable symbols. They can be added together, multiplied by one another, subtracted from each other, and divided by one another, just as any variable symbols such as x and y. Similarly, other operations, such as raising measurement values to powers or finding their roots, require that you treat the measurement units in the same manner as you would any variable symbol as you do the calculation

--> Rule for addition and subtraction: When adding or subtracting measurements, the units of measure must be the same.

i.e. Two masses, a 5 kg mass and a 10.5 kg mass, are being added together. These two measurements have the same unit so the numerical values can be added resulting in a sum having the same unit as the individual measurements.
 

Problem - add masses Measurements Operation
1st mass   5 kg addition
2nd mass 10.5 kg sum = mass(1) + mass(2)
Answer 15.5 kg

--> Rule for multiplication and division: When multiplying or dividing measurements, the units do not have to be the same. However, all units, except those that legitimately divide out (sometimes referred to as canceling), must be retained.

i.e. An object travels 3,000 m in a time interval of 15 seconds. Using the equation for speed of an object, v = d / t, calculate the speed of the object.
 
 

Problem - calculate speed Measurements operation
distance 3,000 m division
time 15 s speed = distance / time
Answer 200 m / s

--> Other operations: Always perform other math operations, when using measurement unit symbols, as you would any variable symbol values such as x and y. All units must remain in the answer along with the numerical value arrived at, unless they actually divide out, and are legitimately gone from the answer.

An Example: A rectangular shaped object is placed upon a scale and found to have a mass of 25 grams. The dimensions of the object, measured with a ruler, were found to be 5 cm long, 3 cm high, and 2 cm tall. Calculate (1) the volume of the object and (2) the density of the object.
 

(1st) Volume is found by the equation V = L x W x H (length, width, and height, respectively). so the calculation looks like:
 
 

In this problem we are using a problem solving format to find volume.*
Problem step labels Problem statements Comments
Given statement (G:) Length is 5 cm, Width is 3 cm, and Height is 2 cm write what you are given
Find statement (F:) Find the volume of the object write what you are to find
Equation statement (E:) V = L x W x H write the equation
Substitution Statement (S:) V = 5 cm x 3 cm x 2 cm write the substitution of values into the equation
Answer statement (A:) V = 30 cm^3 write the answer

(2nd) Density is found by dividing the mass of the object by its volume. The equation is D = M / V.
 
 

In this problem we are using a problem solving format to find density.*
Problem step labels Problem statements Comments
Given statement (G:) Mass is 25 g, Volume is 30 cm^3 write what you are given
Find statement (F:) Find the density of the object write what you are to find
Equation statement (E:) D = M / V write the equation
Substitution statement (S:) D = 25 g / 30 cm^3 write the substitution of values into the equation
Answer statement (A:) D = 0.83 g / cm^3 write the answer

*The problem solving format places emphasis on displaying the solution to a problem in a logical series of steps. This approach emphasizes critical thinking skills and over time will improve your ability to solve problems quickly. Always remember that, if you cannot get the units of measurement in your solution (not the correct one) to work out correctly, there is no point picking up your calculator, because when the units of measurement do not work out correctly, there is something wrong in your choice of math operations you have chosen to solve the problem. This applies to all kinds of problems involving the use of math, not just problems using the metric system.

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2. Working with LARGE and small measurements: Measurements can have reasonably manageable numbers and be relatively easy to manipulate. On the other hand, especially in the sciences, numbers associated with measurements can be quite large, as in the case of Astronomical data, or numbers associated with measurements can be quite small, as in the case of Quantum Theory. To deal with the very large and the very small, mathematics has given us exponential notation. Exponential notation as used in the sciences is often called scientific notation. Scientific notation is defined as "A method of writing or displaying numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10. The scientific notation of 10,492, for example, is 1.0492 × 10^4. The basics of scientific notation is modeled in the following table.
 
 
Basic examples of Scientific Notion
Numerical Value Scientific Notation format Comment
1,000,000,000 1 x 10^9 one billion
1,000,000 1 x 10^6 one million
1,000 1 x 10^3 one thousand
100 1 x 10^2 one hundred
10 1 x 10^1 ten
1 1 x 10^0 one
0 ---* ---
0.1 or 1/10 1 x 10^-1 one tenth
0.01 or 1/100 1 x 10^-2 one hundredth
0.001 or 1/1000 1 x 10^-3 one thousandth
0.000001 or 1/1,000,000 1 x 10^-6 one millionth
0.000000001 or 1/1,000,000,000 1 x 10^-9 one billionth

*The number 0, itself, is not a power of ten. It can't be written as 10 to the something power.

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3. Displaying and processing data: Gathering, recording, and processing data are skills associated with serious research into any human endeavor. An important data chart compiled by using accurate measurements produced by astronomers is the Planetary Data Chart. A version of this chart appears below.
 
 
Planetary Data Table
Name of Object Pronunciation Average distance from the sun (m) Mass of the object (kg) Size of object Average radius (m)
Sun Sun 0 1.991 x 10^30 6.96 x 10^8
Mercury Mer qu ree 5.8 x 10^10 3.2 x 10^23 2.43 x 10^6
Venus Vee nus 1.081 x 10^11 4.88 x 10^24 6.073 x 10^6
Earth Earth 1.4957 x 10^11 5.979 x 10^24 6.3713 x 10^6
Mars Mars 2.278 x 10^11 6.42 x 10^23 3.38 x 10^6
Jupiter Joo pi ter (short i) 7.781 x 10^11 1.901 x 10^27 6.98 x 10^7
Saturn Sat ern 1.427 x 10^12 5.68 x 10^26 5.82 x 10^7
Uranus Yoo ray nes 2.870 x 10^12 8.68 x 10^25 2.35 x 10^7
Neptune Nep toon 4.5 x 10^12 1.03 x 10^26 2.27 x 10^7
Pluto Ploo toh 5.9 x 10^12 1.2 x 10^22 1.15 x 10^6

Data Charts such as the one above are referred to as data tables. A table provides a means of organizing and displaying data that has been collected through observations.

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4. Categories of Observations: Observations fall into two categories: One of the types of observation is the qualitative observation, while the other is the quantitative observation.

--> The qualitative observation is a descriptive observation. It involves a clear and concise description of an object or circumstance. It does not involve the making of measurements (quantities).  Some examples of qualitative measurements are:
 
 

Example 1 The sky is blue
Example 2 The leaves on the tree are green
Example 3 The metal is soft enough to bend
Example 4 The surface of the wood is rough
Example 5 The liquid is very thick

--> The quantitative observation is a measurement. It involves an accurate measurement expressed with as much precision as the measuring tool being used will allow. Some examples of quantitative measurements are:
 
 

Example 1 The length of the table is 2.0000 m
Example 2 The person's height is 150 cm
Example 3 The mass of the truck is 900 kg
Example 4 The volume of orange juice is 1.89 L
Example 5 The speed of the object is 1.5 m/s

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5. Accuracy and Precision: Accuracy and Precision always come into play when anyone is making quantitative observations (measurements).

Accuracy is defined as "The ability of a measurement to match the actual (correct) value of the quantity being measured." It is a concept that deals with whether a measurement is correct when compared to the known value or standard for the particular measurement being made. When this comparison is being made using a percent, it is referred to a percent error.

--> Example of accuracy. Suppose that a carpenter is making a cabinet to fit in an exact space having a width of 1.15 m. Now what happens if in his effort to cut the materials he makes his cuts 1.14 m. What effect does this have on the making of the cabinet? Well on the first look, it may seem that this 0.01 difference is quite small. However, upon closer inspection, 0.01 m is a whole centimeter, which is close to one half of an inch. The cabinet will now show a gap of almost a half an inch, because the carpenter was not accurate in measuring. This why in terms of measurement, carpenters, etc., have always said "measure twice and cut once".

In the above situation, more careful measurement would reveal the true (correct) width, either by measuring more carefully or by using a more precise ruler, perhaps one that measured to the 3rd or even the 4th decimal place should be used. However, when measurements are being made of the very large (astronomical distances) or of the very small (subatomic scale) for the very first time, accuracy is difficult to determine, because no one has access to correct answer. In this kind of a situation, we turn to precision and statistics to help us ascertain the usefulness of our measurements.

Precision is defined as "The ability of a measurement to be consistently reproduced." and "The number of significant digits to which a value can be reliably measured." The first definition is the main focus of scientists and engineers who are looking at the work of others. After studying what the other people had done, they set out to test the shared (often in the form of published reports) information to see whether they can reproduce the results. Reproducible results are much more likely to be accurate (correct) than results that cannot be reproduced. The second definition is the main focus of anyone using a measuring device. How precise or exact the measuring device is that you are using is based upon the degree to which the measurement units are broken down (subdivided).  Precision is the concept which deals with the degree of exactness when using a specific measuring tool and expressing a measurement made with that tool. The precision of any measuring tool and the exactness of the measurement made with that tool is limited by how precise the instrument is capable of being. the precision of a measuring tool is based upon the smallest defined unit of measure on the devise. The number of digits used to express a measurement when using a tool whose degree of precision (smallest unit available) are called significant digits. See the example below:
 
 

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6. Handling Measurements in Calculations: The use of significant digits in problem solving.

In all problem solving situations where measurements are involved, not only should you carry units along with each math operation performed, but you need to take into consideration the number of significant digits being expressed in each measurement being used, so each and every calculation using measurements is never expressed with more precision than the measurements themselves. That would be absurd (or, if you prefer, ridiculous). In working with measurements expressed in significant digits, there are three things you always need to consider. They are:
 
 

Recognizing the number of significant digits in another persons measurement This requires that you can properly identify the use of zeros in measurements
Using significant digits in both the addition and the subtraction operations This requires that you can identify the number of significant digits in measurements
Using significant digits in both the multiplication and and the division operations This requires that you can identify the number of significant digits in measurements

The Rules of zeros (0's) used in identifying the number of significant digits in a measurement. The rules are as follows:
 
 
 

Rules for determining when Zeros (0's) are Significant
Rule #1 All none zero digits in a number are significant. Example: 25.34 cm has four significant digits
Rule #2 All zeros between any two non zero digits are significant Example: 100.052 g has six significant digits
Rule #3 When no decimal point is designated, all zeros to the right of the last non zero digit are not significant
(Writing in a designated decimal point would indicate that all of the zeros to the right of the last non zero digit are significant)
(A bar can be placed over a zero indicating that it and any other zeros between it and the last non zero digit are significant)
Example 1: 23,300 km has three significant digits the 2, 3, and 3.

Example 2: 23,000. km (notice placement of the decimal) has five significant digits

Example 3: If a bar were written above the 2nd zero following the 3 in the value 23,000 km, there would be four significant digits

Rule #4 All zeros that come before the first non zero digit, which, itself,  comes after a designated (written) decimal point, are only place holder zeros and place holder zeros are not considered to be significant. Example 1: 0.000052 Gm has only two significant digits

Example 2: 0.00205 cm has only three significant digits.

Rule #5 All zeros after a designated decimal which are also to the right of a non zero digit are significant. Example: 0.050400 km has five significant digits
Rule #6 When writing numbers in scientific notation, all non significant digits, including non significant zeros are dropped from the value being written. Example 1: 
0.0060720 kg = 6.0720 x 10^-3 kg
Example 2:
705,200 km = 7.052 x 10^5 km

Significant Digits in Calculations: The precision in the answer to a calculation where new information is derived from existing information cannot be greater than the precision of the least precise measurement.

Now the handling of this sort of situation where measurements are used in simple math operations is divided into two categories of math problems. They are the Addition-Subtraction problems and the Multiplication-Division problems.

--> 1. Addition and Subtraction Problems: When adding or subtracting measurements where precision and significant digits are being used, the answer can be no more precise than the measurement with the least amount of precision. In the following example you will be rounding off to the second decimal place (the one-hundredths column), because nothing is known about the 3rd decimal value in the measurement 3.11 kg. This makes the 3rd decimal in the final answer too uncertain to be recorded. So we round off correctly to the 2nd decimal place. In Addition and subtraction it is the measurement with the least number of decimal places that affects how much you are to round off to.
 
 

Example of Sig. Digs. in Addition
55.656    kg Always
14.3430  kg    Align
                         + 03.11        kg <---- least significant Your
         ANSWER 73.109    kg --> 73.11 kg Numbers

 
 
An Example of Subtraction
10.9 cm - 8.364 cm = 2.536 cm = 2.5 cm 
Note: 10.9 cm has only one decimal place

--> 2. Multiplication and Division Problems: When multiplying or dividing measurements where precision and significant digits are being used, the answer can be no more precise than the measurement with the least amount of precision. In the following example you will be rounding off to only two digits, because in multiplication (and division) you round off to the number of digits in your answer equal to the value with the least amount of significant digits being used in the problem. In multiplication and division it is the measurement with the least number of significant digits that affect how you are to round off.
 
 
 

Example of Sig. Digs. in Multiplication
4.11         m
                    x   3.2         m <----least significant
  822                      <-- Watch your
1233                      <-- Alignment
         ANSWER 13.152     m --> 13 m^2 Here

 
 
An Example of Division
3.14165 g / 11.9 cm^3 = 0.2640042 = 0.264 g/cm^3
Note: 11.9 has only 3 significant digits

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