measurements an introduction to all things quantitative

MeasurementsAn Introduction to All

Things Quantitative

Objectives After this presentation, you should be able

to:Describe the need for measuring things.Define quantitative in your own words.Name and describe five properties that

can be measured.Name the two systems of measurement

in use and how they differ.

Why Measure?

Measurements are among the earliest tools created by human beings.

They were created to accomplish many tasks including:The construction of dwellings and

monumentsThe fashioning of clothingThe bartering of food or raw

materials

What Can be Measured?

There are many physical properties that can be measured in the world around us.

Any property that can be measured and assigned a numerical value is said to be a quantitative property (think quantity).

Ex:

More Quantitative Properties

Time Mass Amount Electrical current Area Speed Velocity Density

Systems of Measurement

There are two major systems of measurement in use today:

The Imperial (standard) units

The Metric system

History of Measurement

LengthCubit - the length of the forearm

from the tip of the elbow to the tip of the middle finger.The cubit was divided into two spans

(the distance between the tip of the thumb to the tip of the pinky in an outstretched hand).

It was also divided into six hands (the width of the hand)

It was also dived into 24 digits (the width of the middle finger)

History of Measurement

The inch, foot, and yard were derived from these units.The roman foot was divided into 12

unciae and 16 digits.The roman mile consisted of 5000

feet, or 1000 paces.Queen Elizabeth I changed the mile

to what we know it as today, 5280ft.

The Need for Standardization

During the development of measurements, the same unit came to measure slightly different amounts depending upon:Where in the world it was used

By whom it was being used

Who the ruler was of the country/empire at the time

This discontinuity led to trade disputes and made life difficult for merchants…a change was needed.

The Metric System (SI)

The metric system (often abbreviated as the SI system due to its French roots) is the measurement system preferred by scientists around the world.

It was created by the French and is used by all but three nations (Burma, Liberia, and The United States) as their official system of measurement.

The Metric System

Perhaps the most significant advantage to the metric system is the fact that it address the seemingly arbitrary relationship between different units for measuring the same property within the imperial system.

Examples: Why are there 12 inches in a foot, but three feet in a yard? Why are their 2 pints in a quart, but 4 quarts in a gallon?

The Metric System

Because it is based on powers of 10, the metric system makes it easy to go from small units to large units within the base (length, mass, volume…etc).

For example, there are 10 millimeter in 1 centimeter, there are 1000 meters in 1 kilometer, there are 10 grams in one decagram.

Wrap up

Why measure? What is a quantitative property? What are five properties that can be

measured? What are the two dominant systems

of measurement in use today and how do they differ?

Taking Measurements

READING TO PRECISION

Accuracy Verse Precision

We often use the terms accurate and precise to mean the same thing, when in actuality, they have different meanings.

When working in the lab, it is important to be both accurate AND precise.

Accuracy

Accuracy refers to how close a measurement is to an established value.

Ex.) If you are playing darts and you come really close to the bull’s-eye on each of your throws, you would be accurate.

Accuracy

Notice that the “hits” are close to the center bull’s-eye, but they are not necessarily close to each other.

Example:

Imagine using a tape measure to measure a football field. You know the field is 100 yds long and when you measure it, you find it to be 99.3 yds. Your measurement would be considered accurate.

Percent Error

Percent error determines how accurate you are.

Equation:

% Error = | Accepted Value – Exp. Value | x 100

Accepted Value

Percent Error

A student reports the density of a pure substance to be 2.83 g/mL. The accepted value is 2.70 g/mL. What is the percent error for the student’s results?

Equation:

% Error = | Accepted Value – Exp. Value | x 100

Accepted Value

Precision

Precision refers to the repeatability of measurements (how close measurements are to each other).

Back to the dart analogy:

If you throw your darts at the board and all of your darts are grouped very close to one another, you are precise.

Notice that I made no reference as to how close the darts were to the bull’s-eye.

These marks would be considered very precise.

Notice that the marks are all close to each other, but not to the bull’s-eye.

Precision

What would the dartboard look like after a player who is both accurate and precise

had a turn?

Relating Accuracy & Precision to Measuring the Lab

In lab, we will have to make many measurements using a number of different tools.

Each tool should be read to the proper precision in order to be successful in lab.

In general, measurements should be made to have one smaller place than the smallest increment on the measuring device.

Determining Increments on Measuring Devices

First, subtract two adjacent numbered markings from each other.

In this example, when we subtract 8 mL from 6 mL, we get 2 mL. This means that 2 mL

of a liquid can fit between the marks on this graduated cylinder.

Next, count the number of marks, or increments, between the numbered marks.

When doing this, we also count the top numbered mark.For this example,

there are 10 marks.



Lastly, divide the amount measured between numbered marks by the number of increments between numbered marks.

For our example, we have 2 mL between 10 marks, so each mark represents 0.2 mL (2/10 = 0.2).

Example

This ruler has marks every tenth of a centimeter.

One place smaller than a tenth of a centimeter is a hundredth of a centimeter (millimeter).

cm

Certain vs. Uncertain Digits

The smallest increment on a measuring device is known as the smallest certain digit.This is because there is no estimating

involved in determining that number, it is just read from the device.

cm

Certain vs. Uncertain Digits

The uncertain digit is also called the estimated digit.In the meter stick example, the number

that would fall in the 1/100th of a cm place would be the estimated digit.

All measurement must end with 1, and only 1, estimated or uncertain digit!

Example:

• Imagine that the numbers listed on the ruler above represent millimeters. The measurement at letter A would be 0.50 mm.

• The first “0” is a certain digit because we can see that letter A lies before the 1 mm mark.

• The “5” tells us that letter A falls at a point between 0.5 and 0.6.

• It appears that letter A is directly on the 0.5 mark, but we still have to take our measurement to one estimated digit. Since it appears to be directly on the 0.5 mark, we show that to anyone who might look at our data by writing this measurement as 0.50 mm.

• We can tell by the measurement B is between the 1 and the 2 mm mark, so we know the first certain digit is going to be 1.

• We can also tell that letter B is at least 3 marks in from the 1 mm mark, but is not as far as the 4th mark, so the smallest certain digit is 3.

• Now we have to estimate the last digit. The arrow looks like it is not quite half way between the 3rd and 4th mark, so a reasonable estimate for the last digit would be 4.

• When we put these digits together, our measurement becomes 1.34 mm.

Practice!

What measurements are represented by letters C, D, E, and F?

– C

– D

– E

– F

DETERMINING SIGNIFICANT

FIGURESCrunching numbers with

accuracy & precision

WHAT IS A SIGNIFICANT FIGURE?

Significant figures =All the digits in a measurement that are known with certainty plus a last digit that must be estimated.

WHICH NUMBERS ARE SIGNIFICANT?

For the purposes of significant figures there are two major categories:

Nonzero digits:1,2,3,4,5,6,7,8,9

Zero digits: 0

38

NONZERO DIGITS

•All nonzero digits are significant

3269 cm – 4 significant figures

257 L – 3 significant figures

1.234567 mm – 7 significant figures

39

ZEROES

Zeroes take three forms:Leading zeroes

Trapped zeroes

Trailing zeroes

40

LEADING ZEROES•Leading zeroes are zeroes that come before the nonzero digits in a number. They are place holders only and are never considered significant.

0.123 L – 3 significant digits

0.000012 m – 2 significant digits

0.003578 mL – 4 significant digits

41

TRAPPED ZEROES• Trapped zeroes are zeroes between two nonzero digits. Trapped zeroes are always significant.

101 s – 3 significant figures

20013 m – 5 significant figures

0.3006 cm – 4 significant figures (the leading zero is not significant)

42

TRAILING ZEROES• Trailing zeroes are zeroes that follow nonzero digits. They are only significant if there is a decimal point in the number.

0.1200 mm – 4 significant figures

3000 s – 1 significant figure

250. mL – 3 significant figures

50.000 g – 5 significant figures

43

Exact Numbers

•Any number which represents a numerical count or is an exact definition has an infinite number of sig figs and is NOT counted in the calculations.

•12 inches = 1 foot

•25 desks in the room

More Examples of Exact Numbers•2 in 2r

•3 and 4 in ¾ r3

•Avogadro’s number is exactly 6.02 x 1023

•One inch is exactly 2.54 cm

Now you try!How many significant digits are in each of the following:

1.034 s

0.0067 g

12 apples

3000 m

72 people

Answers

•1.034 s - 4 significant figures

•0.0067 g - 2 significant figures

•12 apples - exact number

•3000 m - 1 significant figure

•69 people - exact number

Process for Addition/Subtraction

•Step #1: Determine the number of decimal places in each number to be added/subtracted.

•Step #2: Calculate the answer, and then round the final number to the least number of decimal places from Step #1.

Addition/Subtraction ExamplesExample #1:Round to tenths place.

Example #2:Round to hundredths place.

Example #3:Round to ones place.

23.456+ 1.2+ 0.05-------------- 24.706Rounds to: 24.7

3.56- 0.14- 1.3501--------------- 2.0699Rounds to: 2.07

14+ 0.735+ 12.0-------------- 26.735Rounds to: 27

Process for Multiplication/Division•Step #1: Determine the number of sig figs in each number to be multiplied/divided.

•Step #2: Calculate the answer, and then round the final number to the least number of sig figs from Step #1.

Multiplication/Division Examples

Example #1:Round to 1 sig fig.

Example #2:Round to2 sig figs.

Example #3:Round to 3 sig figs.

23.456x 1.2x 0.05-------------- 1.40736Rounds to: 1

3.56x 0.14x 1.3501---------------0.67288984Rounds to: 0.67

14.0/ 11.73

--------------1.193520887Rounds to: 1.19

Practice

Example #1: Example #2: Example #3:.

23.456x 4.20x 0.010--------------Rounds to: ?

0.001+ 1.1+ 0.350---------------Rounds to: ?

17/ 22.73

--------------Rounds to: ?

Scientific NotationMaking large and small numbers more manageable.

Scientific notation is a system for representing a number as a number between 1 and 10 multiplied by a power 10.

Scientific notation is most useful for presenting very large or very small numbers in a form that is easier to use.

For example:

253,000,000,000,000,000,000 = 2.53 x 1020

0.0000000000000000000253 = 2.53 x 10-20

In scientific notation, numbers are expressed as a number between 1 and 10 multiplied by 10 raised to an exponent.

120.3 = 1.203 x 100

100 is the same as 102

So….

120.3 = 1.203 x 102


1203 = 1.203 x 1000

1000 is the same as 103

So…

1203 = 1.203 x 103


An easy way to determine the exponent of ten is to count the decimal positions you move.

0.000001203 = 1.203 x 10-6

The decimal has moved backwards:

6 positions


An easy way to determine the exponent of ten is to count the decimal positions you move.

1203000000 = 1.203 x 109

The decimal has moved forwards: 9 positions

Practice Questions:

74,390,000 = __________ x 10_

0.000009998 = _________ x 10_

-0.0000623 = __________ x 10_

5.466 x 106 = __________

2.3 x 10-4 = ___________

Concept Question

•How would you express a number such as 1.5 or 0.2 in scientific notation?

Using Scientific Notation with the

Calculator

Use the “E” button to enter scientific notation as an exponent

Example: Enter the number 2.64 x 104

1) Type 2.64

2) Hit the 2nd button in the top left corner

3) Hit the , button with the “EE” above it

4) Then type the exponent – in this case type the number 4

Entering Numbers in Scientific Notation

• The “E” that appears on screen stands for the “x10” part of the number

• Try performing the following calculation on your calculator

1.45 x 1013 ÷ 6.67 x 105 =



• Now try this problem involving negative exponents:

5.13 x 10-9 3.48 x 10-1 =

• You can set your calculator to scientific mode which means it will give you your answer in scientific notation

• To switch between normal and scientific mode:

1) Hit MODE

2) Hit the right arrow to highlight “SCI”

3) This will change your calculator from normal to scientific mode

Scientific Notation vs. Normal Mode

Metric System Basics

Metric System• The metric system is based on a base unit

that corresponds to a certain kind of measurement

• Length = meter (m)• Volume = liter (L)• Weight (Mass) = gram (g)

• Prefixes plus base units make up the metric system – Example:

• Centi + meter = Centimeter• Kilo + liter = Kiloliter

Metric System• The three prefixes that we will use

the most are:– kilo– centi– milli

kilo hecto deca

Base Units

metergramliter

deci centi milli

Metric System• So if you needed to measure

length you would choose meter as your base unit– Length of a tree branch

• 1.5 meters – Length of a room

• 5 meters – Length of a ball of twine stretched out

• 25 meters

Metric System• But what if you need to measure a

longer distance, like from your house to school?– Let’s say you live approximately 10 miles

from school• 10 miles = 16093 meters

– 16093 is a big number, but what if you could add a prefix onto the base unit to make it easier to manage:• 16093 meters = 16.093 kilometers (or

16.1 if rounded to 1 decimal place)

Metric System• These prefixes are based on powers of 10.

What does this mean?– From each prefix every “step” is either:

• 10 times larger or

• 10 times smaller– For example

• Centimeters are 10 times larger than millimeters• 1 centimeter = 10 millimeters

kilo hecto deca

Base Units

metergramliter

deci centi milli

Metric System

– Centimeters are 10 times larger than millimeters so it takes more millimeters for the same length

1 centimeter = 10 millimetersExample not to scale

1 mm

1 mm

1 mm

1 mm

1 mm

1 mm

1 mm

1 mm

1 mm

1 mm

1 cm

40

41

41

40

Metric System• For each “step” to the right,

you are multiplying by 10• For example, let’s go from a base unit to centi

1 liter = 10 deciliters = 100 centiliters

2 grams = 20 decigrams = 200 centigrams

kilo hecto deca

meterliter

gramdeci centi milli

( 1 x 10 = 10) = (10 x 10 = 100)

(2 x 10 = 20) = (20 x 10 = 200)

Metric System• An easy way to move within the metric

system is by moving the decimal point one place for each “step” desired

Example: change meters to centimeters

1 meter = 10 decimeters = 100 centimetersor

1.00 meter = 10.0 decimeters = 100. centimeters

kilo hecto deca

meterliter


Metric System• Now let’s try our previous example from

meters to kilometers:16093 meters = 1609.3 decameters = 160.93

hectometers = 16.093 kilometers

• So for every “step” from the base unit to kilo, we moved the decimal 1 place to the left (the same direction as in the diagram below)

kilo hecto deca

meterliter


Metric System• If you move to the left in the diagram,

move the decimal to the left

• If you move to the right in the diagram, move the decimal to the right

kilo hecto deca

meterliter


Metric System• Now let’s start from centimeters and convert

to kilometers

400000 centimeters = ??? kilometers

400000 centimeters = 4.00000 kilometers

kilo hecto deca

meterliter


Metric System• Now let’s start from meters and convert to

kilometers

4000 meters = ??? kilometers

kilo hecto deca

meterliter


kilo hecto deca

meterliter


• Now let’s start from centimeters and convert to meters

4000 centimeters = ??? meters

Metric System• Now let’s start from meters and convert to

centimeters

5 meters = ??? centimeters

kilo hecto deca

meterliter


kilo hecto deca

meterliter


• Now let’s start from kilometers and convert to meters

.3 kilometers = ??? meters

Metric System• Now let’s start from kilometers and convert

to millimeters

4 kilometers = ??? millimetersor

4 kilometers = 40 hectometers = 400 decameters = 4000 meters = 40000 decimeters = 400000 centimeters = 4000000

millimeters

kilo hecto deca

meterliter


Metric System• Review

– What are the base units for length, volume and mass in the metric system?

– What prefix means 1000? 1/10?, 1/1000?– How many millimeters are in 12.5 hm?– How many Kiloliters are in 4.34 cL?

kilo hecto deca

meterliter


Dimensional AnalysisCONVERSIONS MADE EASY

Objectives Understand the need to convert units. After this lesson, you will be able to:

◦ Convert between units within the metric system.◦ Convert between historical units.◦ Convert between U.S. Customary units.◦ Convert from one system to another.

Why convert? Most of the world works within the metric system

◦ The U.S.A. is among only three countries that do not make extensive use of the metric system.

When working with numbers from a system that is not familiar to you, being able to place them into a system you are familiar with will help you understand their significance.

Understanding Fractions is Key! Often times, the relationship between two systems is expressed as a fraction we call a conversion factor.

These take the form of an expression consisting of two equal values expressed in different units.

Examples of Conversion Factors

1 mile is equal to 5280 feet, so a the possible conversion factors that we can make from this relationship are:

Take Note! Since the values in the denominator and the numerator represent the same thing, the conversion factor is a fraction that equals 1 (even though it doesn’t look like it at first).

What happens any time we multiply a number by 1?◦ We end up with a value that is equal to the value of

the original number!

Using Conversion Factors If we were given the task of finding how many miles there are between us and the ground when we are flying in an airplane, we would need two things:◦Our height in feet.◦The conversion factor that relates feet to miles.

The Problem When flying across the country, a Boeing 767 cruises at an altitude of 33,000 feet. If there are 5,280 feet in a mile, how many miles above the ground is a 767 flying?

The Importance of Units The key to properly setting up and solving this problem lies with the units.

The units tell us how to set up the conversion factor!

The goal when setting up the conversion factor is to get the units of the measurement given to us in the problem to cancel out, leaving only the units that we desire our answer to be expressed in.

The Set-up We begin by writing our given value (do not forget the units!)

33,000 feet We then draw an empty fraction being multiplied which gives us a place to place the conversion factor.

33,000 feet

Using the Correct Conversion Factor

Now that we have our empty fraction in place, we are ready to fill up the upper and lower spaces on the right hand side of the fraction with the conversion factor.

We can either place 5,280 ft over 1 mile, or 1 mile over 5,280 ft…only one way is correct!

Because our given value is in feet, we have to set our conversion factor up so that feet cancel.

The only way to get this to happen is to place feet in the denominator.

The Solution

• Since the units “feet” are both in the denominator and the numerator, they cancel out.

• This leaves us with “miles” as our only remaining unit, which just happens to be the unit we are looking for!

33,000 𝑓𝑒𝑒𝑡×1𝑚𝑖𝑙𝑒

5,280 𝑓𝑒𝑒𝑡=¿

Solving To solve the previous expression, we multiply all the numbers across the top and divide by the numbers on the bottom.

To prevent errors, make sure that you hit “enter” when you are finished multiplying and are ready to divide

33,000 x 1 “enter” ÷ 5,280 = 6.25 miles (don’t forget the units!!)

Things To Consider Since all conversion factors are represented by different numbers whose values are the same, we are never changing the value of our given number when we use conversion factors.

In order to ensure the proper use of conversion factors, place the value with the units that we desire in our answer on the top of the fraction, and the given units on the bottom.

Practice! Convert the following: 16,280 inches to miles

Practice! Convert the following: 1.5 years to seconds

measurements an introduction to all things quantitative

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