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• Measurement is a crucial process involving the comparison of an unknown quantity to a known standard.
• The word "measurement" comes from the Greek word "metron," meaning a limited proportion.

Nonstandard Units of Measure

• Nonstandard units include handspans, cubits, arm spans, paces, and foot spans.
• These units are based on the human body (e.g., dangkal, dipa, hakbang).

US Customary or English Systems of Measurement

• The U.S. Customary System is based on the British Imperial System.
• Its units evolved from nonstandard measures during medieval times.

Common Units in the Customary System of Measurement:

• Length:
• Inch (in or ") is approximately 1/2 of a foot.
• Foot (ft or ') is 12 inches, about the length of an adult's foot.
• Yard is 3 feet.
• Mile (mi) is 5,280 feet.
• Weight:
• Ounce (oz) is 1/16 of a pound, about the weight of a pen.
• Pound (lb) is 16 ounces, about the weight of three oranges.
• Ton is 2,000 pounds, about the weight of a container van.
• Capacity:
• Pint (pt) is 2 cups, about the weight of a pen.
• Quart (qt) is 2 pints, about the weight of three oranges.
• Gallon is 4 quarts, about the weight of a container van.
• Note: 2" means 2 inches, and 2' means 2 feet.

Metric System

• It is based on decimals, like the numeration and monetary system.
• Greek and Latin prefixes facilitate easy conversion between units.

Metric System Prefixes:

• Tera (T): 10^12 (1,000,000,000,000)
• Giga (G): 10^9 (1,000,000,000)
• Mega (M): 10^6 (1,000,000)
• Kilo (k): 10^3 (1,000)
• Hecto (h): 10^2 (100)
• Deka (da): 10^1 (10)
• Deci (d): 10^-1 (0.1)
• Centi (c): 10^-2 (0.01)
• Milli (m): 10^-3 (0.001)
• Micro (μ): 10^-6 (0.000001)
• Nano (n): 10^-9 (0.000000001)
• Pico (p): 10^-12 (0.000000000001)

Conversion Metric System for Length

• Units increase or decrease by a factor of 10 (Milli, Centi, Deci, Meter, Deca, Hecto, Kilo).

Equivalent Units in Customary and Metric System:

• Inch (in) multiplied by 2.54 equals centimeter (cm).
• Yard (yd) multiplied by 0.9144 equals meter (m).
• Mile (mi) multiplied by 1.609 equals kilometer (km).
• Pound (lb) multiplied by 0.4536 equals kilogram (kg).
• Gallon (gal) multiplied by 3.7854 equals liter (L).

SI Base Unit of Measure

• SI (Système International d'Unités) is based on the metric system.

SI Base Units in the Metric System:

• Length:
• Kilometer (km) is 1000 meters.
• Meter (m) is the length from the left shoulder to the tip of the outstretched right hand.
• Centimeter (cm) is 1/100 of a meter, about the length of a peanut.
• Millimeter (mm) is 1/1000 of meter, about the width of a period.
• Mass:
• Kilogram (kg) is 1000 grams, mass of a medium-sized watermelon.
• Gram (g) is the mass of a paper clip.
• Milligram (mg) is 1/1000 of gram, mass of a drop of water.
• Volume:
• Liter (L) is the amount of water in a pitcher.
• Milliliter (mL) is 1/1000 of a liter, about a half drop of water from a dropper.

Fundamental Quantities and Their Units

• Length: Measure of distance; basic unit is meter (m).
• Mass/Weight: Mass refers to the amount of matter, Sl unit is Kilogram (kg).
• Time: Time is measured by the amount of the second(S).
• Temperature: Degree of hotness and coldness; metric unit is Celsius (°C).
• Area: The surface included within a particular dimension; basic unit is square meter (m^2).
• Volume and Capacity: Volume measured in length x width x height.
• Speed: The rate of distance traveled per unit time.

Length

• Length is a physical quantity and is a measure of distance.
• The basic unit for length is meter (m).

Mass/Weight

• Weight relates to the gravitational pull on a mass and varies by location.
• Mass refers to the amount of matter in an object and remains constant.
• The SI standard unit of mass is the kilogram (kg).
• The gram (g) is 1/1000 of a kilogram.

Time

• The unit of time is the second (S).

Units of time and their equivalence

• 1 century = 100 years
• 1 score = 20 years
• 1 decade = 10 years
• 1 year = 12 months or 365 1/4 days
• 1 week = 7 days
• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds

Temperature

• Temperature measures hotness and coldness
• The metric unit is degrees Celsius (°C)
• Another unit of temperature is degree Fahrenheit (°F).

Area

• Area is the measurement of a surface within a defined dimension.
• The basic unit for area is square meter (m^2).
• For small areas use square centimeter (cm^2).
• For large land area use square kilometer (km^2) or hectare (ha).

Volume and Capacity

• Capacity refers to how much a container can hold.
• Volume refers to how much space a region takes up.
• Volume is measured in three dimensions (length x width x height).

Pyramid

• A pyramid is a three-dimensional solid consisting of a polygonal base and triangular faces that converge at a single point (apex).

Rectangular Pyramid

• The base is a rectangle with 4 triangular faces meeting at the apex.

Triangular Pyramid

• A three-dimensional figure with a triangular base.
• Features three congruent triangular faces with each face intersecting the base at the same angle.

Speed

• The rate of distance traveled per unit of time.
• Kilometers per hour (kph) or (km/h) "kilometer per hour"
• Meters per second (m/s) means "meter per second"

Average Speed

• Total distance traveled by an object divided by the total time taken.
• Units are usually meters per second, miles per hour, or kilometers per hour

The Language of Sets

• Set: Is a group or collection of objects.
• Element: Each object/member in a set.
• Cardinality: The number of elements in a set.
• Finite Sets: Sets with countable numbers of elements.
• Infinite Sets: Sets with an infinite number of elements.

Three ways in naming a set

• List or Roster Method: List all the element of a set within a paid of braces {}
• Descriptive Method: Described by wiriting a description of its elements.
• Set Builder Notation: Described using variables.

Set Theory Symbols

• Set brackets: {}
• x is an element of set A: x∈A
• x is not an element of set A: x∉A
• There exists, or there does not exist: ∃
• Such that: | or :
• Equal sets: A=B
• Cardinality or order of set A: n(A) or |A|
• Power set of A: P(A)
• for all: ∀
• Set: {}
• Intersection: ∩
• Union: ∪
• element of: ∈
• not an element of: ∉
• a subset of: ⊆
• a subset of or equal to: ⊆
• not a subset of: ⊄
• neither a subset of nor equal to: ⊈
• a superset of: ⊃
• superset of or equal to: ⊇
• not a superset of: ⊅
• neither a superset nor equal to: ⊉
• empty set: Ø
• number of elements in set A: n(A)
• complement of A: A' or A
• power set of A: P(A)

Types of sets

• Empty Set or Null Set: A set containing no elements. (ф or {})
• Singleton Set or Unit Set: A set containing only one element. {5}
• Finite Set: A set containing a countable number of elements. {1, 8, 27, 64, 125, 216}
• Infinite Set: A set with an uncountable number of elements. {1, 2, 3, 4,...}
• Equivalent Set: Sets with the same cardinality.
• Equal Sets: Sets containing exactly the same elements.
• Unequal Set: Sets with at least one different element.
• Overlapping Set: Sets with at least one common element.
• Disjoint Set: Sets with no common element.
• Subset: A set where all elements are also members of another set. Ex. {4, 6, 8} is a subset of {0, 2, 4, 6, 8, 10}
• Power Set: The set containing all the possible subsets of a given set.
• Universal Set: A set containing all the elements related to a particular subject.

Numbers

• Counting Numbers: Natural number used in counting. Example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...
• Natural Numbers: Part of the number system, including all the positive numbers from 1 to infinity. Example: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12...
• Prime Numbers: A number greater than 1 with exactly two factors, 1 and the number itself. Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...
• Even Numbers: Numbers that cam e divided into two equal groups of pairs and are exactly divisible by 2. Example: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22...
• Odd Numbers: Those numbers that are not completely divisible by 2. Example: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23...

Relations on Sets

• Equal Sets: Two sets that have the same elements in them.
• Equivalent Sets: If two sets have the same number of elements, though the elements are different.
• Universal Sets: Contains all the elements being considered in a given situation.
• Subsets: A set ‘A' is said to be a subset of B if every element of A is also an element of B, denoted as A⊆ B. Even the null set is considered to be the subset of another set.

Operations on Sets

• Intersection (A ∩ B): The intersection of two sets is the set containing the elements that are common to both sets.
• Disjoint Sets (A ∩ B) = {}: When A and B have no common elements, we say that A and B are DISJOINT SETS.
• Union of Sets: The union of two sets written by A U B, is the sets of all elements that are in A on in B, or both A and B.
• Difference of Two Sets: If A and B are two sets, then their Difference is given by A-B or B-A
• Complement of a Set: The complement of a set and is is written as A'. Whreas A' ia asetof elements thta are not in A.