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Euclidean Definitions, Axioms and Postulate Video Lecture - Class 9
FAQs on Euclidean Definitions, Axioms and Postulate Video Lecture - Class 9
| 1. What are Euclidean definitions, axioms, and postulates? |  |
Euclidean definitions, axioms, and postulates are the fundamental concepts and principles that form the basis of Euclidean geometry. Euclid, an ancient Greek mathematician, established these principles in his famous work called "Elements." Definitions describe the basic terms used in geometry, axioms are self-evident truths that do not require proof, and postulates are assumptions accepted as true without proof.
| 2. What is the importance of Euclidean definitions, axioms, and postulates in geometry? | |
Euclidean definitions, axioms, and postulates play a crucial role in geometry as they provide a logical foundation for reasoning and proving geometric statements. These concepts help establish the properties and relationships of points, lines, angles, and shapes, enabling us to solve geometric problems and derive new theorems based on established principles.
| 3. Can you provide examples of Euclidean definitions, axioms, and postulates? | |
Certainly! Here are a few examples:
- Definition: A line is a straight path that extends infinitely in both directions.
- Axiom: Two distinct points can be joined by a unique straight line.
- Postulate: Given any straight line segment, a circle can be drawn with that segment as its radius and one endpoint as its center.
| 4. How do Euclidean definitions, axioms, and postulates differ from each other? | |
Euclidean definitions, axioms, and postulates are distinct in their nature and purpose. Definitions provide precise explanations of geometric terms, axioms are self-evident truths used as a starting point for reasoning, and postulates are accepted assumptions used to build logical arguments. In essence, definitions clarify what things are, axioms state what is true, and postulates establish what can be assumed without proof.
| 5. Are Euclidean definitions, axioms, and postulates applicable only to Euclidean geometry? | |
While Euclidean definitions, axioms, and postulates were initially developed for Euclidean geometry, they have also been adapted and extended to other branches of mathematics and geometry. Non-Euclidean geometries, such as spherical or hyperbolic geometry, have their own sets of definitions, axioms, and postulates that differ from those of Euclidean geometry. However, the fundamental concepts and principles established by Euclid still hold immense value and are widely used in various mathematical disciplines.
Text Transcript from Video
welcome to euclid's elements
now before we can learn anything we must
first know what exists and what the
rules are in euclid's world of geometry
so I will list some basic definitions to
get us started but i won't list everything
because it's just too tedious and what I
really want to do is give you a quick
general idea for what the elements is
all about
so our first definition is points exist
and we label them with capital letters
definition number two: a line exist
between two points
we call this line AB
definition three: a plane exist between
three or more points
definition number four:
when two lines touch or cross angles are
formed
when all the angles of the two lines are
equal
the angles are called right angles and
the lines are perpendicular lines
definition number five:
with a center A and radius AB we can
create a circle
we call this circle AB
all radii of the same circle are equal
so that's all of our definitions for now
and we move on to some rules used within
euclid's elements and we call these
postulates
postulate number one: between any two
points a line can always be drawn
postulate number two: given a straight
line you can always extend it by adding
another point
postulate number three:
you can always make a circle with any
two points
our last postulate number four
all right angles are always equal
now our last
set of rules are called axioms which are
not only used within euclid's elements
but are rules used throughout all of
mathematics
so our first axiom is things which are
equal to the same thing are also equal
to one another
now an easy way to represent this is
if A is equal to B
and B is equal to C
then A is equal to C
our second axiom says if equals be added
to equals the wholes are equal
so this means that if A is equal to C
and B is equal to D, then A plus B must equal C plus D
axiom three: if equals be subtracted
from equals the remainders are equal
so A is equal to C, B is equal to D
A minus B is equal to C minus D
our last axiom is a special axiom which
says that things which coincide with one
another are equal to one another
we will leave the explanation for this
axiom for later
so this is our list of definitions and
rules which will help us create some
very interesting facts and the next
videos
you are now ready to learn euclid's geometry
now before we can learn anything we must
first know what exists and what the
rules are in euclid's world of geometry
so I will list some basic definitions to
get us started but i won't list everything
because it's just too tedious and what I
really want to do is give you a quick
general idea for what the elements is
all about
so our first definition is points exist
and we label them with capital letters
definition number two: a line exist
between two points
we call this line AB
definition three: a plane exist between
three or more points
definition number four:
when two lines touch or cross angles are
formed
when all the angles of the two lines are
equal
the angles are called right angles and
the lines are perpendicular lines
definition number five:
with a center A and radius AB we can
create a circle
we call this circle AB
all radii of the same circle are equal
so that's all of our definitions for now
and we move on to some rules used within
euclid's elements and we call these
postulates
postulate number one: between any two
points a line can always be drawn
postulate number two: given a straight
line you can always extend it by adding
another point
postulate number three:
you can always make a circle with any
two points
our last postulate number four
all right angles are always equal
now our last
set of rules are called axioms which are
not only used within euclid's elements
but are rules used throughout all of
mathematics
so our first axiom is things which are
equal to the same thing are also equal
to one another
now an easy way to represent this is
if A is equal to B
and B is equal to C
then A is equal to C
our second axiom says if equals be added
to equals the wholes are equal
so this means that if A is equal to C
and B is equal to D, then A plus B must equal C plus D
axiom three: if equals be subtracted
from equals the remainders are equal
so A is equal to C, B is equal to D
A minus B is equal to C minus D
our last axiom is a special axiom which
says that things which coincide with one
another are equal to one another
we will leave the explanation for this
axiom for later
so this is our list of definitions and
rules which will help us create some
very interesting facts and the next
videos
you are now ready to learn euclid's geometry
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