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Analytical Geometry
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14 Mar 2008 22:32:57 IST
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A "point" is an infinitely small entity at a specific location on a
number line, plane, 3-D space, etc. When we talk about points, we are
referring to one specific location.
number line, plane, 3-D space, etc. When we talk about points, we are
referring to one specific location.
For example, along a number line the number "2" exists at just one
point. I said that points are infinitely small because the point at
'2' is different from the point at '2.000000001'. Here's a picture of
a number line:
point. I said that points are infinitely small because the point at
'2' is different from the point at '2.000000001'. Here's a picture of
a number line:
The point 2
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-infinity <==...---(-1)-----0-----1-----2-----3--... ==> infinity
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\/
-infinity <==...---(-1)-----0-----1-----2-----3--... ==> infinity
Okay, so this makes sense: if you want to distinguish one place
along a number line, you "point" at it. You label that place
with the corresponding number, and refer to it with that number.
along a number line, you "point" at it. You label that place
with the corresponding number, and refer to it with that number.
Now, how do you distinguish a location in 2-dimensional space (i.e.
a sheet of paper)? Imagine that we have two number lines, one
horizontal and the other vertical. We are "pointing" at a place "p":
a sheet of paper)? Imagine that we have two number lines, one
horizontal and the other vertical. We are "pointing" at a place "p":
...
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2
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1 p
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...---(-1)-----0-----1-----2-----3--...
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...
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2
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1 p
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...---(-1)-----0-----1-----2-----3--...
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...
How do we describe where the point 'p' is? We can't just say p is
at 2 because we don't know which number line that refers to - is it
at 2 along the horizontal number line, or 1 along the vertical number
line?
at 2 because we don't know which number line that refers to - is it
at 2 along the horizontal number line, or 1 along the vertical number
line?
To describe where 'p' is, you must talk about where it is both
horizontally AND vertically. So, you can say:
horizontally AND vertically. So, you can say:
'p is at 2 horizontally, and 1 vertically'.
However, this is a mouthful to say. Because describing points in 2
dimensions is really useful, people have defined some conventions to
make life easier. They call the horizontal number line the 'x-axis',
and the vertical number line the 'y-axis'. The convention for talking
about 2-dimension points is to write: ( position along x-axis ,
position along y-axis ).
dimensions is really useful, people have defined some conventions to
make life easier. They call the horizontal number line the 'x-axis',
and the vertical number line the 'y-axis'. The convention for talking
about 2-dimension points is to write: ( position along x-axis ,
position along y-axis ).
Therefore,
'p is at (2, 1)'
2-dimensional points can be described by any pair of numbers.
For example, (4,5) (6.23432, 3.14...) and (-12, 4) are all points.
For example, (4,5) (6.23432, 3.14...) and (-12, 4) are all points.
Sometimes people want to describe a point in three dimensions.
To do this, they need to use a triplet of numbers like (1, 2, -5)
- do you see why?
To do this, they need to use a triplet of numbers like (1, 2, -5)
- do you see why?
I hope this helps.
14 Mar 2008 22:43:19 IST
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2 people liked this
If you still have the courage and patience, read on ---
The question you asked is actually quite a subtle one, and in a
certain sense it makes a difference what kind of math you're doing.
If you're doing Geometry, then no amount of physical torture will get
me to define a point; it's an undefined term, and it's undefined on
purpose.
certain sense it makes a difference what kind of math you're doing.
If you're doing Geometry, then no amount of physical torture will get
me to define a point; it's an undefined term, and it's undefined on
purpose.
So this is really about math in general: what is an undefined term?
Well, as you may know, math is about making definitions and proving
theorems. For example, I can define a circle to be a set of points
equidistant from a given point, and I can prove a theorem about that
circle, such as "the perpendicular bisector of a chord of a circle
will pass through the center of the circle." But in order to make
these definitions and prove these theorems, I need to start from
_something_.
Well, as you may know, math is about making definitions and proving
theorems. For example, I can define a circle to be a set of points
equidistant from a given point, and I can prove a theorem about that
circle, such as "the perpendicular bisector of a chord of a circle
will pass through the center of the circle." But in order to make
these definitions and prove these theorems, I need to start from
_something_.
That's the role of the UNDEFINED TERMS and AXIOMS in math. When I
defined the circle as a set of points, I used objects I already had
(points) to define a new kind of object (the circle). So every time
we define a new object we have to have some old object to base it on.
defined the circle as a set of points, I used objects I already had
(points) to define a new kind of object (the circle). So every time
we define a new object we have to have some old object to base it on.
If you think about the structure of math as a tree, there has to be
something at the bottom of the tree, some objects that aren't defined.
A point is one of these objects. It is undefined. It is just an
object. In geometry, people usually think of points, lines, and
planes as undefined objects (also known as undefined terms).
something at the bottom of the tree, some objects that aren't defined.
A point is one of these objects. It is undefined. It is just an
object. In geometry, people usually think of points, lines, and
planes as undefined objects (also known as undefined terms).
So what are axioms? Well, they're statements that we don't have to
prove (much as undefined terms are objects we don't have to define).
A traditional example of an axiom in geometry is the statement "given
any two points, there is one and only one line that passes through
them." This statement is just accepted as true, so that we have a
starting point, something we can use to prove theorems. It also does
something else: it tells us something about points and lines. Any
concepts we have in our heads about points and lines MUST satisfy this
axiom. If we're thinking of points as bottles of beer and lines as
telephone poles, then we have a problem, because I can show you a
couple of bottles of beer that don't have a telephone pole connecting
them.
prove (much as undefined terms are objects we don't have to define).
A traditional example of an axiom in geometry is the statement "given
any two points, there is one and only one line that passes through
them." This statement is just accepted as true, so that we have a
starting point, something we can use to prove theorems. It also does
something else: it tells us something about points and lines. Any
concepts we have in our heads about points and lines MUST satisfy this
axiom. If we're thinking of points as bottles of beer and lines as
telephone poles, then we have a problem, because I can show you a
couple of bottles of beer that don't have a telephone pole connecting
them.
So that's how it goes in geometry. In other parts of math, for
instance when you're using the coordinate axes, a point may not be an
undefined term at all - it can just be a list of numbers, such as
saying "consider the point (3,2)."
instance when you're using the coordinate axes, a point may not be an
undefined term at all - it can just be a list of numbers, such as
saying "consider the point (3,2)."
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but every 1 conisider as wrong ..
plz. comment i m right or wrong!!!