Connecting Geometric and Algebraic Concepts
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Currently studying the graphs section from Spivak's Calculus, and was slightly confused about one part. In the text he writes:
"The rigorous proof of any statement connecting geometric and algebraic concepts would first require a real proof (or a precisely stated assumption) that the points on a straight line correspond in an exact way to the real numbers. Aside from this, it would be necessary to develop plane geometry as precisely as we intend to develop the properties of real numbers ... We shall use geometric pictures only as an aid to intuition"
Later in the text, he derives the equation of a circle from it's definition. The definition is: "A circle with center (a, b) and radius r > 0 contains all the points (x, y) whose distance from (a, b) is equal to r."
Now, here's the part I'm confused about:
Deriving the equation of the circle depended on the definition of the circle, and the definition of distance between two points on the Cartesian plane.
How do we prove that plotting all the points which satisy the equation of a circle will actually be a circle on the Cartesian plane. For example - if the definition of distance were changed, the 'circle' might look completely different on the plane. Or if the plane were 'bent' a circle would also look different. I assume that the definition of a circle does not change in these cases - it's just that the circle 'looks' different from what we expect. So how do we proceed that the circle still looks like a circle on the Cartesian Plane?
In fact, for any equation, how do we prove that the graph is correct? For example, for a parabola, we could plot a large number of points, and then demonstrate that it forms a pattern that looks like a parabola- but the unplotted points could still break this pattern.
I don't know if this confusion seems to stem from what Spivak mentioned earlier in the text- linking the algebraic and geometric concepts, or if I'm just over thinking things. Any help would be appreciated, and if there is a book or resource that explains this in more detail please let me know.
calculus geometry
asked Nov 14 '18 at 8:05
i8pii8pi
334
$endgroup$
add a comment |
$begingroup$
Currently studying the graphs section from Spivak's Calculus, and was slightly confused about one part. In the text he writes:
"The rigorous proof of any statement connecting geometric and algebraic concepts would first require a real proof (or a precisely stated assumption) that the points on a straight line correspond in an exact way to the real numbers. Aside from this, it would be necessary to develop plane geometry as precisely as we intend to develop the properties of real numbers ... We shall use geometric pictures only as an aid to intuition"
Later in the text, he derives the equation of a circle from it's definition. The definition is: "A circle with center (a, b) and radius r > 0 contains all the points (x, y) whose distance from (a, b) is equal to r."
Now, here's the part I'm confused about:
Deriving the equation of the circle depended on the definition of the circle, and the definition of distance between two points on the Cartesian plane.
How do we prove that plotting all the points which satisy the equation of a circle will actually be a circle on the Cartesian plane. For example - if the definition of distance were changed, the 'circle' might look completely different on the plane. Or if the plane were 'bent' a circle would also look different. I assume that the definition of a circle does not change in these cases - it's just that the circle 'looks' different from what we expect. So how do we proceed that the circle still looks like a circle on the Cartesian Plane?
In fact, for any equation, how do we prove that the graph is correct? For example, for a parabola, we could plot a large number of points, and then demonstrate that it forms a pattern that looks like a parabola- but the unplotted points could still break this pattern.
I don't know if this confusion seems to stem from what Spivak mentioned earlier in the text- linking the algebraic and geometric concepts, or if I'm just over thinking things. Any help would be appreciated, and if there is a book or resource that explains this in more detail please let me know.
calculus geometry
asked Nov 14 '18 at 8:05
i8pii8pi
334
$endgroup$
1
$begingroup$
To say that a shape is a circular shape, that shape must adhere to the definition of the circle. We can't rely on our eyes for that. The definition implicitly depends on elementary definitions and assumptions, such as the one you mentioned, like flatness of the space, etc. The important things is that all the parts must be true if otherwise not stated. "If it looks like a duck, walks like a duck and sounds like a duck....it is a duck". More is here: en.wikipedia.org/wiki/Euclidean_geometry
$endgroup$
– NoChance
Nov 14 '18 at 8:17
add a comment |
$begingroup$
Currently studying the graphs section from Spivak's Calculus, and was slightly confused about one part. In the text he writes:
"The rigorous proof of any statement connecting geometric and algebraic concepts would first require a real proof (or a precisely stated assumption) that the points on a straight line correspond in an exact way to the real numbers. Aside from this, it would be necessary to develop plane geometry as precisely as we intend to develop the properties of real numbers ... We shall use geometric pictures only as an aid to intuition"
Later in the text, he derives the equation of a circle from it's definition. The definition is: "A circle with center (a, b) and radius r > 0 contains all the points (x, y) whose distance from (a, b) is equal to r."
Now, here's the part I'm confused about:
Deriving the equation of the circle depended on the definition of the circle, and the definition of distance between two points on the Cartesian plane.
How do we prove that plotting all the points which satisy the equation of a circle will actually be a circle on the Cartesian plane. For example - if the definition of distance were changed, the 'circle' might look completely different on the plane. Or if the plane were 'bent' a circle would also look different. I assume that the definition of a circle does not change in these cases - it's just that the circle 'looks' different from what we expect. So how do we proceed that the circle still looks like a circle on the Cartesian Plane?
In fact, for any equation, how do we prove that the graph is correct? For example, for a parabola, we could plot a large number of points, and then demonstrate that it forms a pattern that looks like a parabola- but the unplotted points could still break this pattern.
I don't know if this confusion seems to stem from what Spivak mentioned earlier in the text- linking the algebraic and geometric concepts, or if I'm just over thinking things. Any help would be appreciated, and if there is a book or resource that explains this in more detail please let me know.
calculus geometry
asked Nov 14 '18 at 8:05
i8pii8pi
334
$endgroup$
Currently studying the graphs section from Spivak's Calculus, and was slightly confused about one part. In the text he writes:
"The rigorous proof of any statement connecting geometric and algebraic concepts would first require a real proof (or a precisely stated assumption) that the points on a straight line correspond in an exact way to the real numbers. Aside from this, it would be necessary to develop plane geometry as precisely as we intend to develop the properties of real numbers ... We shall use geometric pictures only as an aid to intuition"
Later in the text, he derives the equation of a circle from it's definition. The definition is: "A circle with center (a, b) and radius r > 0 contains all the points (x, y) whose distance from (a, b) is equal to r."
Now, here's the part I'm confused about:
Deriving the equation of the circle depended on the definition of the circle, and the definition of distance between two points on the Cartesian plane.
How do we prove that plotting all the points which satisy the equation of a circle will actually be a circle on the Cartesian plane. For example - if the definition of distance were changed, the 'circle' might look completely different on the plane. Or if the plane were 'bent' a circle would also look different. I assume that the definition of a circle does not change in these cases - it's just that the circle 'looks' different from what we expect. So how do we proceed that the circle still looks like a circle on the Cartesian Plane?
In fact, for any equation, how do we prove that the graph is correct? For example, for a parabola, we could plot a large number of points, and then demonstrate that it forms a pattern that looks like a parabola- but the unplotted points could still break this pattern.
I don't know if this confusion seems to stem from what Spivak mentioned earlier in the text- linking the algebraic and geometric concepts, or if I'm just over thinking things. Any help would be appreciated, and if there is a book or resource that explains this in more detail please let me know.
calculus geometry
calculus geometry
asked Nov 14 '18 at 8:05
i8pii8pi
334
asked Nov 14 '18 at 8:05
i8pii8pi
334
asked Nov 14 '18 at 8:05
i8pii8pi
334
asked Nov 14 '18 at 8:05
i8pii8pi
334
asked Nov 14 '18 at 8:05
i8pii8pi
334
334
1
$begingroup$
To say that a shape is a circular shape, that shape must adhere to the definition of the circle. We can't rely on our eyes for that. The definition implicitly depends on elementary definitions and assumptions, such as the one you mentioned, like flatness of the space, etc. The important things is that all the parts must be true if otherwise not stated. "If it looks like a duck, walks like a duck and sounds like a duck....it is a duck". More is here: en.wikipedia.org/wiki/Euclidean_geometry
$endgroup$
– NoChance
Nov 14 '18 at 8:17
add a comment |
1
$begingroup$
To say that a shape is a circular shape, that shape must adhere to the definition of the circle. We can't rely on our eyes for that. The definition implicitly depends on elementary definitions and assumptions, such as the one you mentioned, like flatness of the space, etc. The important things is that all the parts must be true if otherwise not stated. "If it looks like a duck, walks like a duck and sounds like a duck....it is a duck". More is here: en.wikipedia.org/wiki/Euclidean_geometry
$endgroup$
– NoChance
Nov 14 '18 at 8:17
1
1
$begingroup$
To say that a shape is a circular shape, that shape must adhere to the definition of the circle. We can't rely on our eyes for that. The definition implicitly depends on elementary definitions and assumptions, such as the one you mentioned, like flatness of the space, etc. The important things is that all the parts must be true if otherwise not stated. "If it looks like a duck, walks like a duck and sounds like a duck....it is a duck". More is here: en.wikipedia.org/wiki/Euclidean_geometry
$endgroup$
– NoChance
Nov 14 '18 at 8:17
$begingroup$
To say that a shape is a circular shape, that shape must adhere to the definition of the circle. We can't rely on our eyes for that. The definition implicitly depends on elementary definitions and assumptions, such as the one you mentioned, like flatness of the space, etc. The important things is that all the parts must be true if otherwise not stated. "If it looks like a duck, walks like a duck and sounds like a duck....it is a duck". More is here: en.wikipedia.org/wiki/Euclidean_geometry
$endgroup$
– NoChance
Nov 14 '18 at 8:17
add a comment |
2 Answers
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The 1st paragraph emphasizes that he is not including any axiomatically-founded geometry, nor any formal results about the relation of geometry to analysis or algebra, in spite of the fact that those results exist. So any geometric term is to be understood as its analytic or algebraic definition and no more. Let $S={(x,y)in Bbb R^2: d(,(x,y),(0,0),)=1}.$ He could just as well call $S$ a widget instead of a circle. Any relation of $S$ to any geometric meaning of "circle" is NOT co-incidental, but will NOT be used in this book.
The "equation of $S$" is $x^2+y^2=1.$ What this means is that, with $S$ as defined in my previous paragraph, we have $forall (x,y)in Bbb R^2,(,(x,y)in Siff x^2+y^2=1).$
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
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add a comment |
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Geometrically a circle is the locus of all points
on a plane a given distance from a point.
${ (x,y) : (x-a)^2 + (y-b)^2 = r^2}$ is the set of
all points on the $xy$ plane at a distance $r$ from $(a,b)$.
Is there any difference?
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
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add a comment |
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2 Answers
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2 Answers
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$begingroup$
The 1st paragraph emphasizes that he is not including any axiomatically-founded geometry, nor any formal results about the relation of geometry to analysis or algebra, in spite of the fact that those results exist. So any geometric term is to be understood as its analytic or algebraic definition and no more. Let $S={(x,y)in Bbb R^2: d(,(x,y),(0,0),)=1}.$ He could just as well call $S$ a widget instead of a circle. Any relation of $S$ to any geometric meaning of "circle" is NOT co-incidental, but will NOT be used in this book.
The "equation of $S$" is $x^2+y^2=1.$ What this means is that, with $S$ as defined in my previous paragraph, we have $forall (x,y)in Bbb R^2,(,(x,y)in Siff x^2+y^2=1).$
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
add a comment |
$begingroup$
The 1st paragraph emphasizes that he is not including any axiomatically-founded geometry, nor any formal results about the relation of geometry to analysis or algebra, in spite of the fact that those results exist. So any geometric term is to be understood as its analytic or algebraic definition and no more. Let $S={(x,y)in Bbb R^2: d(,(x,y),(0,0),)=1}.$ He could just as well call $S$ a widget instead of a circle. Any relation of $S$ to any geometric meaning of "circle" is NOT co-incidental, but will NOT be used in this book.
The "equation of $S$" is $x^2+y^2=1.$ What this means is that, with $S$ as defined in my previous paragraph, we have $forall (x,y)in Bbb R^2,(,(x,y)in Siff x^2+y^2=1).$
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
add a comment |
$begingroup$
The 1st paragraph emphasizes that he is not including any axiomatically-founded geometry, nor any formal results about the relation of geometry to analysis or algebra, in spite of the fact that those results exist. So any geometric term is to be understood as its analytic or algebraic definition and no more. Let $S={(x,y)in Bbb R^2: d(,(x,y),(0,0),)=1}.$ He could just as well call $S$ a widget instead of a circle. Any relation of $S$ to any geometric meaning of "circle" is NOT co-incidental, but will NOT be used in this book.
The "equation of $S$" is $x^2+y^2=1.$ What this means is that, with $S$ as defined in my previous paragraph, we have $forall (x,y)in Bbb R^2,(,(x,y)in Siff x^2+y^2=1).$
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
$endgroup$
The 1st paragraph emphasizes that he is not including any axiomatically-founded geometry, nor any formal results about the relation of geometry to analysis or algebra, in spite of the fact that those results exist. So any geometric term is to be understood as its analytic or algebraic definition and no more. Let $S={(x,y)in Bbb R^2: d(,(x,y),(0,0),)=1}.$ He could just as well call $S$ a widget instead of a circle. Any relation of $S$ to any geometric meaning of "circle" is NOT co-incidental, but will NOT be used in this book.
The "equation of $S$" is $x^2+y^2=1.$ What this means is that, with $S$ as defined in my previous paragraph, we have $forall (x,y)in Bbb R^2,(,(x,y)in Siff x^2+y^2=1).$
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
answered Nov 14 '18 at 10:05
DanielWainfleetDanielWainfleet
35.3k31648
35.3k31648
add a comment |
add a comment |
$begingroup$
Geometrically a circle is the locus of all points
on a plane a given distance from a point.
${ (x,y) : (x-a)^2 + (y-b)^2 = r^2}$ is the set of
all points on the $xy$ plane at a distance $r$ from $(a,b)$.
Is there any difference?
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
$endgroup$
add a comment |
$begingroup$
Geometrically a circle is the locus of all points
on a plane a given distance from a point.
${ (x,y) : (x-a)^2 + (y-b)^2 = r^2}$ is the set of
all points on the $xy$ plane at a distance $r$ from $(a,b)$.
Is there any difference?
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
$endgroup$
add a comment |
$begingroup$
Geometrically a circle is the locus of all points
on a plane a given distance from a point.
${ (x,y) : (x-a)^2 + (y-b)^2 = r^2}$ is the set of
all points on the $xy$ plane at a distance $r$ from $(a,b)$.
Is there any difference?
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
$endgroup$
Geometrically a circle is the locus of all points
on a plane a given distance from a point.
${ (x,y) : (x-a)^2 + (y-b)^2 = r^2}$ is the set of
all points on the $xy$ plane at a distance $r$ from $(a,b)$.
Is there any difference?
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
edited Jan 23 at 10:52
Thomas Shelby
3,7492525
3,7492525
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
answered Nov 14 '18 at 8:53
William ElliotWilliam Elliot
8,3372720
8,3372720
add a comment |
add a comment |
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$begingroup$
To say that a shape is a circular shape, that shape must adhere to the definition of the circle. We can't rely on our eyes for that. The definition implicitly depends on elementary definitions and assumptions, such as the one you mentioned, like flatness of the space, etc. The important things is that all the parts must be true if otherwise not stated. "If it looks like a duck, walks like a duck and sounds like a duck....it is a duck". More is here: en.wikipedia.org/wiki/Euclidean_geometry
$endgroup$
– NoChance
Nov 14 '18 at 8:17