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Kobu
science forum beginner
Joined: 02 Dec 2005
Posts: 43
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 Posted: Fri Dec 02, 2005 11:30 pm Post subject:
how are all the different types of mathematics related?
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Does anyone know a website or resource that shows the general areas
each type of math deals with and (somewhat) shows how they are related?
I am seeing references to new types of math that I have never heard of
all the type, like:
Vector Calculus
Differential Geometry
Geometric Calculus
Analytic Algebra
Algebraic Geometry
Analysis
Analytic Geometry
etc.
etc.
(even more confusing if I start listing non-pure math subject titles..
like Combinatronics, Finite Math, etc.).
It seems like there are all kinds of combinations in these new subject
titles. Being someone in highschool thinking of a math career, I'd
like to know what these terms mean. Can anyone point me to a resource?
A Venn diagram showing relations would be awesome!!!!
Thank you very much. |
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Kobu
science forum beginner
Joined: 02 Dec 2005
Posts: 43
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| Posted: Fri Dec 02, 2005 11:39 pm Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote:
| Quote: | Does anyone know a website or resource that shows the general areas
each type of math deals with and (somewhat) shows how they are related?
I am seeing references to new types of math that I have never heard of
all the type, like:
Vector Calculus
Differential Geometry
Geometric Calculus
Analytic Algebra
Algebraic Geometry
Analysis
Analytic Geometry
etc.
etc.
(even more confusing if I start listing non-pure math subject titles..
like Combinatronics, Finite Math, etc.).
It seems like there are all kinds of combinations in these new subject
titles. Being someone in highschool thinking of a math career, I'd
like to know what these terms mean. Can anyone point me to a resource?
A Venn diagram showing relations would be awesome!!!!
|
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation give us
the right solution, but doing other things to both sides give us
problems? |
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Randy Poe
science forum Guru
Joined: 24 Mar 2005
Posts: 2485
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| Posted: Sat Dec 03, 2005 12:36 am Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote:
| Quote: | Does anyone know a website or resource that shows the general areas
each type of math deals with and (somewhat) shows how they are related?
I am seeing references to new types of math that I have never heard of
all the type, like:
Vector Calculus
Differential Geometry
Geometric Calculus
Analytic Algebra
Algebraic Geometry
Analysis
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An oversimplification: In the math you've seen, the objects have been
very concrete. Either numbers, or variables representing numbers.
But in other fields of mathematics, the objects under discussion are
more abstract or are other kinds of things (such as vectors).
This you may have seen. This was a subtitle they gave the
Algebra 2 course I took in high school. It includes such things
as the algebraic description of conic sections (parabolas,
ellipses, hyperbolas, circles) and how their geometric properties
(for instance, all points on a parabola are equidistance from
a point and a line) to their algebraic description. This course
probably first awoke my interest in "real" mathematics.
| Quote: | (even more confusing if I start listing non-pure math subject titles..
like Combinatronics, Finite Math, etc.).
|
Combinatorics. In what sense are these not "pure math"?
| Quote: | It seems like there are all kinds of combinations in these new subject
titles. Being someone in highschool thinking of a math career, I'd
like to know what these terms mean. Can anyone point me to a resource?
A Venn diagram showing relations would be awesome!!!!
Thank you very much.
|
I see someone else has actually pointed you to a website with a
map. Some of the subjects you listed above are motivated by
physics. Vector calculus I'm most familiar with as the
language of electromagnetic theory. Differential geometry is
the language of general relativity. (Before working out the
theory of general relativity, Einstein spent a summer learning
differential geometry and I don't know if he ever considered
himself expert).
- Randy |
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Randy Poe
science forum Guru
Joined: 24 Mar 2005
Posts: 2485
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| Posted: Sat Dec 03, 2005 6:06 am Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote:
| Quote: | Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation give us
the right solution, but doing other things to both sides give us
problems?
|
Hard to figure out what you mean with such a vague description.
I suspect what is confusing you is the difference between "p if and
only
if q" and simple implication, "if p then q".
For instance, a = b if and only if a + c = b + c. That means that if
we
add a number c to both sides of an equation, the solution is unchanged.
If the second equation is true, the first is true. If the first is
true, the second
is true. "If and only if" means the two statements are equivalent.
On the other hand, we can say if a = b then a^2 = b^2, but this
implication
is only in one direction. Not all solutions to the second are solutions
to
the first. a^2 = b^2 does not imply a = b. Thus, squaring both sides of
an equation may change the solution set. a = b is NOT equivalent to
a^2 = b^2.
Learning about the difference between one-way and two-way implication
is the kind of thing you'd see when you are exposed to rigorous proof
methods, which might be part of a calculus course if not sooner.
- Randy |
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Kobu
science forum beginner
Joined: 02 Dec 2005
Posts: 43
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| Posted: Sat Dec 03, 2005 5:04 pm Post subject:
Re: how are all the different types of mathematics related?
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Randy Poe wrote:
| Quote: | Kobu wrote:
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation give us
the right solution, but doing other things to both sides give us
problems?
Hard to figure out what you mean with such a vague description.
I suspect what is confusing you is the difference between "p if and
only
if q" and simple implication, "if p then q".
For instance, a = b if and only if a + c = b + c. That means that if
we
add a number c to both sides of an equation, the solution is unchanged.
If the second equation is true, the first is true. If the first is
true, the second
is true. "If and only if" means the two statements are equivalent.
On the other hand, we can say if a = b then a^2 = b^2, but this
implication
is only in one direction. Not all solutions to the second are solutions
to
the first. a^2 = b^2 does not imply a = b. Thus, squaring both sides of
an equation may change the solution set. a = b is NOT equivalent to
a^2 = b^2.
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Thanks for the explanation. One more question, how about the act of
squaring both sides?
a = b if and only a^.5 = b^.5 RIGHT? |
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Dave L. Renfro
science forum Guru
Joined: 29 Apr 2005
Posts: 570
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| Posted: Sat Dec 03, 2005 5:11 pm Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote (in part):
| Quote: | Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
|
This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
Dave L. Renfro |
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Pubkeybreaker
science forum Guru
Joined: 24 Mar 2005
Posts: 333
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| Posted: Sat Dec 03, 2005 5:12 pm Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote:
| Quote: | Randy Poe wrote:
Kobu wrote:
Thanks for the explanation. One more question, how about the act of
squaring both sides?
a = b if and only a^.5 = b^.5 RIGHT?
|
Wrong.
If one restricts to the real numbers, then a^.5 may not exist, so
the statement
is meaningless. It is like saying a = b iff apple = pear,
since a^.5 is
not a number.
If the domain is C, then a^.5 is multivalued. Thus we can have a
= b, but
different values for a^.5 and b^.5. |
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Robert Low
science forum Guru
Joined: 01 May 2005
Posts: 1063
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| Posted: Sat Dec 03, 2005 8:23 pm Post subject:
Re: how are all the different types of mathematics related?
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Dave L. Renfro wrote:
| Quote: | Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
|
This is one of those annoying points.
If f:A -> B is injective (or 1-1) then there is a function
g: B -> A such that gof is the identity function on A, a
left inverse for f. But there isn't necessarily a function
g:B->A such that fog is the identity on B.
I'm accustomed to calling a function invertible if there is a
function g:B->A such that gof is the identity on A and fog is
the identity on B, so that f is both a left inverse and a
right inverse for g, and if simply called the inverse of f:
this also requires the function to be onto. |
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Dave L. Renfro
science forum Guru
Joined: 29 Apr 2005
Posts: 570
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| Posted: Sun Dec 04, 2005 1:29 am Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote (in part):
| Quote: | Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
|
Dave L. Renfro wrote:
| Quote: | This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
|
Robert Low wrote:
| Quote: | This is one of those annoying points.
If f:A -> B is injective (or 1-1) then there is a function
g: B -> A such that gof is the identity function on A, a
left inverse for f. But there isn't necessarily a function
g:B->A such that fog is the identity on B.
I'm accustomed to calling a function invertible if there is a
function g:B->A such that gof is the identity on A and fog is
the identity on B, so that f is both a left inverse and a
right inverse for g, and if simply called the inverse of f:
this also requires the function to be onto.
|
Darn it, I thought I took care of this by specifying
"1-1 _function_", but I didn't actually think it through.
In my 2003 post the actual property I used is "1-1 function",
because we don't care whether the co-domain is equal to the
range. Since we only want the implication "f(x_1) = f(x_2)
implies x_1 = x_2", we only need an appropriate function
from the range back to the domain, not an appropriate
function from the entire co-domain back to the domain.
I too would rather not call f:A --> B invertible if f simply
has the 1-1 property on A, since otherwise there is a disconnect
between the use of "invertible" here and the way "invertible"
is used pretty much everywhere else in mathematics.
Dave L. Renfro |
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Kobu
science forum beginner
Joined: 02 Dec 2005
Posts: 43
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| Posted: Sun Dec 04, 2005 10:28 pm Post subject:
Re: how are all the different types of mathematics related?
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Dave L. Renfro wrote:
| Quote: | Kobu wrote (in part):
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
Dave L. Renfro wrote:
This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
Robert Low wrote:
This is one of those annoying points.
If f:A -> B is injective (or 1-1) then there is a function
g: B -> A such that gof is the identity function on A, a
left inverse for f. But there isn't necessarily a function
g:B->A such that fog is the identity on B.
I'm accustomed to calling a function invertible if there is a
function g:B->A such that gof is the identity on A and fog is
the identity on B, so that f is both a left inverse and a
right inverse for g, and if simply called the inverse of f:
this also requires the function to be onto.
Darn it, I thought I took care of this by specifying
"1-1 _function_", but I didn't actually think it through.
In my 2003 post the actual property I used is "1-1 function",
because we don't care whether the co-domain is equal to the
range. Since we only want the implication "f(x_1) = f(x_2)
implies x_1 = x_2", we only need an appropriate function
from the range back to the domain, not an appropriate
function from the entire co-domain back to the domain.
I too would rather not call f:A --> B invertible if f simply
has the 1-1 property on A, since otherwise there is a disconnect
between the use of "invertible" here and the way "invertible"
is used pretty much everywhere else in mathematics.
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Ohh my. So the theory of composite function, invertibility and such
come into play with algebraic manipulations. OUCH.........
Time to research and review. I just wish my highschool math teachers
would have explained stuff more deeply... :S |
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deniz.bahar@gmail.com
science forum beginner
Joined: 04 Dec 2005
Posts: 32
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| Posted: Mon Dec 05, 2005 3:17 am Post subject:
Re: how are all the different types of mathematics related?
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Dave L. Renfro wrote:
| Quote: | Kobu wrote (in part):
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
|
Interesting, I've never seen functions used to explain this. How would
operations involve the rearrangement of terms of the variable be
defined as a function?
A function can be defined that adds a number to both sides, or
subtracts/multiplies/divides, but what about operations that add a term
involving x to both sides (or other operations)?
4x = x^2 --> 0 = x^2 = 4x
how can there be a function, f, that performs this?
Not questioning your explanation, just haven't seen functions used to
explain irreversible steps and can't seem to wrap my head around this
type of step via your theory. |
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deniz.bahar@gmail.com
science forum beginner
Joined: 04 Dec 2005
Posts: 32
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| Posted: Mon Dec 05, 2005 3:17 am Post subject:
Re: how are all the different types of mathematics related?
|
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Dave L. Renfro wrote:
| Quote: | Kobu wrote (in part):
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
|
Interesting, I've never seen functions used to explain this. How would
operations involve the rearrangement of terms of the variable be
defined as a function?
A function can be defined that adds a number to both sides, or
subtracts/multiplies/divides, but what about operations that add a term
involving x to both sides (or other operations)?
4x = x^2 --> 0 = x^2 - 4x
how can there be a function, f, that performs this?
Not questioning your explanation, just haven't seen functions used to
explain irreversible steps and can't seem to wrap my head around this
type of step via your theory. |
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| Back to top |
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Kobu
science forum beginner
Joined: 02 Dec 2005
Posts: 43
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| Posted: Tue Dec 06, 2005 10:21 pm Post subject:
Re: how are all the different types of mathematics related?
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deniz.bahar@gmail.com wrote:
| Quote: | Dave L. Renfro wrote:
Kobu wrote (in part):
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
Interesting, I've never seen functions used to explain this. How would
operations involve the rearrangement of terms of the variable be
defined as a function?
A function can be defined that adds a number to both sides, or
subtracts/multiplies/divides, but what about operations that add a term
involving x to both sides (or other operations)?
4x = x^2 --> 0 = x^2 - 4x
how can there be a function, f, that performs this?
Not questioning your explanation, just haven't seen functions used to
explain irreversible steps and can't seem to wrap my head around this
type of step via your theory.
|
Hmm let me get this straight.
If we impart a 1-1 function to both sides of an equation, we are
definitely okay (<==> equivalence ?).
Well f(x) = root(x) is 1-1, but taking the root of both sides causes
problems (can eliminate solutions because it restricts the domain).
I've been looking for a book on the "true theory" of algebraic
manipulation GOOGLE BOOKs for a couple days now with no luck. Can
anyone suggest a proper mathetmatical term to search for? |
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deniz.bahar@gmail.com
science forum beginner
Joined: 04 Dec 2005
Posts: 32
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| Posted: Wed Dec 07, 2005 5:44 am Post subject:
Re: how are all the different types of mathematics related?
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Kobu wrote:
| Quote: | deniz.bahar@gmail.com wrote:
Dave L. Renfro wrote:
Kobu wrote (in part):
Also, which course in university will teach me how algebra really
works? ex: Why does doing something to both side of an equation
give us the right solution, but doing other things to both sides
give us problems?
This post might help with your last question:
sci.math thread "curriculum: precalculus vs. college algebra"
(February 23, 2003)
http://groups.google.com/group/sci.math/msg/c38c3d4c82e2383c
Incidentally, invertible functions (a key notion in that post)
are also called one-to-one functions (or 1-1 functions).
Interesting, I've never seen functions used to explain this. How would
operations involve the rearrangement of terms of the variable be
defined as a function?
A function can be defined that adds a number to both sides, or
subtracts/multiplies/divides, but what about operations that add a term
involving x to both sides (or other operations)?
4x = x^2 --> 0 = x^2 - 4x
how can there be a function, f, that performs this?
Not questioning your explanation, just haven't seen functions used to
explain irreversible steps and can't seem to wrap my head around this
type of step via your theory.
Hmm let me get this straight.
If we impart a 1-1 function to both sides of an equation, we are
definitely okay (<==> equivalence ?).
Well f(x) = root(x) is 1-1, but taking the root of both sides causes
problems (can eliminate solutions because it restricts the domain).
I've been looking for a book on the "true theory" of algebraic
manipulation GOOGLE BOOKs for a couple days now with no luck. Can
anyone suggest a proper mathetmatical term to search for?
|
http://groups.google.ca/group/sci.math/browse_thread/thread/4fac86aa56fa303e/e8a70c9daba0ec2b?q=squaring+both+sides+reversible&rnum=1&auth=DQAAAHAAAABJyh66zIyxancr7FjHude88h7XfTDgDQoLsubGC_EklqsfBhomGez7TfdmuEKHbRvCSiwE5cFccRgxYWf7hwZKcIc0C2_Oftc_W2zhzhohQKmhZ7kcvMStr2_oKHfsYdKl58HUCW8ELDu3o1R07ig3 |
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