Unique Algebra Problem
- Thread starter Jason76
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\(\displaystyle 5x = 10x\)
\(\displaystyle \dfrac{5}{10} x = x\)
\(\displaystyle \dfrac{1}{2} x = x\)
\(\displaystyle \dfrac{1}{2} \dfrac{x}{x} = \dfrac{x}{x}\)Where is this going?
You cannot divide your relation by x, because you can do this only if you are sure that \(\displaystyle x \neq 0 \).
\(\displaystyle 5x=10x => 10x-5x=0 => 5x=0 => x=0 \) .
\(\displaystyle 10x-5x=0 \rightarrow 5x(2 - 1) = 0\)
\(\displaystyle 10x-5x=0 \rightarrow 5x(2-1)=0 \rightarrow 5x=0 \rightarrow x=0 \)
Suppose
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\(\displaystyle 5x = 10x\)
\(\displaystyle \dfrac{5}{10} x = x\)
\(\displaystyle \dfrac{1}{2} x = x\)
\(\displaystyle \dfrac{1}{2} \dfrac{x}{x} = \dfrac{x}{x}\)
Double click image, then again, and again for largest version
Sure
Way back when, because all I ever ran into were equations that had solution sets I imagined that all mathematical statements, if you followed the rules, resolved to something "truthful" or reasonable" , certainly not "1 = 2". Jason's statement was transparently absurd, based on a falsity, because it was so simple. I was demonstrating that a more "knotted up" version of the same thing, one not so transparently absurd, might, when resolved to its basic proposition be just so.
In the end, the point is to run down every thoughtful confusion that one experiences even if you are asking why everything looks upside down just because you are standing on your head, how else are you going to get right side up. Obviously I believe that.
(of course I might be standing on my head, in which case, somebody here will surely set me straight.)
I think so. I remember a long time ago being puzzled about the same thing for several days. Isn't this an analog of those questions in naive set theory, "Can God make a rock so big that "he" cannot lift it." Just because you can frame something in a logical statement doesn't mean it makes sense.Any purpose to this silliness, Jason and Dale
Way back when, because all I ever ran into were equations that had solution sets I imagined that all mathematical statements, if you followed the rules, resolved to something "truthful" or reasonable" , certainly not "1 = 2". Jason's statement was transparently absurd, based on a falsity, because it was so simple. I was demonstrating that a more "knotted up" version of the same thing, one not so transparently absurd, might, when resolved to its basic proposition be just so.
In the end, the point is to run down every thoughtful confusion that one experiences even if you are asking why everything looks upside down just because you are standing on your head, how else are you going to get right side up. Obviously I believe that.
(of course I might be standing on my head, in which case, somebody here will surely set me straight.)
Last edited:
HallsofIvy
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Okay, so there was "purpose to this silliness". But it was still silliness!
Yes
It is, I think, because too often the math education process removes a student from thinking about what one is really talking about and re-focuses attention on correctly applying sets of rules. This is not a rap on math teachers, or the math educational establishment. Most likely it the dark side of education. The educated mind learns to think effectively within a wider context but reflexively, does not question the context, does not even consider that there is a context - the difficult acceptance of a non-flat world, or quantum mechanics, or relativity, or the existence of the unconscious mind, blah,blah and so on, this is not a philosophy forum.
I suppose the point to recognize here is that an equal sign between two expressions does not constitute an equality, an "equation". I do think that math courses, math books in particular would be enhanced by challenging students to debunk certain forms of silliness that pose as paradoxes. Anyway, more then enough about this, I for one AM focusing on learning to classify and apply rules.
Yes, it was all silliness from the beginning. 10x = 5x. Would one say in words, 10 cows is the same thing as 5 cows and try and reason to some conclusion from there? No, of course not, so why would one write 10x =5x anyway? THAT is a serious question.
It is, I think, because too often the math education process removes a student from thinking about what one is really talking about and re-focuses attention on correctly applying sets of rules. This is not a rap on math teachers, or the math educational establishment. Most likely it the dark side of education. The educated mind learns to think effectively within a wider context but reflexively, does not question the context, does not even consider that there is a context - the difficult acceptance of a non-flat world, or quantum mechanics, or relativity, or the existence of the unconscious mind, blah,blah and so on, this is not a philosophy forum.
I suppose the point to recognize here is that an equal sign between two expressions does not constitute an equality, an "equation". I do think that math courses, math books in particular would be enhanced by challenging students to debunk certain forms of silliness that pose as paradoxes. Anyway, more then enough about this, I for one AM focusing on learning to classify and apply rules.
D
Deleted member 4993
Guest
\(\displaystyle 5x = 10x\)
\(\displaystyle \dfrac{5}{10} x = x\)
\(\displaystyle \dfrac{1}{2} x = x\)
\(\displaystyle \dfrac{1}{2} \dfrac{x}{x} = \dfrac{x}{x}\)
This is not unique - just bad algebra!!
5x = 10x → 10x - 5x = 0 → 5x = 0 → x = 0
What is there to get confused about?!!
Laughing
Tsk, Tsk. Sir, I think you have lost the ability to see what others do not see. Alas, you probably lost your ignorance at an early age, but hey, don't be embarrassed, it happens! (I hope you have a sense of humor.)
To no one in particular ... How about this for a junior/intermediate algebra problem?
---
"Given that x = 1 is the solution to an equation, what is THE equation that produces that solution?"
Correct the above sentence so that it makes sense, illustrate with examples.
(Oblique clue, what a difference an article makes.)
---
Really, I think that such a question, offering as it does, a "reverse engineering approach" leads to a significant insight, but, you know, maybe not.
(Mr Denis, thanks, and, x<>0, yeah!)
Tsk, Tsk. Sir, I think you have lost the ability to see what others do not see. Alas, you probably lost your ignorance at an early age, but hey, don't be embarrassed, it happens! (I hope you have a sense of humor.)
To no one in particular ... How about this for a junior/intermediate algebra problem?
---
"Given that x = 1 is the solution to an equation, what is THE equation that produces that solution?"
Correct the above sentence so that it makes sense, illustrate with examples.
(Oblique clue, what a difference an article makes.)
---
Really, I think that such a question, offering as it does, a "reverse engineering approach" leads to a significant insight, but, you know, maybe not.
(Mr Denis, thanks, and, x<>0, yeah!)
D
Deleted member 4993
Guest
What I mean is ..
[
To no one in particular ... How about this for a junior/intermediate algebra problem?
---
"Given that x = 1 is the solution to an equation, what is THE equation that produces that solution?"
Correct the above sentence so that it makes sense, illustrate with examples.
(Oblique clue, what a difference an article makes.)
---
[/QUOTE]
Ahem, the real problem was how to correct the question !
"Given that x = 1 is the solution to an equation, what is THE equation that produces that solution?" should be written ...
"Given that x = 1 is the solution to an equation, what is A equation that produces that solution?"
There is no one equation that has x = 1 as a solution.
vx = v where v is any variable...
x=1 => 2x = 2
x=1 => x^3 = 1
x=1 => x^3 + 2x = 3 => x^3 +2x -3 = 0
I found it very enlightening when I found that a function could be be differentiated one, two, three times and the results used to construct a myriad of different differential equations ... it sort of to took the mystery out of differential equations and allowed me to focus on solving them (with mixed success) ... and checking them.
The point of the question is to introduce, with explanation, a change of perspective, an equation as an accretion, so to speak, of its solution. I mean, after you have assembled an engine you begin to think of it as a sum of its parts rather than a single semi-mysterious thing ... which is my goal with mathematics, to review the things I learned well enough to get through the time crush of the earning a diploma (not in mathematics) and to see how it is all put together ... doing problems, but also learning to see where I am going before I begin pushing symbols around the paper.
So that is the explanation of the problem and its possible import.
(PS. I hope my occasional, perhaps, odd sense of humor is not taken amiss, no disrespect intended.)
[
To no one in particular ... How about this for a junior/intermediate algebra problem?
---
"Given that x = 1 is the solution to an equation, what is THE equation that produces that solution?"
Correct the above sentence so that it makes sense, illustrate with examples.
(Oblique clue, what a difference an article makes.)
---
[/QUOTE]
Ahem, the real problem was how to correct the question !
"Given that x = 1 is the solution to an equation, what is THE equation that produces that solution?" should be written ...
"Given that x = 1 is the solution to an equation, what is A equation that produces that solution?"
There is no one equation that has x = 1 as a solution.
vx = v where v is any variable...
x=1 => 2x = 2
x=1 => x^3 = 1
x=1 => x^3 + 2x = 3 => x^3 +2x -3 = 0
I found it very enlightening when I found that a function could be be differentiated one, two, three times and the results used to construct a myriad of different differential equations ... it sort of to took the mystery out of differential equations and allowed me to focus on solving them (with mixed success) ... and checking them.
The point of the question is to introduce, with explanation, a change of perspective, an equation as an accretion, so to speak, of its solution. I mean, after you have assembled an engine you begin to think of it as a sum of its parts rather than a single semi-mysterious thing ... which is my goal with mathematics, to review the things I learned well enough to get through the time crush of the earning a diploma (not in mathematics) and to see how it is all put together ... doing problems, but also learning to see where I am going before I begin pushing symbols around the paper.
So that is the explanation of the problem and its possible import.
(PS. I hope my occasional, perhaps, odd sense of humor is not taken amiss, no disrespect intended.)