- #1
What the heck is this paragraph saying?
- Thread starter Miike012
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Homework Help Overview
The discussion revolves around the concept of symmetry in mathematical expressions, particularly focusing on the interchangeability of variables and how this relates to their values. Participants are examining examples of expressions and questioning the implications of symmetry in algebraic contexts.
Discussion Character
- Conceptual clarification, Assumption checking
Approaches and Questions Raised
- Participants explore the idea of symmetry in expressions, questioning what is meant by "symmetry" and how it applies to different algebraic forms. There are attempts to clarify the distinction between symmetric and non-symmetric expressions based on variable coefficients.
Discussion Status
The conversation is ongoing, with participants providing insights into the nature of symmetry and its relevance in mathematics. Some have offered examples and analogies, while others express uncertainty and seek further clarification on the topic.
Contextual Notes
There is a mention of the potential for misunderstanding due to the complexity of the term "symmetry" and its application in various mathematical scenarios. Participants also note their amateur status, indicating a learning environment.
- #2
Mark44
Mentor
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In the first example, x + y + z, it's saying that if you swap, say, x and z, you get an different expression that has the same value.
IOW, x + y + z = z + y + x
In the second, bc + ca + ab, you can swap, say b for c, to get a new expression with the same value.
IOW, bc + ca + ab = cb + ba + ca.
bc = cb due to commutivity of multiplication, and ab = ba for the same reason. There's also some commutivity of addition going on, as ca + ab = ab + ca.
- #3
Miike012
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so its not talking about x, y, and origin symmetry?
- #4
HallsofIvy
Science Advisor
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No, its is not. It is talking about interchanging the letter variables themselves:
3x+ 3y is "symmetric" in this sense but x+ 2y is not.
- #5
Miike012
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HallsofIvy said:
and x+ 2y is not symmetric because the x and y coefficients are not equal?
- #6
Dembadon
Gold Member
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The type of symmetry being discussed here reminds me of binomial expansions.
[itex](x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}[/itex]
Notice that the coefficients aren't the only cause of symmetry!
- #7
Miike012
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what do they mean by "symmetry?" what will knowing symmetry of an expression help me with?
- #8
Dembadon
Gold Member
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Miike012 said:
If you can perform some type of transformation to an expression or equation (without murdering it!), then there is a relational symmetry. In other words, the output of a function does not vary when the arrangement of any or all of the variables within the function are changed.
I'm an amateur, though, so I'm sure HoI or Mark can put it more succinctly or correct anything I've said that is ambiguous, imprecise, or simply incorrect.
Miike012 said:
You will encounter situations where things can be assumed or proven because of symmetrical relationships, and it will often save you a lot of work!
- #9
Dembadon
Gold Member
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- 88
This video might be helpful to you:
https://www.youtube.com/watch?v=ylAXYqgbp4M
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