Complicated equation and Simple equation for the Same Curve?

  • #1
Summary:
Complicated and simple representation of same curve.
curve.png
If I draw some arbitrary curve then that curve can be represented convolutedly in mathematical elements and it can also be represented in simple mathematical elements?

Thanks!
 

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  • #2
berkeman
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Summary:: Complicated and simple representation of same curve.

View attachment 280855If I draw some arbitrary curve then that curve can be represented convolutedly in mathematical elements and it can also be represented in simple mathematical elements?

Thanks!
Um, could you maybe define your terms more precisely, and give some examples (using LaTeX) for what you mean?
 
  • #3
FactChecker
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It really depends on how closely you want the equation to fit the curve. A gross approximation to the curve you drew is given by 1/x. You can get an early guess because that curve does not have a lot of "wiggles". If that is not good enough, there are some very sophisticated methods (called "interpolation methods") that will approximate the curve, going through a set of points exactly and smoothly.
 
  • #4
curve-png.png

I could write the equation of the above curve by beginning with first term as x^6 and with more terms I could inch closer and closer to the precise equation of the curve - this is what I am calling convoluted equation.
 
  • #5
berkeman
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View attachment 280867
I could write the equation of the above curve by beginning with first term as x^6 and with more terms I could inch closer and closer to the precise equation of the curve - this is what I am calling convoluted equation.
Um, no.

1617464411814.png
 
  • #6
I could add, subtract, multiply, divide more terms after the first term to get the correct graph.
 
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  • #7
Complicated:
x = 1-3+2-1+2

Simple:
x = 1
 
  • #8
FactChecker
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You can make high-order polynomials that go exactly through a set of points on the curve, but they tend to wiggle around a lot in between those points.
 
  • #9
curve-png-png.png

How do I know if a graph is made out of complicated terms/many terms or simple terms?
Can the equation of the above curve be intricate?
Can the equation of the above curve be simple?

Can the equation for the above curve be intricated or simple?
 
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  • #12
For the above curve, can the equation be both intricate, and simple? The equation might have been intricate but it was later simplified?
 
  • #13
berkeman
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View attachment 280870
How do I know if a graph is made out of complicated terms/many terms or simple terms?
Can the equation of the above curve be intricate?
Can the equation of the above curve be simple?

Can the equation for the above curve be intricated or simple?
Your Profile Page says you are working on your BS in math. What year are you at university?
 
  • #14
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Of course imagine taking a basic trig identity subbing in other identities to make a very complex equation that you now know can be simplified.

##y = sec^2(x)= 1 + tan^2(x) ##

## = sin^2(x) + cos^2(x) + tan^2(x)##
 
  • #15
hutchphd
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I can always make an equation more complicated. For instance ##y=x## can be rewritten as $$y=(x+1)(sin^2x+cos^2x) -1.$$ I cannot always make an expression "less complicated" without sacrificing arithmetic fealty. Your question really needs to be more specific
 
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  • #16
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'x' can be a copy of a book.
Okay, what' the sum of Newton's Principia and Averson's "An Introduction to C*-Algebras?" Is it greater than 6?
 
  • #17
Your Profile Page says you are working on your BS in math. What year are you at university?
I am self-studying. I haven't decided to go to a university. I am trying to find out what I will be doing in a university once I join it.

Complicated:
x = 1-3+2-1+2

Simple:
x = 1

I have one book then I lost three books then I acquired two more books then I lost one book then I acquired two more books.
Total books acquired: five.
Total books lost: four.
Total books in custody: one.

Okay, what' the sum of Newton's Principia and Averson's "An Introduction to C*-Algebras?" Is it greater than 6?

Why/how sum of names of two books be greater than 6?

curve-png-png-png.png

If I have a curve then can its equation be complex, and also simple?
The equation of the curve can become complicated anytime?
Can this curve have complicated equation?
 
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  • #18
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Why/how sum of names of two books be greater than 6?

Sorry, I was just trying to get you to think more clearly about your own statements. I haven't had much success. You said 'x' , an integer in the example you provided, could be the copy of a book. My comment is intended to bring out the absurdity of your statement. I realize you're just waving your hands trying to express different levels of complexity. This is not the same as providing a working definition.

So, consider, is 3 more complicated, less complicated, or equally complicated than 7? One could define a concept of "complicated" for integers in any number of ways. We could use the number of prime factors, size, or some combination of these. To me, 4 - 2 is the same as 2 but you seem to think the extra step of reducing to 2 makes 4 - 2 more complicated. In terms of integers, your question is we could always reduce our expressions or choose not to. Uh, okay. So what?
 
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  • #19
In terms of integers, your question is we could always reduce our expressions or choose not to. Uh, okay. So what?

Complicated:
x = 1-3+2-1+2

Simple:
x = 1

If I reduce the expression then I could know number of books in my custody.
If I don't reduce the expression then I could know what happened to the books over time.
curve-png-png-png-png.png


A gross approximation to the curve you drew is given by 1/x.
As FactChecker points out the curve can be as simple as y = 1/x but what I am trying to know is if this curve can have a complicated equation; something more than just y = 1/x and I would get the same curve.
 
  • #20
FactChecker
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If I reduce the expression then I could know number of books in my custody.
I don't know what you mean here. Do you mean that the point on the right is your current number of books?
If I don't reduce the expression then I could know what happened to the books over time.
So this is a plot of books in custody versus time?
View attachment 280920


As FactChecker points out the curve can be as simple as y = 1/x but what I am trying to know is if this curve can have a complicated equation; something more than just y = 1/x and I would get the same curve.
Sure. There are more complicated expressions that you might get to fit your curve better by adjusting parameters. The general shape of your curve (1/x) is a division by zero at x=0, so a better expression would probably need to have that aspect in it some way.
 
  • #21
I don't know what you mean here. Do you mean that the point on the right is your current number of books?

Complicated:
x = 1-3+2-1+2

Simple:
x = 1

If I solve x = 1 - 3 + 2 - 1 + 2 then I will get number of books in my custody.
If I don't solve x = 1 - 3 + 2 - 1 + 2 then I get information like number of books acquired, and number of books lost over time.
 
  • #22
FactChecker
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It sounds like you want something like ##B_{total} = B_1+B_2+B_3+B_4+...+B_n##
 
  • #23
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Complicated:
x = 1-3+2-1+2

Simple:
x = 1
Your complicated and simple in this example are subjective.
'x' can be a copy of a book.
No.
From what you wrote later, x represents the number of books. It does not represent "a copy of a book." There is a big difference between how many things you are talking about, and the things themselves.

Your "complicated" equation above would not be considered complicated by anyone who has a grasp of very elementary arithmetic. If your intention is to get a degree in mathematics, you have a very long way to go.
I have one book then I lost three books then I acquired two more books then I lost one book then I acquired two more books.
Total books acquired: five.
Total books lost: four.
Total books in custody: one.
Again, no. This would necessarily be the net change in the number of books you have, not the total number of books you have. In other words, the 1 indicates that you now have one more book than you originally started with, which you didn't state. If you have only one book, you cannot possibly lose three books.
 
  • #25
WWGD
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I think the/a main issue here is that of parametrization of a curve up to equivalence ( change of coordinate). I think it is used often in algebraic geometry. Cc @mathwonk
 

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