B Complicated equation and Simple equation for the Same Curve?

[pairofstrings]

TL;DR Summary

Complicated and simple representation of same curve.

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!

 

[berkeman]

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?

 

[FactChecker]

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.

 

[pairofstrings]

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.

 

[berkeman]

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.

 

[pairofstrings]

I could add, subtract, multiply, divide more terms after the first term to get the correct graph.

 

[pairofstrings]

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

Simple:
x = 1

 

[FactChecker]

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.

 

[pairofstrings]

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?

 

[Paul Colby]

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

Simple:
x = 1

Your complicated and simple in this example are subjective.

 

[pairofstrings]

complicated and simple in this example are subjective

'x' can be a copy of a book.

 

[pairofstrings]

For the above curve, can the equation be both intricate, and simple? The equation might have been intricate but it was later simplified?

 

[berkeman]

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?

 

[jedishrfu]

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.

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

$= sin^2(x) + cos^2(x) + tan^2(x)$

 

[hutchphd]

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

 

[Paul Colby]

'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?

 

[pairofstrings]

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?

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?

 

[Paul Colby]

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?

 

[pairofstrings]

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.

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.

 

[FactChecker]

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 280920As 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.

 

[pairofstrings]

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.

 

[FactChecker]

It sounds like you want something like $B_{total} = B_1+B_2+B_3+B_4+...+B_n$

 

[Mark44]

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.

 

[Paul Colby]

If I have a curve then can its equation be complex, and also simple?

yes.

 

[WWGD]

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

 

[Mark44]

If I solve x = 1 - 3 + 2 - 1 + 2 then I will get number of books in my custody.

You are not "solving" this equation for x, since x is already isolated to one side. All you are doing is simplifying the not-very-complicated arithmetic expression on the right side.

 

[pairofstrings]

If you have only one book, you cannot possibly lose three books.

Thanks!
Is it correct if I replace "lose" with "owe" or "debt" in the statement I wrote?
How do I capture the notion of debt/owe/loss in a mathematical statement?
Sorry, I should have mentioned earlier that I have one book at the beginning.

Thanks!

 

[Mark44]

Is it correct if I replace "lose" with "owe" or "debt" in the statement I wrote?
How do I capture the notion of debt/owe/loss in a mathematical statement?
Sorry, I should have mentioned earlier that I have one book at the beginning.

Let's get rid of all three of these words: "owe", "debt", and "lose." Instead, imagine that you are running a shop that buys and sells books.

Let's also get rid of the equation you wrote -- x = 1 - 3 + 2 - 1 + 2 and focus just on the expression 1 - 3 + 2 - 1 + 2.

During one day, five customers come in. Let's assume that your shop starts the day with N books, where N is a reasonably large number.

Customer 1 sells 1 book.
Customer 2 buys 3 books.
Customer 3 sells 2 books.
Customer 4 buys 1 book.
Customer 5 sells 2 books.

At the end of the day, the number of books in the shop is N + 1 - 3 + 2 - 1 + 2, or more simply, N + 1 books. At the end of the day, the shop has 1 more book than at the start of the day.

 

[pairofstrings]

Can 'N' also be negative??

 

[Mark44]

Can 'N' also be negative??

Not in the scenario as I described it. To be meaningful, the seller could not sell more books at any time than there are on hand.

I'm just elaborating on the example you gave earlier.

 

[pairofstrings]

I can always make an equation more complicated. For instance y=x can be rewritten as y=(x+1)(sin^2x+cos^2x) -1.

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same meaning or is it the same graph but only the expressions are changing?

 

[Mark44]

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same meaning

Yes. The two equations are equivalent -- for a given value of x, both equations produce the same y value.

or is it the same graph but only the expressions are changing?

Yes to that, also. The graphs would be exactly the same. The only difference is that the expression on the right in the 2nd equation is unsimplified.

 

[pairofstrings]

The graphs would be exactly the same. The only difference is that the expression on the right in the 2nd equation is unsimplified.

I didn't think that I have online graphing calculators for producing graphs for these equations. Sorry about that.

Okay, so the unsimplified 2nd equation is destined to become 'x' on simplification? Like this: y = x? Is it possible to get anything else other than 'x' on simplification from this unsimplified 2nd equation?

 

[hutchphd]

Yes. Making something "simpler" is more difficult: you reach the point where you cannot continue but you can never be certain. You can complexify ad infinitum

 

[pairofstrings]

The two equations are equivalent -- for a given value of x, both equations produce the same y value.

the unsimplified 2nd equation is destined to become 'x' on simplification? Like this: y = x? Is it possible to get anything else other than 'x' on simplification from this unsimplified 2nd equation?

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same context?
y = x may have context not similar to y = ( x + 1)( sin^2x + cos^2x ) - 1...

First one is linear equation (straight lines) and the second equation is trigonometric equation (side lengths and angles).

What could y = x be?
What could y=(x+1)(sin^2x+cos^2x) -1 be?

What does it mean if y is x and y is (x+1)(sin^2x+cos^2x) -1?

Thanks!

 

[Mark44]

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same context?
y = x may have context not similar to y = ( x + 1)( sin^2x + cos^2x ) - 1...

I don't know what "same context" means in regard to equations. The two equations are equivalent, meaning that they both represent exactly the same points, and their graphs are exactly the same.

First one is linear equation (straight lines) and the second equation is trigonometric equation (side lengths and angles).

No, they are both linear equations, meaning that the graph of each equation is a straight line. Both graphs pass through the origin, and both graphs have a slope of 1.

What could y = x be?
What could y=(x+1)(sin^2x+cos^2x) -1 be?

Already answered -- both are straight lines through the origin with a slope of 1.

 

[pairofstrings]

I don't know what "same context" means in regard to equations. The two equations are equivalent, meaning that they both represent exactly the same points, and their graphs are exactly the same.

Context here is that there is sine, cosine in y = (x + 1)(sin²x + cos²x) -1 but there is nothing like that in y = x...

 

[Mark44]

Context here is that there is sine, cosine in y = (x + 1)(sin²x + cos²x) -1 but there is nothing like that in y = x...

Right, but "context" is pretty meaningless when you're talking about equations. A specific graph can have any number of equations that represent it, in part due to whether the expressions making up the equation have been fully simplified or not.

So, what reason will make me use sin²x + cos²x in the equation?

Normally you wouldn't. The person who gave this as an example was just making a point that the equation y = x can be written many ways.

In yet another way, the equation $y = \frac 1 2 \ln(e^{2x})$ is equivalent to y = x, if x > 0. IOW, the graph of this equation aligns perfectly with the graph of y = x in the first quadrant, and excluding the origin.

 

[Frabjous]

I might have missed the suggestion, but things could get ugly with an infinite series expansion in terms of your favorite orthogonal functions.

 

[pairofstrings]

The person who gave this as an example was just making a point that the equation y = x can be written many ways.

y = (x + 1)(sin²x + cos²x) -1 is an equation which appears as y = x when simplified.

Identity:
sin²x + cos²x = 1

What was the scenario for which Trigonometric functions like sine, cosine in equation y = (x + 1)(sin²x + cos²x) -1 were used?

I ask this question because y = x and y = (x + 1)(sin²x + cos²x) -1 both have similar graph, but the terms in the expressions are different. The terms are not same but the graph is, that is why please let me know the difference between y = x and y = (x + 1)(sin²x + cos²x) -1.

 

[Frabjous]

y = (x + 1)(sin²x + cos²x) -1 is an equation which appears as y = x when simplified.

Identity:
sin²x + cos²x = 1

What was the scenario for which Trigonometric functions like sine, cosine in equation y = (x + 1)(sin²x + cos²x) -1 were used?

I ask this question because y = x and y = (x + 1)(sin²x + cos²x) -1 both have similar graph, but the terms in the expressions are different. The terms are not same but the graph is, that is why please let me know the difference between y = x and y = (x + 1)(sin²x + cos²x) -1.

I do not understand your question.
sin²x + cos²x =1 so the equation simplifies to (x+1)-1=x

 

[Mark44]

What was the scenario for which Trigonometric functions like sine, cosine in equation y = (x + 1)(sin²x + cos²x) -1 were used?

As I already said, someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value.

I ask this question because y = x and y = (x + 1)(sin2x + cos2x) -1 both have similar graph, but the terms in the expressions are different.

The graphs are not just similar -- they are exactly the same.

The terms are not same but the graph is, that is why please let me know the difference between y = x and y = (x + 1)(sin²x + cos²x) -1.

Again, there is no real difference - the first one is just the simplified form of the latter one.

 

[pairofstrings]

As I already said, someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value.

But what is this equation talking about?: y = (x + 1)(sin²x + cos²x) -1?

What is the phenomenon?

I agree that y = (x + 1)(sin²x + cos²x) -1 is y = x on simplification, and someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value but what is sin²x, cos²x doing in the equation?

 

[Frabjous]

But what is this equation talking about?: y = (x + 1)(sin²x + cos²x) -1?

What is the phenomenon?

What is the equation y=x talking about? What is its phenomena?

The answers to your questions are the same as the answers to mine.

My answers are nothing and nothing. An equation is a mathematical abstraction that we map reality to.

 

[Mark44]

We seem to be going around in circles. You said this in post #20:

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.

Yes it can have a more complicated equation. If I multiply the right side of the equation by any expression whose value is always 1 (such as $\sin^2(x) + \cos^2(x)$) or add some expression whose value is always 0, I will get a new equation that is equivalent to the simpler one I started with -- exactly the same graph.

But what is this equation talking about?: y = (x + 1)(sin²x + cos²x) -1?

Who cares what it is "talking about"? All that matters is that this equation is equivalent to the much simpler equation y = x. To get this equation, we can add 0 in the form of 1 + (-1) to x to get x + 1 - 1, then multiply the x + 1 part by 1 in the form of $\sin^2(x) + \cos^2(x)$.

What is the phenomenon?

It doesn't have to represent any phenomenon. Someone can write down an equation that has no physical significance whatsoever, but so what? You are being overly concerned about something that really isn't worth all of that angst.

I agree that y = (x + 1)(sin²x + cos²x) -1 is y = x on simplification, and someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value but what is sin²x, cos²x doing in the equation?

It is multiplying a part of the right hand side by 1, which is always legal to do.

 

[pairofstrings]

I have equation of heart curve: x⁶ + 3x⁴y - 3x⁴ + 3x²y⁴ - x²y³ - 6x²y² + 3x² + y⁶ - 3y⁴ + 3y² = 1.

1. Heart curve with trigonometric functions:
x = 16 sin³
y = 13cost - 5cos(2t) - 2cos(3t) - cos(4t).

2. Psy curve with trigonometric functions.

Last thing I want to know is can I build any object with trigonometric functions as well?

Thanks.

 

[Frabjous]

I have equation of heart curve: x⁶ + 3x⁴y - 3x⁴ + 3x²y⁴ - x²y³ - 6x²y² + 3x² + y⁶ - 3y⁴ + 3y² = 1.

Heart curve with trigonometric function:
x = 16 sin³
y = 13cost - 5cos(2t) - 2cos(3t) - cos(4t).

Last thing I want to know is can I build any object with trigonometric functions as well?

Thanks.

Yes to within reason. It is called a Fourier series.

 

[pairofstrings]

Yes to within reason.

What is "Yes to within reason"? Does it mean that I can build only few types of objects with Fourier Series, not all objects?

 

[Frabjous]

What is "Yes to within reason"?

For example, a discontinuous function will have ringing near the discontinuity or a multi-valued function. I think that you could get around these by defining regions of applicability by defining regions of applicability for multiple series.

 

[Mark44]

I have equation of heart curve: x⁶ + 3x⁴y - 3x⁴ + 3x²y⁴ - x²y³ - 6x²y² + 3x² + y⁶ - 3y⁴ + 3y² = 1.

1. Heart curve with trigonometric functions:
x = 16 sin³
y = 13cost - 5cos(2t) - 2cos(3t) - cos(4t).

This is a different question from the one you posted in this thread, and is related to another thread you started a month or so ago. Your question of this thread apparently has been answered, so I'm closing this thread.

In your first equation above, you have an equation involving x and y. In the equations involving trig functions, x and y are given as parametric equations. A given curve can have different equations that generate the points on the curve, depending on whether the equations have been simplified or not (discussed in this thread) or depending on whether the curve is define in terms of parametric equations (heart equation with trig functions). A given curve can also have different sets of equations depending on the coordinates being used, such as if it's defined Cartesian coordinates or polar coordinates.

For example, a circle of radius 1/2 with center at (1/2, 0) can be defined by the Cartesian equation $(x - 1/2)^2 + y^2 = 1$ or by the polar equation $r = \cos(\theta)$.