What are the different types of equations and their names?

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In summary, the conversation discusses different types of equations and the potential need for names to categorize them. However, it is argued that context and aim are more important factors in determining the relevance and usefulness of naming equations. The different types of equations mentioned are expression = 1, expression = y, and expression = expression, but it is concluded that these classifications are not useful in categorizing equations. It is also noted that in the context of functions on real numbers, the equation "y" in terms of "x" is called a function.
  • #1
pairofstrings
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TL;DR Summary
There can be a expression on LHS and RHS, 1 on LHS expression on RHS, y on LHS expression on RHS.
Hi.
What is the name of the equation whose LHS and RHS are expressions?
What is the name of the equation whose LHS is constant like '1' and RHS is an expression?
What is the name of the equation whose LHS is a variable like 'y' and RHS is an expression?

What is the name of the equations in the below statements?
1 = expression --?
y = expression --?
expression = expression --?

Are there any other variants that I am missing?
Thanks.
 
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  • #2
Hi,

This is a physics forum. They are all equations. No subspecies like in biology.
Why should they have names ?

##\ ##
 
  • #3
BvU said:
Why should they have names ?
I thought they might have names like several other different types of equations like Logarithmic, Simultaneous, Cubic, Quadratic, Trinomial, Binomial, Monomial, Polynomial, Trigonometric, Radical, Exponential, Rational, Linear, Cartesian, Polar, Algebraic, Nonparametric, Parametric. These are equations. So, if there so many different types of equation I thought those three equations could have three more names. :)
 
  • #4
pairofstrings said:
Summary:: There can be a expression on LHS and RHS, 1 on LHS expression on RHS, y on LHS expression on RHS.

Hi.
What is the name of the equation whose LHS and RHS are expressions?
What is the name of the equation whose LHS is constant like '1' and RHS is an expression?
What is the name of the equation whose LHS is a variable like 'y' and RHS is an expression?

What is the name of the equations in the below statements?
1 = expression --?
y = expression --?
expression = expression --?

Are there any other variants that I am missing?
Thanks.
Names are given if it a) helps to summarize complex conditions, b) an object occurs repeatedly, or c) contributes to clarification. They are normally context-sensitive. E.g., the word ring can have various meanings. Sure, we have named formulas (Pythagoras, Euler, etc.) but these refer to specific equations, not to some sort of equations. ##1=f(x,y)## is a contour line, a level equation. However, note the context-sensitivity: If we are not wandering through calculus, but e.g. through linear algebra, then ##1=f(x,y)## might simply refer to some invertible matrices, a determinant, or whatever.

Hence context is the primary condition to be specified before we talk about names.
 
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  • #5
fresh_42 said:
Hence context is the primary condition to be specified before we talk about names.
The context is that I want to study a situation where there is equation like: expression = expression.

This "expression = expression" may not have a name..
 
  • #6
What changes for you if it has a name ? Do you want to look it up with google ?
 
  • #7
pairofstrings said:
The context is that I want to study a situation where there is equation like: expression = expression.
$$\begin{eqnarray*}
f(x)&=&g(x)\\
\frac{f(x)}{g(x)}&=&1
\end{eqnarray*}$$So isn't it a matter of choice whether you have a pure number on one side or not?
 
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  • #8
pairofstrings said:
The context is that I want to study a situation where there is equation like: expression = expression.
Please define "expression", rigorously! What is your alphabet, what is your syntax, and what is your grammar?
 
  • #9
I believe the OP is searching for the term "implicit function" or "implicitly defined function".
Are we all on nasty pills today?
 
  • #10
hutchphd said:
I believe the OP is searching for the term "implicit function" or "implicitly defined function".
Are we all on nasty pills today?
I don't think so. Definitions of used technical terms are the basis for any scientific discussion. Implicit functions are a guess as any other guess before. Expression isn't well-defined, so it's legitimate to ask what is meant by it.
 
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  • #11
Maybe if he knew the definition he wouldn't be asking the question??
 
  • #12
BvU said:
What changes for you if it has a name ?
Nothing much and I will move on little sad.

BvU said:
Do you want to look it up with google ?
I say that there could be three distinct artifacts:
  • expression = 1
  • expression = y
  • expression = expression
The first is a object confined within 1 unit on a graph.
The second one is a function.
The third -- ?

This the third is what I am after.

fresh_42 said:
Please define "expression", rigorously! What is your alphabet, what is your syntax, and what is your grammar?
I say that "expression" is some finite number of terms that can exist on either side of equals-to and these terms can be numbers, variables, expression(?).

Thanks.
 
  • #13
Let us assume you mean functions on real numbers: ##f(x)=y\, : \,\mathbb{R} \longrightarrow \mathbb{R}##.

Then ##f(x)=1## can be viewed as ##f(x)-1=0##. If ##x\in \mathbb{Z}## and ##f(x)## are linear functions, then the system is called Diophantic.

If we consider the possible values ##x## then we call them - the ##x## - zeros, or roots; algebraic variety under other circumstances (##f## polynomial, possibly other fields than ##\mathbb{R}) .##

Depending on the structure of ##f(x):=g(x)## as a definition of ##f## by ##g##, it is called simply a function, polynomial, exponential, constant, continuous, ##n##-times differentiable, or whatever.

I do not see the difference between expression = y and expression=expression.

It all depends on context and aim.
 
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  • #14
pairofstrings said:
Nothing much and I will move on little sad.
no need !
I say that there could be three distinct artifacts:
  • expression = 1
  • expression = y
  • expression = expression
The first is a object confined within 1 unit on a graph.
The second one is a function.
The third -- ?

This the third is what I am after.
Unfortunately I see no difference at all between the three
  • expression - 1 = 0 ##\qquad\qquad\Leftrightarrow\qquad ## expression = 0
  • expression - y = 0 ##\qquad\Leftrightarrow\qquad ## expression = 0
  • expression - expression = 0 ##\qquad\Leftrightarrow\qquad ## expression = 0
 
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  • #15
There is a name for expression with a colon.
##f(x):=g(x)## (##f## is defined by ##g##) or ##f(x)=:g(x)## (##g## is defined by ##f##) are definitions.
 
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  • #16
BvU said:
Unfortunately I see no difference at all between the three
  • expression - 1 = 0 ⇔ expression = 0
  • expression - y = 0 ⇔ expression = 0
  • expression - expression = 0 ⇔ expression = 0
I agree completely. The original classifications -- expression = 1, expression = y, expression = expression -- are not useful in categorizing equations.
 
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  • #17
 
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  • #18
fresh_42 said:
Let us assume you mean functions on real numbers: f(x)=y:R⟶R.
Here the equation 'y' in terms of 'x' is called a function..
So, shouldn't there be a name if the equation is in both x, y?
 
  • #19
pairofstrings said:
Here the equation 'y' in terms of 'x' is called a function..
So, shouldn't there be a name if the equation is in both x, y?
##y(x)## in ##f(x,y(x))=\ldots## is called an implicit function. ##f(x,y)=\ldots## is still a function.

To make it worse: even function is already an implicit assumption. It could as well be just a relation.
 
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  • #20
fresh_42 said:
y(x) in f(x,y(x))=…
The above equation 'y' has terms in 'x' and it is called function?
Equation even with x, y variables on either sides of the equation is a function?
 
  • #21
pairofstrings said:
The above equation 'y' has terms in 'x' and it is called function?
Equation even with x, y variables on either sides of the equation is a function?
Yes and yes. There is no rule on how many variables a function has. O.k., it's usually a finite number, but even infinitely many variables are sometimes regarded.

If ##y## depends on ##x##, in notation: ##y=y(x)##, and ##y## also occurs as a variable of a function, then ##f(y)=f(y(x))=f(x,y(x))## the function automatically depends on ##x##, too. If only ##f## is given, then we say that ##y(x)## is implicitly given, namely with the help of ##f##.
 
  • #22
But there is something called "Vertical Line Test" that says if the graphs has more than one intersection then it is said that the equation is not a function.
What is your saying on this?

Thanks.
 
  • #23
pairofstrings said:
But there is something called "Vertical Line Test" that says if the graphs has more than one intersection then it is said that the equation is not a function.
What is your saying on this?

Thanks.
Yes. In that case we have appointed to ##y##-values to the same ##x##-value, i.e. ##y_1=f(x)=y_2##. The vertical line at ##x## has two intersections with ##f##, namely at the two heights ##y_1## and ##y_2##.

But even my notation as ##y_1=f(x)=y_2## is already wrong. You cannot write it as a function. If it appoints two values, then it is a relation. Every function is a relation, but not every relation is a function. A relation is a subset of ##X\times Y,## the pairs of all ##x-##values and all ##y-##values. In this case we would call the subset ##R## for relation and write ##(x,y_1)\in R## and ##(x,y_2)\in R.##
 
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  • #24
Is equation of a circle: ##x^{2}+y^{2}=1## also a function?
 
  • #25
It depends on how you define it. It is the contour line of the function ##f(x,y)=x^2+y^2## at level ##1.## In this case our function goes ##f\, : \,\mathbb{R}^2\longrightarrow \mathbb{R}\, , \,(x,y)\mapsto (x^2+y^2).##

However, it is no function of the reals to the reals. The vertical line test fails. We could talk about implicit functions in this case, but I'm afraid that it might confuse you. For short: the circle is locally a function if ##x\neq \pm 1##. In a small neighborhood around a point, we have a function since e.g. the lower half of the circle isn't in that neighborhood. If ##x \in (0,1)## and ##y>0##, then ##y=\sqrt{1-x^2}##. Similar if ##y<0##, then we get ##y=-\sqrt{1-x^2}##. These are local functions. The entire circle as subset of ##\mathbb{R}\times \mathbb{R}## is a relation, no function.
 
  • #26
expression = expression
is the same as
0 = expression - expression
 

1. What are the most commonly used equations in science?

The most commonly used equations in science include Newton's Second Law of Motion, the Law of Universal Gravitation, the Pythagorean Theorem, the Ideal Gas Law, and Einstein's Mass-Energy Equivalence.

2. How are equations named?

Equations are typically named after the scientist who first discovered or developed them, or after the concept or phenomenon they represent.

3. What is the significance of equations in scientific research?

Equations play a crucial role in scientific research as they provide a concise and precise way to describe and quantify natural phenomena, making them easier to study, analyze, and predict.

4. Can equations be modified or combined to create new equations?

Yes, equations can be modified or combined to create new equations that can describe more complex or specific phenomena. This is often done in scientific research to better understand and explain natural processes.

5. Are equations always accurate and reliable?

Equations are based on scientific principles and are constantly tested and refined through experiments and observations. While they may not always be 100% accurate, they are generally considered reliable and are constantly being improved upon.

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