# How do you distinguish between an identity and an equation?

• tahayassen
In summary, an equation becomes an identity when it is true for all values of the variable(s). An equation is also an identity when it is a special case of a function, but not all equations are identities. Identities are statements that are always true, while equations can be conditional, only true for specific values, or even false. In mathematics, a statement can have a logical value and can be used in exercises, mathematical statements, or definitions.
tahayassen
If you're just given x2+y2=1, how would you know if it's an equation or an identity? Functions are identities, right?

It is an equation as soon as there is an "=" Functions are abstract objects that take an input and produce an output. What you have written is not a function, because it would have a multivalued output. One uses the word identity, in two cases. First case: With f(x)=0 can mean either for some x or for all x. In the second case one says f(x) is identically zero and you can use three bars as some kind of super equal sign. Second case you have a complicated term and you want to replace it with another complicated term like $\sin^2 x = 1 - \cos^2 x$ this type of equations as replacement rules are sometimes called identities.

If A is identical to B the we may replace A by B in all cases and all situations, without restriction.

If A equals B then we may only substitute B for A, subject to restrictions.

For example Distance = Speed x Time is an equation, but not an identity since it is subject to the condition that the speed is constant.

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tahayassen said:
If you're just given x2+y2=1, how would you know if it's an equation or an identity?
There is usually some context around the equation if it's an identity. Sometimes people write this symbol, ##\equiv##, when they write an identity.

For example, (x - 2)2 ##\equiv## x2 - 4x + 4.

The equation above is true for any real value of x.

An equation that is true only for specific values of the variable is called a conditional equation. Every equation that you're asked to solve is of this type.

For example, x2 - 2x = - 4.
This equation is true for only one value.
tahayassen said:
Functions are identities, right?

tahayassen said:
Functions are identities, right?
A function is a type of relation so I don't know what you mean by it is an "identity". Perhaps you mean the identity map which is a special case of a function.

Generally I'd say identities are statements, but equations are not.

When we mean sin^2x + cos^2x = 1 as an identity, we're saying something. Namely that this is true for all angles x.

But if I were to put 3x^2 + x = 2, I'm not saying anything. It can be an exercise, used in as a part of a mathematical statement or even a definition, but by itself it says nothing.

I could say "there is an x such that 3x^2+x=2", but I could also say "solve for x when 3x^2+x=2" or "let x be such that 3x^2+x=2". For identities, you'd always begin your sentence with "for all x: sin^2x+cos^2x = 1", quantifying each variable.

I think you have to be be very careful using just equations to show the difference between equality and identity. Several of the equations offered are suspect.

For example is the following equation an equality or an identity?

$$\frac{1}{2} + \frac{1}{2} = 1$$

If I cut a sheep in half and gave you both halves would that be identical to a whole sheep?

If I cut two sheep in half and gave you the back half of each one would that be the same as a whole sheep or the same as the first situation?

Another exmple is the equation

The sum of angles of a triangle = 180.

This equation is true for all plane triangles but does not make all triangles identical. It is not an identity.

Identity is also important without an explicit equation using numbers.

For example every equilateral triangle is similar but not identical. Similarity amongst triangles is not an identity.
However if the triangles are not only equilateral but have one side equal then they are congruent. This means they are the same whatever their orientation. Congruence is an identity.

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Studiot said:
I think you have to be be very careful using just equations to show the difference between equality and identity. Several of the equations offered are suspect.

For example is the following equation an equality or an identity?

$$\frac{1}{2} + \frac{1}{2} = 1$$
This statement is an identity.
Studiot said:
If I cut a sheep in half and gave you both halves would that be identical to a whole sheep?
This has nothing to do with the equation 1/2 + 1/2 = 1, which is purely a relationship with numbers. If you add additional context, such as that 1 represents 1 sheep, you are moving away from the mathematical meaning.
Studiot said:
Another exmple is the equation

The sum of angles of a triangle = 180.

This equation is true for all plane triangles but does not make all triangles identical.
It's not saying that all (plane) triangles are identical - only that their angles add up to the same value. In the sense that this equation applies to all plane triangles, it is an identity - a statement that is always true.

disregardthat said:
Generally I'd say identities are statements, but equations are not.
This is incorrect. Conditional equations and identities are different kinds of statements. Inequalities are also statements.

In mathematics, a statement has a logical (i.e., true or false) value. A statement can be always true, true only under certain conditions, or never true (i.e., always false).
disregardthat said:
When we mean sin^2x + cos^2x = 1 as an identity, we're saying something. Namely that this is true for all angles x.

But if I were to put 3x^2 + x = 2, I'm not saying anything.
You are saying something; namely, that for some number(s), 3 times the square of the number plus the number is the same as 2. I can factor this equation and find the values of x that make it a true statement.

This equation is an example of a conditional equation, a kind of statement.
disregardthat said:
It can be an exercise, used in as a part of a mathematical statement or even a definition, but by itself it says nothing.

I could say "there is an x such that 3x^2+x=2", but I could also say "solve for x when 3x^2+x=2" or "let x be such that 3x^2+x=2". For identities, you'd always begin your sentence with "for all x: sin^2x+cos^2x = 1", quantifying each variable.

Good evening Mark. Your replies differ from the definitions given in my Collins Reference Dictionary of Mathematics. It has quite a bit to say about identities, the identity element, the identity function etc, offering 7 different mathematical cases of the word identity itself in all.

My congruent triangles corresponds to definition 5a.

It is true that a half plus a half can be an identity, and also corresponds to case5a but I gave conditions, I think you are invoking case1 incorrectly, "The property of being (another word for) the same individual". If value is all you are interested in then yes, but identity can be stronger than this.

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Hi Studiot,
I have a mathematics dictionary, but not at hand, so I can't compare. I agree that the word "identity" has different meanings in different contexts (e.g., identity function, identity element, etc.), but the context of the OP was in regard to equations. What I said about identities was specific to mathematical statements in the form of equations or inequalities.

I understand Tayahassen to be asking the very reasonable question what is the difference between an equality (equation) and an identity that we need to distinguish two different properties. And why (and when) do we use = and $\equiv$.

I have always understood that an identity is a stronger statement in some way and reflects the idea I put forward in post#3 that you can replace one side of an identity with the other without visible effect.

It is interesting that you write

There is usually some context around the equation if it's an identity

and then complain when I provide context!

If you don't like counting sheep how about replacing them with a further geometric example?

Cut a triangle in 'halves'.

Call them half-A and half-B.

Let the cut divide the triangle so that half-A has two sides and half-B three.

Do this to a second triangle and add the Halves-A together or in my parlance substitute half A for half B in the original triangle.

You will now have a quadrilateral!

Mark44 said:
You are saying something; namely, that for some number(s), 3 times the square of the number plus the number is the same as 2. I can factor this equation and find the values of x that make it a true statement.

As I explained, that would be if you put the "there exists an x" quantifier in front of the equation. I wouldn't call that part of the equation. Rather, the quantifier is added to it. The equation alone is not a complete mathematical statement, but has uses nevertheless. If you put an "for all x" in front, I'd call it an identity.

To be concise, in the statement (there exists an x: 2x=1), "there exists an x" is the quantifier, and "2x=1" is the equation. That doesn't make "2x=1" a statement.

Mark44 said:
In mathematics, a statement has a logical (i.e., true or false) value. A statement can be always true, true only under certain conditions, or never true (i.e., always false).

What kind of conditions are you talking about here? Statements are either true or false, there are no conditions.

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Studiot said:
I understand Tayahassen to be asking the very reasonable question what is the difference between an equality (equation) and an identity that we need to distinguish two different properties. And why (and when) do we use = and $\equiv$.

I have always understood that an identity is a stronger statement in some way and reflects the idea I put forward in post#3 that you can replace one side of an identity with the other without visible effect.

It is interesting that you write
There is usually some context around the equation if it's an identity

and then complain when I provide context!
My complaint was not that you provided context, but that what you provided was not related to what was asked in the OP, which was about equations.

And I'm not convinced that cutting a sheep in half says anything useful about the equation 1/2 + 1/2 = 1.
Studiot said:
If you don't like counting sheep how about replacing them with a further geometric example?

Cut a triangle in 'halves'.

Call them half-A and half-B.

Let the cut divide the triangle so that half-A has two sides and half-B three.

Do this to a second triangle and add the Halves-A together or in my parlance substitute half A for half B in the original triangle.

You will now have a quadrilateral!

Mark44 said:
You are saying something; namely, that for some number(s), 3 times the square of the number plus the number is the same as 2. I can factor this equation and find the values of x that make it a true statement.
disregardthat said:
As I explained, that would be if you put the "there exists an x" quantifier in front of the equation. I wouldn't call that part of the equation. Rather, the quantifier is added to it. The equation alone is not a complete mathematical statement, but has uses nevertheless. If you put an "for all x" in front, I'd call it an identity.
This is a good point, but might be a smidgen on the pedantic side. I would venture to say that most mathematics instructors would treat the qualifier "there exists an x such that..." as being implied. Logicians might quibble at this, but I doubt that mathematics instructors would.
disregardthat said:
To be concise, in the statement (there exists an x: 2x=1), "there exists an x" is the quantifier, and "2x=1" is the equation. That doesn't make "2x=1" a statement.
I'll go out on a limb and say that most mathematics instructors would call this a conditional equation, which is a kind of statement.
disregardthat said:
Mark44 said:
In mathematics, a statement has a logical (i.e., true or false) value. A statement can be always true, true only under certain conditions, or never true (i.e., always false).

What kind of conditions are you talking about here? Statements are either true or false, there are no conditions.

Clearly we disagree here. I consider the equation 2x = 1 to be a statement. It is true if the variable happens to be 1/2, and is false otherwise.

Clearly no disagreement of substance here.

But in my view you're talking about "For all x, if x = 1/2, then 2x=1", which is a (true) statement using the equation "2x=1". I don't really know of conditional statements, for me, statements are either true or false. "Conditional statements" seems like no more than a label for uncompleted statements. For the sake of clarity in this context uncompleted statements should not be mixed with proper ones.

FWIW, here's some of the wiki article on equations (http://en.wikipedia.org/wiki/Equation).
(I added the underline in two passages, below)
An equation, in a mathematical context, is generally understood to mean a mathematical statement that asserts the equality of two expressions.[1] In modern notation, this is written by placing the expressions on either side of an equals sign (=), for example
$$x + 3 = 5$$

Identities
One use of equations is in mathematical identities, assertions that are true independent of the values of any variables contained within them. For example, for any given value of x it is true that x(x - 1) = x2 - x.

However, equations can also be correct for only certain values of the variables.[2] In this case, they can be solved to find the values that satisfy the equality. For example, consider the following.
x2 - x = 0.

The equation is true only for two values of x, the solutions of the equation. In this case, the solutions are x = 0 and x = 1.

Many mathematicians reserve the term equation exclusively for the second type, to signify an equality which is not an identity. The distinction between the two concepts can be subtle; for example,
(x + 1)2 = x2 + 2x + 1
is an identity, while
(x + 1)2 = 2x2 + 2x + 1
is an equation with solutions x = 0 and x = 1. Whether a statement is meant to be an identity or an equation can usually be determined from its context. In some cases, a distinction is made between the equality sign (=) for an equation and the equivalence symbol (##\equiv##) for an identity.

The example of the equation that is not an identity is what I'm calling a conditional statement.

That's an informal article citing informal sources.

But to address their explanation, an equation is only a statement when there are no variables involved (in which case any quantifier is irrelevant). So asserting that x+3 is equal to 5 makes sense only when x is a predefined value, not when it's a variable. It is essential that all variables in an expression are bounded in order for it to be a statement (with a truth value).

See http://en.wikipedia.org/wiki/Sentence_(mathematical_logic)

In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: As the free variables of a (general) formula can range over several values, the truth value of such a formula may vary.

Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic sentences by applying connectives and quantifiers.

In this context equations such as "2x +3 = 1" for a variable x are atomic sentences.

So "1=1", or "1+1 = 2" are statements, but "x=1" is not.

If, on the other hand, I say "let x = 3", then suddenly "x=3" is a (true) statement, but that's entirely different.

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Hello, disregardthat, you have brought in the issue of 'completedness' in terms of qualifiers.

Equally I might observe that the statement $\forall$x is incomplete.

What x?

$\forall$x $\in$ R?

$\forall$x :→x>0?

etc

So where do you draw the line (stop)?

disregardthat,

In post #15, I said
This is a good point, but might be a smidgen on the pedantic side. I would venture to say that most mathematics instructors would treat the qualifier "there exists an x such that..." as being implied. Logicians might quibble at this, but I doubt that mathematics instructors would.

These definitions are taken from the Oxford Concise Dictionary of Mathematics.

identity: An equation which states that two expressions are
equal for all values of any variables that occur, such as x2 -
y2 = (x + y)(x - y) and x(x - 1)(x - 2) = x3 - 3x2 + 2x.
Sometimes the symbol
is used instead of = to indicate that a
statement is an identity.

equation: A statement that asserts that two mathematical
expressions are equal in value. If this is true for all values of
the variables involved then it is called an *identity, for
example 3(x – 2) = 3x – 6, and where it is only true for some
values it is called a *conditional equation; for example x2 – 2x
–3 = 0 is only true when x = –1 or 3, which are known as the
*roots of the equation.

Best Pokemon said:
These definitions are taken from the Oxford Concise Dictionary of Mathematics.

identity: An equation which states that two expressions are
equal for all values of any variables that occur, such as x2 -
y2 = (x + y)(x - y) and x(x - 1)(x - 2) = x3 - 3x2 + 2x.
Sometimes the symbol
is used instead of = to indicate that a
statement is an identity.

equation: A statement that asserts that two mathematical
expressions are equal in value. If this is true for all values of
the variables involved then it is called an *identity, for
example 3(x – 2) = 3x – 6, and where it is only true for some
values it is called a *conditional equation; for example x2 – 2x
–3 = 0 is only true when x = –1 or 3, which are known as the
*roots of the equation.

Thank you, Best Pokemon. These definitions look pretty close to what I have been saying in this thread.

I always thought the ≡ symbol meant "is defined as". But I guess if it's defined as something, then it's also an identity. I think if they used that symbol more, then there would be less confusion for new students about the concept of an identity.

Thanks for the clear-up. :)

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tahayassen said:
I always thought the ≡ symbol meant "is defined as".
No, the usual meaning is that the expressions on either side have the same value.

I've seen some texts that borrow notation from the Pascal programming language, :=, to indicate that a term or variable is being defined.

tahayassen said:
But I guess if it's defined as something, then it's also an identity. I think if they used that symbol more, then there would be less confusion for new students about the concept of an identity.
Other texts distinguish between identities and conditional equations by using ##\equiv## vs. =.

Wait a second...

$$(x-3)(x+2)(x)\\ ={ (x }^{ 2 }-x-6)(x)\\ ={ x }^{ 3 }-{ x }^{ 2 }-6x$$

should really be...

$$(x-3)(x+2)(x)\\ \equiv { (x }^{ 2 }-x-6)(x)\\ \equiv { x }^{ 3 }-{ x }^{ 2 }-6x$$

High school teachers and professors should really teach the $\equiv$ notation more. It makes things easier to understand by removing ambiguity. In programming, they make that distinction I believe.

I've seen some texts that borrow notation from the Pascal programming language, :=, to indicate that a term or variable is being defined.

Good point.

Pascal is not the only language to use = as the 'assignment operator'
Tayahassen is studying computing.

You then get such programming statements as

x=x+1

Which makes sense in computing terms.

Mark44 or any other authority, this has long time bugged me, though I live with it. Wouldn't you agree that most math e.g. textbooks writes a sort of slang here? That a lot of the 'equations' really should be identities?
[STRIKE][/STRIKE]
As in any old theorem at random - say if you say

If f(x) = g(x)h(x) then f'(x) = g'(x)h(x) + g(x)h'(x)

you are not wanting to say you have this f, g, and h and then it may happen that for some particular values of x that first equation might be satisfied, in which case the second one is also for those particular values?

Or reformulating in a single [STRIKE]equation[/STRIKE] [STRIKE]expression[/STRIKE] relationsionship

[g(x)h(x)]' = g'(x)h(x) + g(x)h'(x)

should really be an identity, as maybe more than half the 'equations' in the average textbook should be?

Studiot said:
Good point.

Pascal is not the only language to use = as the 'assignment operator'
Tayahassen is studying computing.

You then get such programming statements as

x=x+1

Which makes sense in computing terms.

Many programming languages distinguish between assignment and equality by using different symbols. In Pascal-based languages such as Modula-2 and (I think) Ada, the assignment operator is := and the equality operator is =.

C-based languages use = for assigment (only) and == for equality.

Your example, Studiot, can be confusing to beginning programmers, until they understand that x = x + 1 is not saying that x and x + 1 are equal (an impossibility), but that the value of the expression x + 1 is being assigned to the variable x.

epenguin said:
Mark44 or any other authority, this has long time bugged me, though I live with it. Wouldn't you agree that most math e.g. textbooks writes a sort of slang here? That a lot of the 'equations' really should be identities?
Yes on both.
epenguin said:
As in any old theorem at random - say if you say

If f(x) = g(x)h(x) then f'(x) = g'(x)h(x) + g(x)h'(x)

you are not wanting to say you have this f, g, and h and then it may happen that for some particular values of x that first equation might be satisfied, in which case the second one is also for those particular values?
These equations are essentially identities. We are defining f to be the product of g and h, pretty much independent of the value of x (as long as x is in the common domain of g and h, of course). The equation with f' can also be considered an identity.
epenguin said:
Or reformulating in a single [STRIKE]equation[/STRIKE] [STRIKE]expression[/STRIKE] relationsionship

[g(x)h(x)]' = g'(x)h(x) + g(x)h'(x)

should really be an identity, as maybe more than half the 'equations' in the average textbook should be?
I agree. It would probably cut down the confusion.

A different example that comes to mind is the technique of partial fractions decomposition, in which you break down a rational expression that represents a product into the sum of two or more simpler rational expressions.

For example, let's look at ##\frac{2}{x^2 + x} = \frac{2}{x(x + 1)}##

The idea is that we want to write ##\frac{2}{x(x + 1)} ## as ##\frac{A}{x} + \frac{B}{x+1}##

The equation we get is really an identity
$$\frac{2}{x(x + 1)} \equiv \frac{A}{x} + \frac{B}{x+1}$$
meaning that the two expressions on the left and right must be identically equal (barring two obvious values of x).

If we multiply both sides by x(x + 1), we get ## 2 \equiv A(x + 1) + Bx##.
Doing some rearranging, we get ## 2 \equiv (A + B)x + A##.

Where students who are new to this technique have problems is recognizing that the only way two polynomials can be identically equal is when the coefficients of their respective terms are equal.

The polynomial on the left is really 0x + 2, amd the one on the right is (A + B)x + A. With the recognition that this is an identity, it's easy to see that A + B = 0 and A = 2, hence B = -2.

Statements in mathematics take place within some context. Equations and identities are statements. Equations have "solution sets", which are the set of "things" that satisfy them within the given context. (i.e. their variables represent elements of some "universal set"). An identity is an equation who solution set is "everything", in given context.

For example, xy = yx is an identiy in the context of the set of real numbers, but not in the context of the set of 2 by 2 matrices of real numbers.

Mark44 said:
Where students who are new to this technique have problems is recognizing that the only way two polynomials can be identically equal is when the coefficients of their respective terms are equal.

If ≡ was replaced with =, wouldn't it also be true that the coefficients of their respective terms are equal?

tahayassen said:
If ≡ was replaced with =, wouldn't it also be true that the coefficients of their respective terms are equal?

No, because the two equations would have different meanings. Let's look at the equation in my example a couple of posts ago.

As an identity:
2 ##\equiv## (A + B)x + A

As an identity, the equation above has to be true for all values of x. In my previous work I solved for the constants A and B, and found them to be A = 2, B = -2.

As an ordinary equation:
2 = (A + B)x + A
If we interpret the above as an ordinary equation (not an identity), given values of A and B, we could solve for the value of x that makes the equation a true statement.

The work would look like this.
2 - A = (A + B)x
$$\Rightarrow x = \frac{2 - A}{A + B}$$
If someone tells us the values of the constants A and B, we can determine the value of x.

## 1. How do you define an identity?

An identity is a mathematical statement that is always true, regardless of the values of the variables involved. It is an equality that holds true for all values of the variables.

## 2. What is an equation?

An equation is a mathematical statement that shows the relationship between two or more quantities. It consists of an equality sign and expressions on either side that may contain variables.

## 3. How do you distinguish between an identity and an equation?

An identity is always true, while an equation may or may not be true depending on the values of the variables. Additionally, an identity is an equality between two expressions that are identical, whereas an equation may involve different expressions on either side.

## 4. Can an identity also be an equation?

Yes, an identity can also be an equation if the expressions on either side of the equality sign are identical. However, not all equations are identities.

## 5. How do you prove that an equation is an identity?

To prove that an equation is an identity, you must show that it holds true for all values of the variables involved. This can be done through algebraic manipulation or by substituting different values for the variables and showing that the equation remains true.

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