What Does It Mean If an Expression Is Equal to Another Expression?

[pairofstrings]

TL;DR Summary

Equations with: equals to "one", equals to "y", and equals to another 'equation'

Hello.
My question is:
what does it mean if an equation is being equaled to another equation?
Please give me an example.
Thanks.

Thank you for pointing my mistake.
I should have written "expression" instead of "equation".

My question is: what does it mean if an expression is being equaled to another expression? Could you please shed some light on it?

 

[Office_Shredder]

This question doesn't really make sense. An equation is two expressions, with an equal sign between them. You could say two equations are equivalent, which means one is true if and only if the other one is. For example $2x=4$ is an equation, and $x=2$ is an equivalent equation.

I don't think anyone would say those two equations are equal though.

 

[PeroK]

My question is: what does it mean if an expression is being equaled to another expression? Could you please shed some light on it?

It can mean one of two things:

1) The expressions are identically equal. If we have variables $x, y$, then it means that the equation (identity) holds for all values of $x$ and $y$. Examples: $$x + y = y + x$$ $$\sin^2 x = 1 - \cos^2 x$$
2) Or, we are looking for specific values of the variables where the equation holds. Examples: $$x + 2 = 5$$ $$x^2 + y^2 = 1$$
In the second case these equations may or may not have solutions. In the first example, we have a single solution: $x = 3$. In the second example, the solution is the set of points $(x, y)$ that form a circle of radius $1$ centered at the origin.

There are, of course, examples of equations with no solutions: $$x = x + 1$$ $$x^2 + 1 = 0$$

 

[pairofstrings]

We have three things:
1. expression = constant
2. expression = variable
3. expression = expression
Please give every-day life example for 3rd in the above list.
Thanks..

 

[FactChecker]

3. expression = expression
Please give every-day life example for 3rd in the above list.

Joe's age in 5 years will equal twice Alice's age last year:
J+5 = 2(A-1)
Although this is an artificial example, there are countless examples occurring every day in the real world.

 

[pairofstrings]

..there are countless examples occurring every day in the real world.

Please give me two examples, one that has level as Intermediate, and the other that has level as Advanced.
Thanks.

 

[FactChecker]

Please give me two examples, one that has level as Intermediate, and the other that has level as Advanced.

It's hard to know what you would consider "intermediate" and "advanced", but here are two.
The formula for the sine of the sum of two angles: $sin(\alpha+\beta) = sin(\alpha)cos(\beta)+cos(\alpha)sin(\beta)$
Green's theorem: $\oint_{C}(Ldx+Mdy) = \iint_D(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y})dxdy$

 

[pairofstrings]

1. expression = constant
2. expression = variable
3. expression = expression

Is there any name given to 1st type, 2nd type, 3rd type?
Thanks.

 

[FactChecker]

Is there any name given to 1st type, 2nd type, 3rd type?
Thanks.

There is a lot of mixing, matching, and ambiguity of terms, but here are some that often apply in the right situation:
1. expression = constant : constraint
2. expression = variable : definition of the variable
3. expression = expression : identity
In general, these terms are not universal short-hand for that type of equation.

 

[pairofstrings]

3. expression = expression : identity

Can you please tell me what an 'identity' is?

 

[FactChecker]

Can you please tell me what an 'identity' is?

In mathematics and English, it is equation = equation.

 

[Paul Colby]

Can you please tell me what an 'identity' is?

It's a relation between two objects or things. For any given identity, one must define the thing on the right, the thing on the left and, what it means to be identical. For example, $m=2.$ The thing on the left, $m,$ is a symbol I define as any integer. The thing on the right, $2,$ is an example of a specific integer. The equal sign is usually defined as "has the same numerical value."

 

[PeroK]

Can you please tell me what an 'identity' is?

In a moment of inspiration I wondered if there might be something on Wikipedia on this:

https://en.wikipedia.org/wiki/Identity_(mathematics)

 

[PeroK]

Can you please tell me what an 'identity' is?

See also post #3 (in case you missed it the first time):

1) The expressions are identically equal. If we have variables $x, y$, then it means that the equation (identity) holds for all values of $x$ and $y$. Examples: $$x + y = y + x$$ $$\sin^2 x = 1 - \cos^2 x$$

 

[PeroK]

It's a relation between two objects or things. For any given identity, one must define the thing on the right, the thing on the left and, what it means to be identical. For example, $m=2.$ The thing on the left, $m,$ is a symbol I define as any integer. The thing on the right, $2,$ is an example of a specific integer. The equal sign is usually defined as "has the same numerical value."

I'm not sure where you got that idea from, but it's not what a mathematical identity is. See the Wikipedia entry above. In short, it is an equation that holds for all values of the variables - a common example being trigonometric identities, such as the double-angle formulas.

 

[Paul Colby]

I'm not sure where you got that idea from, but it's not what a mathematical identity is.

Perhaps but I still see a need to define what's on the left, what's on the right, and what is meant by equal. Is $m$ real, integer, quaternion? Are we talking mappings? Modular arithmetic? Domains, ranges, equivalence classes, all these really need to be supplied when required. I don't know, is there one and only one concept of mathematical identity? Seems like there is often more to it.

 

[pairofstrings]

In a moment of inspiration I wondered if there might be something on Wikipedia on this: https://en.wikipedia.org/wiki/Identity_(mathematics)

Quote from the above link.
I understand the first statement.
The following is the second statement.
"In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined."

Attempt to understand above quoted statement:
The statement is saying that: if A = B is an identity if A and B define the same function, and identity is also an equality between functions that are differently defined?

Thanks..

 

[PeroK]

Quote from the above link.
I understand the first statement.
The following is the second statement.
"In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined."

Attempt to understand above quoted statement:
The statement is saying that: if A = B is an identity if A and B define the same function, and identity is also an equality between functions that are differently defined?

Thanks..

Like $\tan x = \frac{\sin x}{\cos x}$.

That's an identity. Being the same function is another way of saying the expressions are equal for all values of the variable(s).

 

[PeroK]

Compare that with the equation $$\sin x = \cos x$$ which has only two solutions for $x \in [0, 2\pi)$.

 

[pairofstrings]

So, identities means... facts?

 

[PeroK]

So, identities means... facts?

No. Fact has no specific mathematical meaning. A true statement is what we say in mathematics. But, these are logical constructions and much more than just identities. Examples of true statements are:

$5$ is a prime number

$x^2 + 2x + 1$ is a quadratic expression.

$\sin x$ is a continuous function

$\sin^2 x + \cos ^2 x = 1$ is an identity.

 

[Mark44]

In mathematics and English, it is equation = equation.

No. An equation can't be "equal to" an equation.
This is like saying that $2x = 4 = x = 2$, which is nonsensical.
An identity is an equation that is true for all values of the variable in question.
A couple of examples:
$2(x + 3) = 2x + 6$
$\sin(2x) = 2\sin(x)\cos(x)$

In contrast, some equations are conditional; i.e., true only under some condition, such as:
$2(x + 3) = 9$
This is a true statement only if x = 3.
$(x - 1)(x + 2) = 0$
This is a true statement only if x = 1 or if x = -2.

 

[FactChecker]

No. An equation can't be "equal to" an equation.
This is like saying that $2x = 4 = x = 2$, which is nonsensical.

Sorry. Good catch. I should have said expression = expression.