# Notation convention for distinguishing uses of equal sign when manipulating equations

When reading (or writing, in fact) a proof, or the resolution of an equation, I often find myself in the following situation.

Say, for example, that the writer proves the statement $A=B$, where $A$ and $B$ represent any valid mathematical expressions. Along the course of the proof, the writer manipulates the equation until at some point they use a statement of the form $$X=Y=Z\ \text{,}$$ and then I find myself trying to decipher if it is something like $$A=A'=B$$ or $$A=B=B'$$ (or even $A=B'=B$), where the prime ($'$) symbol represents a "direct" algebraic manipulation of the corresponding expression.

To clarify what I mean, here's a simple example. Suppose we're solving the equation $(1+2)x=x^3$. At some point we write $$(1+2)x=3x=x^3\ .$$ In this example it is trivial to see that the second expression has been obtained by an algebraic manipulation of the first one: this statement is semantically equivalent to the sentence "A, which by the way is the same as A', is equal to B", where—in this particular case—$A$ is $\ (1+2)x\$, $A'$ is $\ 3x\$ and $B$ is $\ x^3\$.

But for much more complicated expressions, it is not so trivial to see, without assuming a convention or using different symbols for each equality. Moreover, often algebraic manipulations are performed on both sides, but only one of them is "expanded" in the format $A=A'$—the other one is written directly ($B'$).

So my question is, is there a convention that allows a direct interpretation of the 'framework' of the statement without comparing the actual expressions?

For example, would the following use of the "defined as" symbol ($\doteq$) be customary?

$$A = A'\doteq B$$

If we were, instead of just solving an equation, proving that $A=B$, would the symbol $\stackrel{?}=$ make sense?

$$A = A' \stackrel{?}= B$$

• $A=B$ says $A$ and $B$ are equal. It says nothing about how we know they are equal. That is usually better explained in text, rather than by notation. But if $A'=B$ is already known (perhaps from a definition of $A'$ or of $B$, perhaps from other results) and then $A$ is obtained by simplification of $A'$, then the most natural way to write this might be $B = A' = A$. – Robert Israel Aug 14 '18 at 19:30
• You might write something like "where the first equality comes from ... and the second from ...". – Robert Israel Aug 14 '18 at 19:34

I agree with your concern, this is a common situation. In cases the chain of identities would not seem obvious, textual explanations and/or splitting the derivation in more steps can be helpful.

E.e.

Given the equation

$$\tag1 (1+2)x=x^3$$ we can rewrite

$$(1+2)x=3x$$

so that from the initial identity

$$3x=x^3.$$

I wouldn't favor the "defined as" convention which doesn't sound logical. $$3x:=x^3$$ is not the intended meaning.

Alternatively, you could annotate the equal sign with a reference to where the identity appeared.

$$(1+2)x=3x\stackrel{(1)}=x^3.$$

• Thanks for your insight; the annotation above the equals sign is a potential solution for enhancing readability while keeping things short. Alternatively, if the context of the initial equation was a proof, specifically (i.e. we wanted to prove $A=B$) would the use of the symbol $\stackrel{?}=$, as in $A=A'\stackrel{?}=B$, make sense? Or would it actually be more confusing? – Anakhand Aug 15 '18 at 8:01
• @Anakhand: this is addressing a different issue. – Yves Daoust Aug 15 '18 at 8:04