How to read chained equalities out loud?

Question

I find that my community-college students are usually very hazy on the status and meaning of chained equality statements (or other relational statements). This seems like a really critical element of mathematical grammar that is almost never directly discussed in books or lectures, and not assessed or tested in any way.

On that point, it occurred to me that I'm not entirely sure (and have no idea where to find) a standard prescription for how to read such statements out loud. For example, given: $$a = b = c$$

I could imagine reading this aloud as:

• "$a$ equals $b$ equals $c$",
• "$a$ equals $b$, and $b$ equals $c$",
• "$a$ equals $b$, which is equal to $c$",

or probably a variety of other constructions. What is most conventional, and what would be most clear for an audience struggling with these statements? Consider also cases that grow very long, possibly broken up over many lines on a single page.

Related: Framework for Compound Inequalities

Answer 1

When an inexperienced student sees $a=b=c$, I'd assume that both $a=b$ and $b=c$ are clear but the transitivity that yields $a=c$ might not be obvious.

That's why I'd focus on this hidden equality when reading it out loud, by not just reading the equation (your first option), but rather saying "a, b, and c are all equal" or "a, b, and c are the same [number/term/equation/set...]"

For longer terms between the equal signs, I'd also emphasize that all of them are equal (borrowing from Ethan's example): "The following terms are all equal: x to the fourth power minus y to the fourth power and the difference of ... times the sum of ... and finally ..."

Answer 2

I'll suggest that the best verbalization is the last one in the question:

$a$ equals $b$, which is equal to $c$

Here's why I think so: It's the closest to a literal symbol-by-symbol reading of the chained equation, while still being a grammatically-correct English sentence. Note that this matches the top-voted (but not accepted) answer to the analogous closed question on SE English Language & Usage.

Contrast this with a purely symbolic reading, like "$a$ equals $b$ equals $c$". Unfortunately, this doesn't qualify as valid English, being an example of a run-on sentence. There are two separate valid statements here ("$a$ equals $b$" and "$b$ equals $c$"), but in English we need a conjunction to join them.

On the other hand, recall that we're looking for a direct reading, not an interpretation. In this regard, "$a$, $b$, and $c$ are the same number" is an interpretation, not a literal verbalization; for example, the verb clause ("are the same") only appears once, whereas the analogous relation ("$=$") appears twice, it's in a different order, etc. Consider also that such an interpretation/summary would not extend well to chained relations using a variety of symbols, such as $a = b < c = d \le e$.

The root of the problem is that algebraic chained relations have a structure that simply isn't permitted (as a run-on sentence) in English. Personally I choose to verbalize something that is proper English, but as close to the written symbols as possible, by inserting the single word "which" -- thereby turning the relations after the first into nonrestrictive relative clauses (see definition three here).

Answer 3

Personally, I often add a variety of different expressions. I think doing so not only makes the presentation marginally less dry, but also perhaps increases the fluency with which students will be able to deal with symbols by understanding their meaning.

In particular, instead of just "equals" I would often use expressions like "the same as", "is nothing but", or "is equivalent to". Similarly, when dealing with things like logical equivalences, instead of always using "if and only if", I've found it natural to say things like "which happens exactly when", but also "is equivalent to" and "the same as" when the meaning is clear.

The idea that we can easily switch between the different expressions makes it clear that equality is really a relation that just claims that two things are equal and equivalent, and is not some kind of voodoo symbol that is devoid of any meaning. This could also aid in solving the problem of some students thinking of the equal sign as some indication of "the next step", leading to problems like $2x=10=x=5$ occurring. If you read this out in the fluent and natural way, one gets "$2x$ is equal to $10$ which is the same as $x$", and any student would (hopefully) be able to catch the error here.

This verbalisation makes the most sense when talking through the process of solving a problem or going through a derivation, of course. It would be unnatural and unnecessary to use this sort of thing in a problem statement by saying things like "the number of real roots of the polynomial $p(x)$ is nothing but $0$". It is also probably unnatural when discussing a long chain of equal terms without any logical relation, e.g. "twice of the number of apples that Alan has is the same as thrice the number of apples that Bob has which is equal to seven times the number of apples David has" could be confusing; in this case I would go with Jasper's suggestion. Else, in the case where there is clear logical connection between terms in the equality, varying the expression from just "equals" can in my experience be very helpful.

Answer 4

I fully agree with the self-accepted Answer, and am adding an orthogonal point:

in mathematics, $$p \Leftrightarrow q \Leftrightarrow r$$ is commonly understood as $$(p \Leftrightarrow q) \;\text{ and }\; (q \Leftrightarrow r),$$ while in formal logic, $$p \leftrightarrow q \leftrightarrow r\tag{*}$$ is commonly understood as $$p \leftrightarrow (q \leftrightarrow r),\tag1$$ which is not logically equivalent to $$(p \leftrightarrow q) \;\text{ and }\; (q \leftrightarrow r)\tag2.$$

(Notice that for the assignment $(p,q,r)=$ (false, false, true), $(1)$ is true but $(2)$ is false.)

So, in fact, for $(*),$ the purely symbolic (albeit grammatically unfortunate) verbalisation “$p$ iff $q$ iff $r$”, is actually is more accurate than the grammatically-correct “$p$ iff $q,$ and $q$ iff $r$”, as the latter is a misinterpretation!

Here, any grammatically-correct verbalisation necessarily involves some interpretation; in fact, since $(*)$ is in some quarters understood as $(p \leftrightarrow q) \leftrightarrow r\tag3,$ disambiguation is even necessary in the proess of avoiding a purely-symbolic reading.

Answer 5

I think how you read (and write) them depends on how they are being used. $$ x^4 - y^4 = (x^2- y^2)(x^2 + y^2) = (x - y )(x+y)(x^2 + y^2) $$ is fine but pedagogically $$ \begin{align} x^4 - y^4 &= (x^2- y^2)(x^2 + y^2)\\ &= (x - y )(x+y)(x^2 + y^2) \end{align} $$ is better.

More important is dealing with this common misuse of chained equals when solving equations: $$ 2x+6 = 10 = 2x = 4 = x = 2 $$ where every other equals sign is really a logical implication:

If

$$ 2x+6 = 10 $$ then $$ 2x = 4 $$ and so $$ x = 2 . $$

That common error suggests avoiding chained equalities even when they are correct.