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In arithmetic, I'm sure we've all used the informal notation where we write a list of numbers above each other with an operator, find the answer, then use that number and chain another operation, etc. For example:

$$\begin{array} {rr} & 123 \\ & 157 \\ + & 34 \\ \hline & 314 \\ \times & 2 \\ \hline & 628 \end{array}$$

Currently I'm in a logic-oriented calculus series, and the professor says anything that isn't pedantically correct, no matter how minor, is wrong, and points will be deducted. I know I can use $\Rightarrow$ and re-write the 314 in this example, but then the beneficial terseness of the equation chaining is lost. Is there a pedantically accurate way of chaining equations like this, especially for "sidebars"/scratch-work and substitutions?

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    Also, is there a way of removing the column space separations in MathJax? I'd normally use tabular for something like this, but MathJax doesn't seem to have an equivalent. – 2017-01-19
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    You could make it 2 columns and right align them. I'll edit the post to do that, you can revert if it isn't what you like. – 2017-01-19

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It's a universal convention that whenever we write "$a = b = c$" we mean "$a = b \land b = c$". Note that this applies for binary relations in general such as "$x \in S \subseteq T$" or "$p \le q < r$" or "$C_n \cong H \unlhd G$".

So it is perfectly valid (no sane professor should reject) to write:

$(123+157+34) \times 2 = 314 \times 2 = 628$.

Note that for equality or inequality chains we essentially never explicitly write down the easily deducible conclusion about the relation between the first and last in the chain, which in this case is:

$(123+157+34) \times 2 = 628$.

Either way, note that presentation of the computation as an equality chain can be easily made totally rigorous in a formal system, even though most programming languages do not have such constructs (because they are designed to capture not reasoning but processes).

Finally, note that it is actually logically incorrect to use the "$\Rightarrow$" symbol in-between statements each of which follows from the previous one. "$A \Rightarrow B$" has a very precise meaning.

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    And note that this convention means that "$1 < 3 > 2$" is true and does not make any claim about the relation between $1$ and $2$. – 2017-01-19
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    My concern was in cases where, let's say, I wouldn't know that `123 + 157 + 34 = 314` until I've done the vertical arithmetic. Isn't the vertical arithmetic chaining really saying `[123 + 157 + 34 == 314] \times 2 == 628`? And, if left as that, wouldn't it be a type error `(\times) Bool Num`? – 2017-01-20
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    @BrainFRZ: It would indeed be a type error if your language does not support chaining. But we can specify the language to allow it. Precisely, given valid expressions $x,y,z$ we can allow "$x = y = z$" to mean "$x = y$ and $y = z$", but "$(x = y) = z$" to mean that the truth value of $(x=y)$ is the same as $z$. In mathematics we seem to have done exactly this because it's very troublesome to repeat ourselves in the case of relation chains. In natural language we also have this feature in the form of unrestrictive relative clauses. – 2017-01-20
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    Thanks, that makes a lot of sense! Just to clarify on your comment on my use of logical implication, I was meaning something like $\int \pi sin(\pi \theta) d \theta, u = \pi \theta \Rightarrow du = \pi d \theta \Rightarrow \int sin(u) du = -cos(\pi \theta) + C$. Are you saying this would be an incorrect usage? – 2017-01-20
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    @BrainFRZ: Yes this is incorrect. A lot of professors use it, but that does not make it correct. It is usually understood, but in logic it has a precise meaning contrary to the popular incorrect usage. In particular $( A \Rightarrow B \Rightarrow C )$ is at worst ambiguous if it refers to the material conditional and would be true if $C$ is true regardless of $A,B$, or at best referring to the logical implication relation, in which case it is misused in normal proofs because it is defined to be external to any proof. As for your example, it has many errors besides that. [continued] – 2017-01-20
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    [continued] "$du = π\ dθ$" is simply meaningless in normal real analysis; one cannot treat $du,dθ$ as real numbers. What you should write is: Let $u = πθ$. Then $\frac{du}{dθ} = π$ and hence $\int π \sin(πθ)\ dθ = \int \sin(u) \frac{du}{dθ}\ dθ = \int \sin(u)\ du = -\cos(u) + C$ for some constant $C$, where the middle equality is by the substitution theorem. – 2017-01-20
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A chain of identities of equations such as $E1=E2=E3=E4$ translates to the following sequence of implications in logic:

  1. $E1=E2$

  2. $E2=E3$

  3. $E1=E3$ from 1 and 2 (as a formal inference, this is typically called $= Elim$)

  4. $E3=E4$

  5. $E1=E4$ ( $= Elim$ on 3 and 4)

So, for your example, we would get:

  1. $123+157+34=314$

  2. $2*314 = 628$

  3. $\therefore 2*(123+157+34)=628$ ($= Elim 1,2$)

Is that the kind of thing you are looking for?

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    I was really hoping to avoid having to re-write results. It just dawned on me that what I'm trying to do is very similar to `*=` (in this example) in some programming languages, which allows you to multiply and reassign the state of a variable. I realize state doesn't make any sense in maths, but that's basically what vertical arithmetic seems to be doing. I was really hoping there was some notational shortcut that I could use. It's possible I'm trying to mix precision and terseness in a way that doesn't work and I just have to deal with it. :) – 2017-01-19
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    Even in a programming language you can't write "if (a == b == c)" and expect the compiler to translate that into "if (a == b && b == c)" . However, even in (most) logics, you are free to give a name to an expression and use that name instead of the expression for brevity. Check with your prof. – 2017-01-19