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I have a long inequality in a single line

$(1+a)^{k+1}=(1+a)^k(1+a)\ge(1+ka)(1+a)=1+(k+1)a+ka^2\ge 1+(k+1)a$

But I find it difficult to show when the width of my paper is short. What can I do to make the lines readable for short width papers.

Veak
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  • https://latex-tutorial.com/align-equations/ – angryavian Aug 09 '23 at 01:06
  • I do not mean how to align in latex, but to align for readability of proof. Otherwise I start wondering where the equality or inequality applies to which expression. – Veak Aug 09 '23 at 01:14
  • For instance, if I align $=1+(k+1)a+ka^2$ with the first equal sign, one might think that the first expression is equal to the fourth expression, which they are not. – Veak Aug 09 '23 at 01:19
  • @Rrinpoche Even on wide paper your chain of equations and inequalities is a bit hard to read. A typical style is that you align every sign with the first sign so that it is relatively easy to spot the $\geq$ signs. If the $=$ signs are confusing, maybe you don't need so many equations: $(1+a)^{k+1}\ge(1+ka)(1+a)\ge 1+(k+1)a$ seems clear enough to me. – David K Aug 09 '23 at 03:25
  • You can also break your chain into pieces interspersed with a few words so that you don't have so much alternation of equality and inequality. – David K Aug 09 '23 at 03:26
  • Right. You decided to keep just the inequalities. That makes things easier as you say. And intersperse with some words is also of same value. I find the usual presentation of mathematics hard to read (not like a novel). I am trying to see whether I can change the presentation so that for the most part I do not have to write anything on paper, but just follow the document by looking at the symbols only. – Veak Aug 09 '23 at 04:01
  • I thought the question was about writing math, not reading it. This question gets less and less clear with each comment. But I think removing the words from a math paper and using only symbols would make it much harder to read in most cases. (There are some exceptions for very short works called "proofs without words", but I've never seen that style successfully used for anything longer.) – David K Aug 09 '23 at 06:57
  • It is about writing it so one can read it better. – Veak Aug 09 '23 at 07:49

1 Answers1

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It is common to chain equalities and inequalities together line by line, especially in analysis proofs. See for example this proof of the triangle inequality on Wikipedia:

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I think this usage is common enough that you don't need to worry about someone misinterpreting an inequality for equality.

For your example specifically, since you asked in the comments:

\begin{align} (1+a)^{k+1} &= (1+a)^k(1+a) \\ &\ge (1+ka)(1+a) \\ &= 1+(k+1)a + ka^2 \\ &\ge 1 + (k+1)a. \end{align} I think mathematicians would unequivocally interpret this as $(1+a)^{k+1} \ge 1+(k+1)a$, despite the equality signs in the intermediate steps. I can't speak for other fields, where perhaps other conventions might exist.

angryavian
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  • How would you do it for my example ? – Veak Aug 09 '23 at 02:31
  • I see that the last equality looks like it is equality with $||u+v||^2$. But it is not. I agree that the usage for triangle inequality ins not something to worry about, and generally people will understand it. But for complicated mathematical physics one is not very acquainted with, I am not so sure. – Veak Aug 09 '23 at 02:37
  • @Rrinpoche Updated my answer to show how I would write your example specifically. – angryavian Aug 09 '23 at 03:56
  • I'm not entirely happy about the word "interpret", but certainly anyone with any experience of inequalities should be able to deduce from the four steps given that $(1+a)^{k+1}\ge1+(k+1)a$, and that, surely, is the point. – David Aug 09 '23 at 04:34
  • I am saying something else though. Suppose you do not have experience of the material (and the expressions of the inequalities are unfamiliar). Would one interpret them correctly ? Or could it be ambiguous ? – Veak Aug 09 '23 at 04:57