## a mathematical sequence whose verb is equal

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## The Grammar According to West

### by Douglas B. West

## Summary

I have been accumulating observations about writing mathematics for many years.

These conclusions arose both from writing textbooks and from noting writing errors commonly made by my thesis students and in papers submitted to journals.

My first objective was to train my students, thereby reducing the time needed to edit their theses.

As the document grew, I made it publicly available in the hope that others may find it useful. I have received a broad range of responses, mostly positive.

If you don’t find it useful (or if you object to it on principle), then please ignore it. I hope to make some writers of mathematics

(especially students and non-native speakers of English) aware of issues they may not have considered;

small changes can produce mathematical writing that is easier to read by wider audiences.

### why care in writing mathematics is needed

After an introductory explanation of why care in writing mathematics is needed, I discuss (1) mathematical style, (2) notation and terminology, (3) punctuation and English grammar as used in mathematical writing, and (4) English usage for non-native speakers.

Some points are minor distinctions, but even these make mathematical writing clearer when used consistently.

My intent is not to make writing rigid, but rather to make it transparent to avoid distracting the reader by ambiguities or awkwardness in the flow of the narrative.

I note that a number of other articles and even books have been published on writing mathematics;

see some of these here.

I list these just to make alternative viewpoints available, without judgment pro or con. One contrast to note is that my list below focuses on relatively specific items.

I have not yet included much discussion of the general structure of an article; some of these references discuss that.

## Introduction and Motivation

Live mathematical conversations use many shortcuts that are inappropriate in precise mathematical writing. The context is known by all participants, and shortcuts evolve to save time. Furthermore, the speaker can immediately clarify ambiguity.

Without immediate access to the author, written mathematics must use language more carefully.

Also, mathematical concepts are abstract, without context from everyday experience, so the writing must be more consistent to make the meaning clear.

Outside mathematics, imprecise writing can still be understood because the objects and concepts discussed are familiar.

Some mathematicians object to some of my recommendations. Many time-honored practices in the writing of mathematics are grammatically incorrect.

These mistakes in writing cause no difficulty for readers with sufficient mathematical sophistication or familiarity with the subject,

but it is unnecessary to restrict the audience to such readers. A bit of care leads to clearer writing that makes mathematics more easily accessible .

and readable to a wider and less specialized audience.

### usage or grammar

Some languages have conventions of usage or grammar that lead to typical errors in English mathematical writing by their native speakers.

### Habits die hard

I apologize in advance for my own grammatical errors. Habits die hard, and it is easy to err in applying principles of writing.

In particular, there are inconsistencies between what I propose here and what I wrote in my earlier books. Those books were written in the previous millennium.

and I have learned many things about writing since then.

Also, I am a speaker of American English, and some points are consistently different in British English (such as the treatment of “which” vs. “that” and the aversion to serial commas).

Some of my conclusions conflict with manuals of English style.

My conclusions are intended to produce clear mathematical writing that is more logically consistent than publishers’ conventions.

This applies especially to punctuation and to words that serve as logical connectives.

I welcome corrections, suggestions/inquiries, and “pet peeves” that may lead to further items in later versions of this guide.

### Mathematical style

*Abstract, Introduction, and Conclusion.*

We begin with the overall structure of a research article in mathematics.

The abstract states the results as fully as possible in a brief presentation.

Crucial specialized terms the reader needs to know to understand the statements should be defined.

The abstract stands on its own, especially in the age of electronic communication where it may be separate from the rest of the paper.

and hence it contains no numbered reference to the bibliography.

The first section of the paper is an “Introduction” that should motivate the problem.

discuss related results, state the results more completely, and perhaps summarize the techniques or the structure of the paper or crucial definitions.

### concluding remarks

The introduction should also contain any concluding remarks or key conjectures.

There is generally little or no value in a separate section of concluding remarks. Such remarks either are redundant or contain information that readers usually seek in the introduction.

Readers who study the full details of the proofs have no need of a summary in a concluding section. Readers who do not read the full details have no desire to go on to a concluding section.

A mathematical research article is not read like a novel or even like an essay that seeks to “persuade” the reader; it does not need an epilogue.

### Terminology and notation (especially in discrete mathematics)

*Definition symbol* “:=”. Some mathematicians use this symbol to indicate that the preceding symbol is being defined to mean the subsequent object. If this occurs in a sentence like “Let *[n]:={*

*1,…,n}*“, then the verb states that the notation is being defined, and the special notation is unnecessary. If it occurs in a sentence about the object being defined.

“Consider a coloring of *[n]:={1,…,n}*“, then it is an improper Double-Duty Definition and should be rewritten: “Consider a coloring of *[n]*, where *[n]={1,…,n}*.”

Reading “:=” requires thinking “be defined to be” when preceded by “let”,

and it requires even more convoluted phrases when placed in a Double-Duty Definition. This awkward notation is never needed and encourages grammatical errors.

## English usage in mathematical writing

*Introductory words.* Words or phrases like “nevertheless”, “for example”, “to the contrary”, and “on the other hand” usually should be separated by commas from the rest of the sentence.

Introductory prepositional phrases are a bit different. I am told that a phrase with one preposition (“In 1995”) does not require a comma.

but a phrase with two prepositions (“In August of 1995”) does. Another reader tells me that an introductory phrase with at least five words.

(perhaps we should say five syllables) should be followed by a comma. I would use the comma unless the intent is to lead into what follows as a single thought (see Hence/Thus/Therefore).