Proper Usage of Less Than and Greater Than Symbols in LaTeX Math Expressions

As an indispensable document preparation system for science and technology fields, LaTeX provides robust tools for typesetting complex mathematical expressions. One of the most ubiquitous elements in any math or computing context is the set of inequality signs denoting less than (<) and greater than (>). Correctly formatting these relation symbols in LaTeX requires awareness of the subtle syntax rules involved.

This comprehensive guide covers proper usage of less than and greater than operators in LaTeX math mode, including less than or equal (≤), greater than or equal (≥), along with troubleshooting common issues that arise. Best practices are provided for both inline and display style formulas to ensure symbols are spaced and rendered appropriately by the LaTeX compiler.

LaTeX Math Mode and Plain Text Pitfalls

LaTeX encloses mathematical expressions within specifically delimited regions called math mode. This switches the LaTeX interpreter from regular text processing rules into special mathematical typesetting logic that handles superscripts, subscripts, Greek letters, and all technical symbols appropriately in a document.

Without math mode delimiters, LaTeX does not recognize any math-related syntax like:

• Symbols: x, y, +, – etc.
• Relations: <, >, =, ≤, ≥ etc.
• Markup: superscripts, subscripts, fractions, etc

In fact, the plain less than < and greater than > characters have special meaning outside of math mode:

• < – Begin text formatting mode
• > – End text formatting mode

If you simply type x < y with no other delimiters, LaTeX will interpret the < as an unmatched text format start, causing an error:

\begin{center}
\includegraphics[width=0.8\linewidth]{less-than-error}
\end{center}

To instruct LaTeX to properly handle inequality relations, we must enclose them within math mode blocks, using either:

• Inline math: Surround with single $ delimiters
• Display math: Surround with double $$ delimiters (or \begin{equation}, \end{equation})

With math mode activated, LaTeX will correctly set spacing, kerning, and semantic meaning for the less/greater than and related operators.

Inline Math Mode Usage

For simple inline mathematical statements within paragraphs, single $ dollar signs are used. For example:

Let $x$ and $y$ be integers with $x < y$.

Renders as:

Let $x$ and $y$ be integers with $x < y$.

Note that within math mode blocks, the plain < and > work correctly as LaTeX has switched into its mathematical typesetting rules instead of applying text mode special meaning.

When using less than and greater than in inline math regions, LaTeX handles spacing and kerning automatically to ensure neat, professional appearance aligned to the current text line height.

Use cases: Inline math mode is great for short statements or asides within sentences or summaries where readability is paramount:

• Inline math blends well into sentences without causing disruption
• Easy way to denote relations between variables, numbers, etc
• Convenient for adding statistic figures ($65\%$), research quantities ($N = 30$)

For more complex multiline expressions, display math style is preferred.

Display Math Mode Syntax

To typeset important equation blocks that require visual prominence from body text, LaTeX showcases a display math mode. This is delimited using either:

• Double dollar signs: $$...$$

• Begin/end tags:

    \begin{equation}
    ...
    \end{equation}

• Begin/end tags with optional equation number:

    \begin{equation} 
       ...
    \end{equation}

For example:

$$
\begin{aligned}
x &< y \\  
y &> x
\end{aligned}
$$

Renders as:

$$
\begin{aligned}
x &< y \
y &> x
\end{aligned}
$$

Note the use of the alignment environment inside the display math zone to align the relation statements cleanly via the ampersand (&) symbols.

Display style formulas are necessary for mathematical expressions:

• Spanning multiple lines
• Containing complex line breaks
• Needing visual weight beyond inline usage

Scientific reports, statistics textbooks, academic papers, and computing software manuals frequently rely on properly formatted display formulas.

Use Cases:

Any mathematical concept requiring detailed explanation benefits from display mode visibility:

• Theorem statements
• Matrix transformations
• Multi-line equations
• Algorithm steps overview
• Statistic visualizations

By calling attention to the critical formulas in this way, LaTeX produces enhanced technical documents.

Less Than Or Equal and Greater Than Or Equal

The plain less than (<) and greater than (>) relations form only part of the common inequality expressions needed for mathematics and computer science. Just as frequently used are the compound relations:

• Less than or equal to (≤)
• Greater than or equal to (≥)

LaTeX provides dedicated math mode commands for typesetting these compound symbols properly:

• \le – ≤ (Less than or equal to)
• \ge – ≥ (Greater than or equal to)

For example:

$0 \le x \le 10$

Outputs:

$0 \le x \le 10$

And in display style:

\[
x \ge 0 \\
y \le 100
\]

Outputs:

[
x \ge 0 \
y \le 100
]

Note when using these compound symbols in inline math mode, manual spacing tuning with \ thin spaces may be required to prevent variables like x and y from colliding with the larger multi-character relations. LaTeX does not automatically adjust spacing for \le and \ge commands within inline usage.

Therefore:

Good: $x \le 100$

Bad: $x\le 100$

The thin space fixes collision.

Semantic Meaning and Math Style Consistency

From a semantic standpoint, LaTeX renders \lt and \gt as relation symbols carrying mathematical meaning associated with ordering and analysis. Using the plain text </> in math mode works syntactically but may not always carry the same semantic association depending on the context.

Similarly, \le and \ge convey discrete conceptual meaning denoting "less than or equal to" and "greater than or equal to" for logical comparison.

Beyond visual consistency, using the dedicated LaTeX relation commands where possible improves semantic clarity. For technical writing, choosing strict semantic accuracy over mere syntax typically results in higher quality documents.

In terms of style conventions, using \lt \gt \le and \ge shows deeper LaTeX proficiency compared to relying only on text-based symbols. Just like preferring \times over plain x or \infty over oo, invoking LaTeX‘s suite of math operators signals formula fluency.

Troubleshooting Common Symbol Issues

While LaTeX handles most math typesetting automatically, inequality relations can demonstrate pitfalls for novice or casual users.

1. Unescaped Plain Text Symbols

As covered earlier, plain < or > characters used outside math mode will trigger LaTeX processor errors due to ambiguity with text-formatting tags.

Solution: Enclose all relation statements with either inline or display math mode via $ $ or $$ $$ delimiter symbols.

2. Inconsistent Math Mode Blocks

Even if starting with correct math mode delimiters, equations can still fail if intermediate blocks are unbalanced.

Bad Inline Math Block:

$x < y 
y $> x$

Fails because LaTeX expects math mode termination after y before starting fresh inline block on following line.

Solution: Audit math mode usage to ensure every opening delimiter ($) matches the proper closing ($). Check for unescaped characters breaking open existing blocks.

3. Space Omission Collisions

As seen earlier with \le and \ge, compact variable names adjacent to large compound symbols may collide in dense inline math expressions unless spacing is manually inserted.

Solution: For inline math with ##### collisions, add \ thin space tokens between variables and relations.

Good Spacing: $x \le y$

Poor Spacing: $x\le y$

This improves readability at the micro typography level.

4. Mixing Plain Text and Symbol Commands

When first learning LaTeX math, new users may try using a hybrid approach of mixing text greater than > and less than < characters alongside \lt, \gt commands and custom spacing which can fail or look messy depending on factors like theme, compiler, etc.

Solution: Standardize on either text-based inequality signs or formal LaTeX relational commands like \le, \ge for consistency across expressions. Mixing notation introduces fragility.

5. Overnesting Math Mode Blocks

Nesting inline math ($...$) within display math ($$) may complicate spacing, alignment, and render malformed expressions:

$$
$x < y$
$$ 

Solution: Avoid nested math blocks spanning modes. Separate inline statements from display layouts into their own blocks for cleanliness:

Inline math: $x > y$

Display math:
$$ 
\begin{aligned}
   x &< y
\end{aligned}
$$

Adhering to best practices for math block delimiting prevents chaotic nested expressions.

Comparison to Markdown Math Engines

As a pioneer formal document processing system, LaTeX establishes conventions that downstream formats like Markdown continue to follow. Even modern math typesetting engines in Markdown such as MathJax and KaTeX adopt LaTeX practices regarding math delimiters and inequality symbol handling.

For example, MathJax standardizes on \\(, \\) inline math notation along with \\[,\\] for display equations. KaTeX implements $ inline and $$ display delimiters by default.

Both MathJax and KaTeX render plain < > correctly within math blocks while applying special syntax consideration outside delimited regions just like LaTeX. This maintains broad compatibility with decades of existing mathematical writing reliant on LaTeX symbol semantics.

Thus while Markdown allows more human-readable source formatting, LaTeX principles permeate technical typesetting across preparing complex documents. Learning LaTeX math foundations transfers directly to scientific writing in many other formats.

Emergence of LaTeX Preprocessors

To boost author productivity given LaTeX‘s syntactic density, various preprocessing tools have emerged like Literate that allow writing more natural Markdown/Python syntax that compiles into LaTeX.

Here is an example Markdown math statement that Literate converts into valid LaTeX:

Math expressions in {math} delimiters compile to proper LaTeX syntax with {math}x < y{/math}. No need to directly encode LaTeX math mode markers.

This shorthand approach lowers the barrier for accurately authoring math-heavy content while harnessing LaTeX for high-quality composition.

As LaTeX usage continues growing yearly according to arXiv.org traffic analysis, expect more intelligent preprocessors to bridge the readability gap without losing semantic rigor.

Best Practices for Production LaTeX Documents

Based on the guidelines presented, here are best practices for working with less/greater than symbols in professional LaTeX outputs:

• Delimit all math expressions/inequalities via inline or display math modes
• Standardize on \lt, \gt for semantic meaning beyond plain <, > text
• Include manual spacing around \le, \ge in dense inline statements
• Proofread for colliding symbols requiring negative space
• Validate all math mode blocks have matched open/close delimiters
• Avoid nested inline/display math mixtures
• Consider LaTeX preprocessing if managing complex documents
• Use macros/shortcuts for repetitive statements
• Consult latexerrors.com for troubleshooting help

Following these recommendations when authoring LaTeX documentation will ensure correctly rendered inequality relations as well as beautifully typeset formulas across technical writing needs ranging from academic journals to software manuals.

Conclusion

Mastering LaTeX‘s math mode syntax provides authors, scientists, and engineers with excellent control over typesetting complex mathematical ideas involving inequality relations. While the < and > operators function correctly within math mode blocks, the semantically richer \lt, \gt commands add deeper meaning compared to plain text. Similarly, \le and \ge convey "less/greater than or equal to" with proper spacing adjustments.

By understanding LaTeX‘s capabilities along with common symbol pitfalls, technical writers can produce precise formulas as needed for research reports, statistics textbooks, computer science notes, and any mathematical discourse. Even as easier preprocessing methods emerge, LaTeX‘s enduring standards for math layout and semantics will continue guiding technical document preparation for decades more.