Checkpoint 17: Mathematical Expressions

Mathematical expressions must be properly tagged with Formula tags and made accessible through alternative text or Unicode character mappings. This ensures that users of assistive technology can understand mathematical content.

What This Means

Mathematical expressions in PDFs present unique accessibility challenges. Unlike regular text, math uses:

• Special symbols and operators
• Spatial arrangements (fractions, exponents, roots)
• Greek letters and specialized notation
• Complex structures (matrices, integrals, summations)

PDF/UA requires that mathematical content:

1. Be tagged with the Formula tag: Identifies content as mathematical
2. Have alternative text (Alt attribute): Provides a text description, OR
3. Use proper Unicode mappings: Characters map to meaningful Unicode values

The Formula tag tells assistive technology that the content is mathematical, enabling specialized handling. The alternative text or Unicode mappings provide the actual accessible representation.

Why It Matters

Mathematics is central to education, science, engineering, finance, and many other fields. Users who cannot access mathematical content are excluded from:

• Academic studies and research
• Technical documentation
• Scientific publications
• Financial reports and analyses
• Engineering specifications

Screen readers face significant challenges with math:

• Symbols may be announced incorrectly or not at all
• Spatial relationships (numerator/denominator) are not conveyed
• Complex expressions become incomprehensible strings of characters
• Standard reading mode cannot interpret mathematical structure

With proper tagging and alternatives:

• Screen readers can announce "x squared plus y squared equals r squared"
• Users understand the mathematical meaning, not just character sequences
• Complex expressions can be navigated and understood
• Mathematical content is searchable and processable

Common Violations

The Matterhorn Protocol defines three failure conditions for mathematical expressions.

17-001: Mathematical Expression Not Tagged with Formula (Human Testing)

What's Wrong: Content that is a mathematical expression is not tagged with a Formula tag. It may be untagged, tagged as a figure, tagged as text, or have another inappropriate tag type.

How to Identify:

• Look for mathematical content in the document:
  • Equations and formulas
  • Mathematical symbols and operators
  • Variables with subscripts or superscripts
  • Fractions, roots, integrals
• Check the Tags panel for how this content is tagged
• Mathematical content should use the Formula tag

Common Scenarios:

• Equations tagged as Figures without proper Formula identification
• Inline math (like "x = 5") tagged as regular text
• Complex expressions tagged as images
• Math created as images without any structure tags

17-002: Formula Tag Missing Alt Attribute (Machine Testable)

What's Wrong: A Formula tag exists but does not include an Alt (alternative text) attribute. Without alt text, screen readers cannot provide a meaningful description of the mathematical expression.

How to Identify:

• PDF/UA validators will detect this automatically
• In Acrobat, find Formula tags in the Tags panel
• Check Properties for the Alt attribute
• Missing or empty Alt attribute triggers this violation

Note: This violation can be avoided if the Formula content uses proper Unicode mappings (see 17-003). However, Alt text is the most reliable method for complex expressions.

17-003: Unicode Mapping Requirements Not Met (Machine Testable)

What's Wrong: The characters within the Formula do not map to appropriate Unicode values. This is an alternative to providing Alt text; if Unicode is used, each character must map to a meaningful Unicode code point.

How to Identify:

• PDF/UA validators check Unicode mappings
• Copy mathematical text and paste; check if meaningful characters appear
• Look for undefined characters or private use area mappings
• Verify that symbols map to correct Unicode math characters

Technical Context: PDF fonts can use custom encodings. For accessibility, each glyph must map to standard Unicode. Mathematical symbols should map to Unicode math blocks:

• Mathematical Operators (U+2200-U+22FF)
• Supplemental Mathematical Operators (U+2A00-U+2AFF)
• Mathematical Alphanumeric Symbols (U+1D400-U+1D7FF)

How to Fix in Adobe Acrobat

Identifying Mathematical Content

1. Visually scan the document for math:
   • Equations (displayed on their own line)
   • Inline expressions (within text)
   • Variables, symbols, operators
   • Fractions, roots, matrices
2. Open the Tags panel and locate corresponding content
3. Note what tag type is currently used

Creating Formula Tags

1. In the Tags panel, find the mathematical content
2. If not tagged as Formula:
   • Right-click on the tag > Properties
   • Change Type to Formula
3. If math is untagged:
   • Go to Tools > Accessibility > Reading Order
   • Draw around the math content
   • Click Formula button

Adding Alternative Text

1. In the Tags panel, click on the Formula tag
2. Right-click > Properties
3. Click the Tag tab
4. In the Alternate Text field, enter a text description
5. Click Close

Writing Good Math Alt Text:

Handling Different Math Formats

Math as Images:

1. If math is an image, it needs the Figure tag with alt text
2. Or convert/recreate as actual Formula content
3. Add comprehensive alt text describing the math

Math from MathML:

1. Some PDFs include MathML (structured math markup)
2. Ensure MathML is associated with the Formula tag
3. Screen readers may be able to read MathML directly

Math from LaTeX-generated PDFs:

1. LaTeX with proper packages creates accessible math
2. Verify Formula tags are generated
3. Add alt text if not automatically included

Checking Unicode Mappings

1. Select and copy mathematical text from the PDF
2. Paste into a text editor or Unicode analyzer
3. Verify characters are standard Unicode, not custom glyphs
4. If mappings are wrong, you may need to recreate the PDF

Using Touch Up Reading Order for Math

1. Go to Tools > Accessibility > Reading Order
2. Select the math region
3. Click Formula to tag as formula
4. After tagging, add alt text through Tags panel

How to Fix in Source Documents

Microsoft Word

Using the Equation Editor:

1. Click Insert > Equation
2. Build equations using the Equation toolbar
3. Word's equation editor creates structured math
4. Add alt text:
   • Click on the equation
   • Right-click > Edit Alt Text
   • Enter a description

Best Practices:

• Always use the Equation Editor, never type math as plain text
• Use built-in templates for fractions, radicals, etc.
• Add alt text before exporting to PDF
• Test PDF export to verify Formula tags are created

LaTeX

Accessible LaTeX packages:

• Use pdfx package for PDF/UA-compliant output
• Use axessibility package for auto-generated alt text
• Use accessibility package for tagging support

Example:

\usepackage{pdfx}
\usepackage{axessibility}

\begin{document}
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
\end{document}

Manual alt text:

\pdftooltip{$E = mc^2$}{E equals m c squared}

Adobe InDesign

1. Use native InDesign math features or MathML import
2. After creating/importing math:
   • Select the math object
   • Go to Object > Object Export Options
   • In Alt Text tab, enter description
3. Export to PDF with accessibility settings enabled

MathType

MathType (standalone or Word add-in):

1. Equations created in MathType can include accessibility data
2. Use MathType's accessibility features
3. Export settings should preserve structure
4. Add alt text within MathType before export

Writing Effective Math Alt Text

General Guidelines

1. Be complete: Include all parts of the expression
2. Be unambiguous: Use words that have clear meaning
3. Follow conventions: Use standard mathematical English
4. Consider context: Provide enough context for understanding
5. Test with screen reader: Verify it makes sense when read aloud

Common Patterns

Operations:

• "plus", "minus", "times" (or "multiplied by"), "divided by"
• "equals", "not equal to", "greater than", "less than"

Powers and Roots:

• "squared", "cubed", "to the power of n"
• "square root of", "cube root of", "nth root of"

Fractions:

• "x over y" or "x divided by y"
• "the fraction with numerator x and denominator y"

Grouping:

• "the quantity", "end quantity" for parenthetical groups
• "open parenthesis... close parenthesis"

Subscripts and Superscripts:

• "x sub i", "x subscript i"
• "x to the power of n", "x superscript n"

Complex Expression Example

Expression: ∫₀^∞ e^(-x²) dx = √π/2

Alt text: "The integral from 0 to infinity of e to the power of negative x squared dx equals the square root of pi divided by 2"

Or more technically: "The definite integral from zero to infinity of e to the negative x squared with respect to x equals square root pi over 2"

Testing Your Fix

Automated Testing

PAC (PDF Accessibility Checker):

1. Run PDF/UA validation
2. Check for Formula tag issues
3. Review alt text warnings
4. Check Unicode mapping errors

Adobe Acrobat:

1. Go to Tools > Accessibility > Accessibility Check
2. Run full check with PDF/UA settings
3. Look for figure/formula alt text issues

Manual Verification

1. Find all mathematical content visually
2. For each piece of math:
   • Verify it is tagged as Formula
   • Check that Alt text exists and is meaningful
   • Or verify Unicode mappings are correct
3. Document any gaps

Screen Reader Testing

1. Open PDF with NVDA, JAWS, or VoiceOver
2. Navigate to mathematical content
3. Listen to how the math is announced
4. Verify:
   • All parts of the expression are read
   • The reading is comprehensible
   • Mathematical meaning is conveyed
   • Complex expressions are understandable

Copy-Paste Test

1. Select mathematical content
2. Copy and paste into a text editor
3. If alt text is present, you may see it
4. If Unicode is used, verify meaningful characters appear
5. Garbled or empty results indicate accessibility problems

Validation Checklist

• All mathematical expressions are tagged as Formula
• Each Formula tag has Alt text OR proper Unicode mappings
• Alt text accurately describes the mathematical expression
• Complex expressions are described completely
• Screen readers announce math understandably
• Mathematical meaning is preserved in accessible form
• Inline math and display math are both addressed

Special Cases

Inline Math

Simple variables or expressions within text:

• May use Formula tag with alt text
• Or ensure proper Unicode for the characters
• Example: "The value of x is 5" needs Formula tag for "x" if styled as math

Display Equations

Centered equations on their own line:

• Should definitely use Formula tag
• Alt text should describe complete expression
• Consider equation numbers in alt text if present

Equation Numbers

If equations are numbered (1), (2), etc.:

• Include the number in the accessible content
• Alt text might be "Equation 1: x equals y plus z"
• Or tag the number separately

Chemical Formulas

Chemical notation (H₂O, CO₂):

• May use Formula tag or appropriate science notation
• Alt text: "H 2 O" (water) or "carbon dioxide C O 2"
• Consider whether chemical name helps understanding

Additional Resources

Official Standards and Guidelines

• W3C WCAG 2.1 Success Criterion 1.1.1: Non-text Content
• W3C WCAG 2.1 Success Criterion 1.3.1: Info and Relationships
• PDF Association Matterhorn Protocol 1.02

Math Accessibility

• W3C Math Accessibility
• MathML Accessibility
• Benetech DIAGRAM Center: Math Accessibility

Screen Readers and Math

• NVDA MathCAT Add-on
• JAWS Math Support
• VoiceOver and Math (macOS/iOS)

Tools

• MathType - Equation editor with accessibility features
• LaTeX Project - Document preparation system
• MathJax - Math rendering (for web, can inform PDF practices)

This documentation is based on the Matterhorn Protocol 1.02, the definitive reference for PDF/UA validation. One math violation requires human testing (identifying what is math); two are machine testable (missing alt text and Unicode mapping issues). For the most current information, consult the PDF Association and W3C WCAG guidelines.