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[a11y TF] ambiguous notation

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Peter Krautzberger edited this page Feb 8, 2019 · 42 revisions

As per minutes to the TF meeting on 2018-08-28, the TF wants to gather examples of ambiguous notation.

although a definition cannot be false it may be improper; and the impropriety may arise either from its inducing ambiguity or from its offending against received principles.

- Charles Babbage in The Edinburgh Encyclopedia, Vol. 14, 1832

Fraction-like notations

Fraction-like notation look similar to a vertically stacked fraction notation. They are usually created in equation authoring systems using constructs based off of fraction. A typical example is MathML's mfrac element and TeX's \over, although tables might be used for stacking the elements in some of the notations shown.

Fraction

Example: x + y over 2

The numerator and denominator can be any expression and can nest.

Note: fractions might be spoken in many different ways, particularly for numeric fractions such as one half. Another special case is rates, such as kilometers per second ("kilometers per second").

Continued fraction

Example: continued fraction a_0 + fraction with numerator 1 and denominator a_1 + fraction ... till a_n in denominator

Technically, this is very similar to the meaning for a fraction but the diagonal ellipsis is what makes it technically not a regular fraction.

There are a number of alternative notations as well.

Binomial coefficient

Example: binomial coef image of n choose k

The top and bottom can be any expression and can nest, but nesting is very rare. Typically, they are relatively simple expressions.

These typically are generated as either a fraction with a zero-width fraction line enclosed in parens that stretch or as a 2x1 table/ 2 column vector (enclosed in parens that stretch)

Note that it appears that Euler denoted coefficients of a binomial with parenthesis and a fraction line, cf. Babbage in The Edinburgh Encyclopedia, Vol. 14, i.e., like the Legendre Symbol.

Legendre symbol

Example: legengdre symbol a over p -- fraction notation enclosed in parens

The denominator should be a prime, which means typically it is either a prime integer or a variable (possibly subscripted) that represents a prime. Potentially, it could be an expression, but that would be very rare.

Note that because the Legendre symbol is an expression, it can be raised to a power as shown in the Jacobi symbol example below.

Jacobi symbol

Example: Jacobi symbol a over n defined in terms of Legendre symbols factorization of n

The example includes Legendre symbols on the right, which represents the prime factorization of n. Hence, knowing which is the Jacobi symbol and which is the Legendre symbol requires knowing with a number is prime and/or with a symbol represents a prime.

Derivative

Examples:

  • dx by dy,
  • d squared x by d y squared equals d by dy of d y by d x,
  • partial y by partial x,
  • i+j+k th partial with respect to x to the i, y to the j, z to the k followed by other notations for the same partial derivative

Total and partial derivatives can be written in several different notations. The Leibniz notation uses the fraction notation shown above. The function/variable being differentiated can be in the numerator or can appear after the fraction (second example). Higher order derivatives use superscripts on the 'd'/'∂' and on the variables of differentiation (second and fourth examples).

Because the differential d operator can be confused with a variable named d, the Unicode value U+2146 (ⅆ) might be used instead of d.

2-Dimensional vector notation

Example: 2 by 1 column vector

2 row vectors/2x1 matrices may use parens or brackets to surround the fraction like notation. As noted above, with parens, these are identical notation to the binomial notation.

Logical deduction

Example: inference rule for A and B

More examples and an explanation of this notation: Wikipedia

Roots vs radical sign

Authors usually do not differentiate between square roots, generic roots and the radical sign. Usually, they will use notation for square roots, e.g., check the notation used near "radical sign" at https://en.wikipedia.org/wiki/Nth_root.

Implicit operations

Equational content often omits implict operations which are supposed to be derived from the context. While Unicode provides such tools as invisible times or invisible function operation, these are often missing in real life content.

  • multiplication
    • e.g., ab + c vs id + c
  • function application
    • e.g., (f+g)(x) vs (f+g)(h)
  • mixed fractions
    • 4(1/2), 4+1/2 vs 4·(1/2)

Variables

Even when dealing with variables some ambiguities exist:

  • Measure units can be easily confused by variables. km is kilometer or "k times m"
  • φ is the letter "phi" or the constant "Euler phi"
  • Multiple letter variables: velocity is "velocity" or "v times e times t times o ..." This issue appears both in poorly written LaTeX or MathML.

Decimal numbers

  • 4,5 decimal point in French, Spanish and other Romanic cultures.
  • 4E10 scientific notation

Precedence

  • sin x^2 it is not well defined whether it is sin(x^2) or (sin x)^2
  • sin^{-1} x is not well defined. It usually means asin(x) (arcsine, the inverse function) and is frequently confused with (sin x)^{-1} (the inverse of the value, cf. the reciprocal function cosecant usually denoted by csc x).
  • a∧b∨c it is not clear the order precedence. Usually ∧ has precedence over ∨, but not always.
  • Typically bracketing operators/fences have distinct open/close symbols, but | is an exception. That leads to ambiguities like |a|b|c| - is this (|a|) b (|c|) or |a (|b|) c|?
  • | also has issues with bra-ket notation: <ϕ|Ψ>, <ϕ|, |Ψ> in that | need to be a separator or fence.

Multiplication

  • ab
  • a*b
  • a x b
  • a(b)
  • a • b

less / greater signs and relationships

Various fields will interpret less and greater relationships differently. Some examples:

numbers, in particular decimial markers

Decimal markers vary greatly across the world, cf. https://en.wikipedia.org/wiki/Decimal_separator#Examples_of_use. Voicing numbers with decimal markers requires the context (i.e., 10 million one hundred and seventeen).

In some cases, it is not desired that AT users receive a simplified voicing of a number (e.g., in testing situations when learning number systems).

custom characters / glyphs

Equation layout can include custom characters that do not have a corresponding Unicode point; they can be represented as PUA codepoints in a custom font (cf. Wiley Chemistry fonts, St Mary Rd fonts), as an image (e.g., knots cf mglyph examples in MathML) or as a layout "hack" (e.g., \raise{2pt}{\rlap{\scriptstyle\nabla}}\nabla to create a faux blackboard bold nabla [found in a published paper]).

Such characters have no obvious meaning.

vertical bars

Equation layout will often use vertical bars to denote a variety of concepts.

  • absolute value
  • norm
  • determinant
  • calculation of definite integrals
  • BraKet notation

Pairs of vertical bars are predominantly used to describe absolute values as well as generalized concepts such as magnitude, distance and norms.

Commonly used heuristics will often be used on pairs of vertical bars. While these cases often work individually they quicky fail when combining them, e.g., see this equation from the Wikipedia entry for the Determinant:

where heuristics usually guess the left-hand side of the equation to be an absolute value whereas the middle part is guessed to be a determinant.

Literature

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