# Math Symbol Hierarchy

The Unicode Standard 7.0 has 2311 math symbols not including the ASCII letters and the standard combining marks like tilde, which are also used in math zones. Such a large number of symbols can be confusing if not intimidating to people who don’t use mathematics professionally. Even those who use math a lot don’t use most of the advanced symbols. The reason so many symbols are in Unicode is to fulfill the promise that all math symbols in modern print be defined, thereby allowing the Unicode math symbol set to be used as a basis for technical publications. This is a very worthy goal, but we don’t want it to get into the way of students learning math or more advanced folks that only need to use a fraction of the symbols. Accordingly it’s desirable to define some subsets that work for people of varying levels of mathematical expertise.

Three general levels seem pertinent:

1)      Basic (arithmetic, fractions, i.e., pre algebra):
+-/÷×*±∓([{|}])=≠≈~≅≃∝<>≤≥≪≫←↑↓→↔

2)      Intermediate (basic + algebra, calculus, set theory)

 Alphabetics a..z A..Z ℂℕℙℚℝℤℵ Γ∆∇ΘΛΞΠΣΦΨΩ αβδγεϵζηθϑκλμνξπρστυϕφχψω Math alphabetic styles Italic, bold, bold italic, Fraktur, open-face, script Operators •∘⊗′†!∴∵⇐⇒⇓⇑⇔∞≡∀∃∄∂∅∠∡∢∟∧∨¬∩∪∊∍∈∋⊂⊃⊆⊇≶⟨⟩∥⊥ N-ary operators ∫∮∏∐⋀⋁⋂⋃ Accents ′  ̀  ́  ̂  ̃  ̅  ̆  ̇  ̈ ⃗ Not symbols Many symbols can be negated by overwriting them with a /, e.g., = → ≠

3)      Unicode (all Unicode math symbols)
see Unicode Technical Report #25 Unicode Support for Mathematics and associated data files

The basic set is useful generally and should suffice for K-6 students, who have no need of integrals, summations, products and the like. The intermediate set should suffice for high school and college students, who have no need for most of the rarer mathematical symbols. In our user interfaces we can have options to expose only basic symbols, intermediate, or on demand, the full set. For any level, a list of the most recently used symbols is helpful since people almost always use a subset of the Unicode math symbols.

Within the sets, other classifications can be used to show symbols that have substantial similarity to one another. For typographical and syntactic purposes, math operators can be classified into the following general categories:

 Class Meaning Example N Normal - includes all digits and symbols requiring only one form 0..9 § A Alphabetic a..z B Binary × ÷ C Closing - usually paired with opening delimiter )]} D Diacritic ̂   ̃ F Fence - unpaired delimiter (often used as opening or closing) | G Glyph_Part - piece of large operator ⎩ L Large - n-ary or large operator, often takes limits ∫ O Opening - usually paired with closing delimiter ([{ P Punctuation ; : R Relation - includes arrows = → S Space U Unary - operators that are only unary ∀ ∃ V Vary - operators that can be unary or binary depending on context + −

In addition, these classes can be subclassed, e.g., grouping all the symbols that have equality in them like =≠≈~≅≃≤≥. There are often multiple ways to group symbols, e.g., ≮ would be included in a <-like symbols group like <≮≤≪∈∉∊⊂⊄⊆ as well as in a not-like symbols group like ≮≠≯∄∉∌≁≄≇≉≭≢≰≱≸≹⋡⊄⊅.

You can see a variety of math symbol collections by searching the Internet for mathematical symbols. For example, in addition to Unicode Technical Report #25 and its associated data files, there’s the article Mathematical operators and symbols in Unicode. That article displays the Unicode code charts for all the math symbol blocks in Unicode along with math symbols in other blocks. Scanning through it provides a quick way to view the Unicode math symbols. Other articles like Math Symbols list many math symbols and include descriptions of how the symbols are used.

Subclasses of math symbols are helpful for organizing symbol galleries such as in the Office math ribbon and in math touch screen keyboards. The math ribbon symbol gallery offers a superset of the intermediate class above broken into seven subclasses:  Basic Math, Greek Letters, Letter-Like Symbols, Operators, Arrows, Negated Relations, Scripts, and Geometry. The ribbon’s Basic Math class is more extensive than the basic level given here. Touch-screen keyboards can include both symbol galleries as done for emoji collections as well as surround menus that appear when you hold down a key like <. The possibilities that such keyboards offer for efficient math input are very intriguing. A keyboard based on the basic level set defined here would be ideal for many scenarios in addition to K-6 math. Keyboard galleries could be used for more advanced symbols. In principle the entire Unicode math symbol set could be accessed in a way that’s not intimidating to K-6 students.