This page includes the math-related write-ups from mine short-lived blog Symbolism. The non-mathematical posts are collected here.

You are watching: What is the backwards e in math

Table of contents:

Weierstrass p function

Mathematicians don’t often readjust the format of letters to represent special functions, yet the Weierstrass ℘ function is a remarkable exception. The price is defined in Unicode together U+2118 (SCRIPT resources P). It has a named HTML entity ℘. The LaTeX because that ℘ is wp.

The mathematical definition of the Weierstrass ℘ role is that all elliptic functions can it is in expressed as rational functions of this duty and the derivatives.

Rotated T’s

Logic offers a symbol that looks choose a sans serif T (⊤, U+22A4) to represent “true.” The very same symbol turned upside under (⊥, U+22A5) is offered for “false.” An advantage of making use of this symbol rather than some form of F is the it makes the the contrary of part formulas much more apparent.

Logic also uses signs that look prefer a T rotated 90 ° clockwise or counterclockwise, ⊢ (U+22A2) and also ⊣ (U+22A3). The expression xy method that y is provable indigenous x. Likewise xy method that x is provable indigenous y. Both expressions with each other are provided for equivalence, i.e. x ⊣⊢ y way x and y are each provable native the other.

The icons ⊢ (U+22A2) and ⊣ (U+22A3) space vdash and dashv in LaTeX.

The symbols ⊤ (U+22A4) and also ⊥ (U+22A5) op and ot in LaTeX.

Cyrillic letters in math

I discussed in a conversation the math rarely offers Hebrew or Russian letters. Andres Caicedo discussed that the Cyrillic letter sha (Ш, U+0428) is an exception. It is offered for the Tate–Shafarevich team of an Abelian variety. I’d never heard of together a point before.

The letter sha is additionally used in distribution theory because that the “Dirac comb,” an infinite sum of same spaced delta functions. I’d seen the Dirac comb before, however I don’t recall Ш being provided as a symbol for it.

Sha is apparently an exception that proves the preeminence that Cyrillic letters room seldom used in mathematics (at the very least in the West). Wikipedia claims “Ш has actually the distinction of being the only distinctly Cyrillic letter internationally used in mathematics.” additionally according come Wikipedia, sha most likely comes type the modern Hebrew Shin letter (ש, U+05E9).

By the way, the phrase “the exception that proves the rule” doesn’t make sense as frequently interpreted. If anything, exceptions disprove a rule. I think the original knowledge of the phrase was the if exceptions are remarkable, that mirrors that the dominion often holds.

Planck’s constant

In physics, Planck’s continuous is the proportionality constant between energy and particle frequency: E = hν. Once working with angular frequency ω = 2πν, the is convenient to introduce a new constant ħ equal to h/2π so that E = ħω. The price ħ is simply pronounced “h bar” and also is sometimes called the reduced Planck constant.

The correct Unicode worth for h is U+210E (PLANCK CONSTANT), not just the common h U+0068 (LATIN little LETTER H). Also, there is a character ħ in the Latin Extended-A section of Unicode, U+0127 (LATIN little LETTER H through STROKE), yet the exactly code allude for ħ is U+210F (PLANCK constant OVER 2 PI). If you’re wonder why anyone have to care around making such distinctions, check out this post.

The LaTeX command because that ħ is hslash. LaTeX go not have actually a special command because that h since it does not make semantic distinctions between visually identical characters.

If you look in ~ the source for this page, you’ll check out that i deliberately usage the wrong characters for h and also ħ. That’s because font support for Unicode is poor. I’m betting that much more people will be able to see glyphs for the Latin letters than for the exactly symbols.

Epi and also mono arrows

Mathematicians sometimes decorate arrows in diagrams come encode much more information about what the arrowhead represents. This decorations aren’t entirely standard, for this reason it’s tough (at least for me) to remember what they mean.

One-to-one functions are denoted through an arrow with a separation tail. This is U+21A3 or ightarrowtail in LaTeX. (These are additionally called injective functions or monomorphisms.)


Onto features are denoted v an arrow with 2 heads. This is U+21A0 or woheadrightarrow in LaTeX. (These are also called surjective features or epimorphisms.)


I don’t know whether the following pattern was intentional, however it’s at the very least a mnemonic. If you recognize the an interpretation of the arrows above, you can use this to remember the group theory definitions. Or if you have the right to remember the category theory definitions, you can use this come remember the an interpretation of the arrows.

In category theory, monomorphisms are defined using an diagram that has two arrows top top the left. You might imagine these turning into the two lines in the tail of one arrow.


Epimorphisms are identified using a diagram through arrows on the right. You could imagine putting these 2 arrows on peak of each various other then slide one horizontally so that you have two arrowhead heads.


There exists

In logic, a behind E is an abbreviation because that “there exists.” The symbol is called the “existential quantifier.” its code suggest in Unicode is U+2203. The TeX command to produce the symbol is exists.

There are a couple common variations on the symbol. One is to monitor the price by an exclamation point to signify that the thing whose presence is being asserted is unique, i.e. You deserve to read the pair of signs as “there exist a unique.”

Another sport is to attract a slash through the symbol to signify that something does no exist. The Unicode code suggest for this sport is U+2204 and the TeX command is exists.

Long S

Gottfried Leibniz offered the initial ‘s’ that summa, the Latin for sum, to signify integration. He used a “long S,” a form of the letter generally used in his day.


A famous instance of the lengthy S is the word “Congress” composed atop the invoice of Rights.


The lengthy S dropped out of use in print shortly after the invoice of legal rights was written, despite it persisted in handwriting a couple of decades longer.

The Unicode code suggest for the lengthy S is U+017F. The code suggest for the integral authorize is U+222B.


It’s hard to imagine math without an same sign, yet our symbol because that equality was invented relatively recently loved one to the history of mathematics. Robert Recorde invented the prize in 1557 and also it was not widely provided until the 1700’s. Recorde used parallel present to symbolize equality because “noe 2 thynges can be moare equalle.”

The definition behind an same sign deserve to be subtle. We need to be careful around in what feeling we consider the things on either next to be equal. Barry Mazur created a sophisticated article ~ above this with the deceptively straightforward title as soon as is one thing equal come some other thing?

Equality have the right to be subtle in programming languages together well. The statement a = b have the right to mean different things in different languages. In C it method to place the worth b in the attend to a. In Python it method assign the surname a come the worth b. In C++, the assignment operator can be overloaded, and so a = b can cause arbitrary password to execute. In a functional programming language, a = b means to assign the value of b to a permanently. In other languages, it means that a has or refers to the value b for now.


The just Hebrew letter typically used in mathematics is aleph, א, the very first letter that the Hebrew alphabet. The is offered with subscripts to denote the cardinality of various boundless sets. The most typical use is v a subscript 0, pronounce “aleph naught” or “aleph null,” to denote the cardinality that the integers. The TeX command because that aleph is, unsurprisingly, aleph.

When used as a Hebrew letter, its Unicode code point is (U+05D0). When provided as a mathematical price it has a unique code point, U+2135. I supposed aleph null (ℵ0) to have its very own Unicode value but that doesn’t it seems to be ~ to be the case.

א is regularly transliterated aleph on the end; Wikipedia, for example, adheres to this convention. The Unicode standard, however, supplies alef because that the letter and many sports on it.


I’ve never ever used a Hebrew letter in HTML before writing this post and apparently there space things I should learn around HTML and around WordPress. Here’s a tiny HTML paper I created, beginning the letter aleph three ways: together an HTML entity, using the Unicode value, and also directly start the letter:


Here’s just how this calculation in my browser:


I’m surprised that entering HTML escape sequences because that aleph produces different output than straight entering the aleph character. And also I’m surprised the א0 place the subscript shows up on the left but ℵ0 put the subscript top top the right.

When I paste the exact same code right into WordPress, this is what ns see:


Now the escaped Unicode value and directly beginning the character create the exact same output, however the HTML entity ℵ to produce a bigger symbol. Likewise the subscript on the last item relocated from the best side to the left side.

(The screenshots over were taken indigenous Firefox to run on Ubuntu. Ns get comparable results on Windows making use of Firefox or web Explorer other than that א and ℵ render in various fonts.)

Could someone please leave a comment explaining why the three various ways of entering an aleph use different fonts and follow various subscript rules? Thanks.

See more: Can You Eat Octopus And Squid Ink (Which One Is Squid Ink Bad For You

Update: I had assumed the the HTML reality ℵ was an abbreviation for the Hebrew letter aleph, however it is one abbreviation because that the mathematical symbol aleph. This describes the various subscripting behavior. Also, assumed that the default character set for HTML was UTF-8. However when i explicitly add to mine HTML record I gain the outcomes I had actually expected, i.e. א and also א work-related the same.