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

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Weierstrass ns function

Mathematicians don’t often adjust the layout of letter to denote special functions, but the Weierstrass ℘ function is a remarkable exception. The prize is defined in Unicode together U+2118 (SCRIPT funding P). It has a named HTML reality ℘. The LaTeX for ℘ is wp.

The mathematical meaning of the Weierstrass ℘ duty is that all elliptic functions can it is in expressed together rational attributes of this function and its derivatives.

Rotated T’s

Logic uses a symbol the looks prefer a sans serif T (⊤, U+22A4) to signify “true.” The same symbol turned upside down (⊥, U+22A5) is supplied for “false.” An advantage of making use of this symbol rather than some form of F is that it renders the the contrary of part formulas more apparent.

Logic additionally uses icons that look like a T rotated 90 ° clockwise or counterclockwise, ⊢ (U+22A2) and ⊣ (U+22A3). The expression xy method that y is provable from x. Likewise xy means that x is provable native y. Both expressions together are offered for equivalence, i.e. x ⊣⊢ y way x and also y room each provable native the other.

The symbols ⊢ (U+22A2) and also ⊣ (U+22A3) are vdash and also dashv in LaTeX.

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

Cyrillic letters in math

I discussed in a conversation the math rarely supplies Hebrew or Russian letters. Andres Caicedo stated that the Cyrillic letter sha (Ш, U+0428) is one exception. It is provided 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 for the “Dirac comb,” an unlimited sum of same spaced delta functions. I’d checked out the Dirac comb before, however I don’t recall Ш being offered as a symbol for it.

Sha is supposedly an exemption that proves the dominance that Cyrillic letters are seldom supplied in mathematics (at the very least in the West). Wikipedia states “Ш has actually the distinction of gift the just distinctly Cyrillic letter around the world used in mathematics.” additionally according to Wikipedia, sha more than likely comes kind the modern-day Hebrew Shin letter (ש, U+05E9).

By the way, the phrase “the exemption that proves the rule” doesn’t make sense as commonly interpreted. If anything, exceptions disprove a rule. Ns think the original understanding of the expression was the if exceptions are remarkable, that shows that the dominion often holds.

Planck’s constant

In physics, Planck’s consistent is the proportionality consistent between energy and particle frequency: E = hν. When working with angular frequency ω = 2πν, that is practically to introduce a new constant ħ same to h/2π so that E = ħω. The price ħ is merely pronounced “h bar” and also is sometimes referred to as the reduced Planck constant.

The correct Unicode worth for h is U+210E (PLANCK CONSTANT), not simply the typical h U+0068 (LATIN tiny LETTER H). Also, over there is a character ħ in the Latin Extended-A portion of Unicode, U+0127 (LATIN tiny LETTER H with STROKE), however the correct code allude for ħ is U+210F (PLANCK constant OVER two PI). If you’re wondering why anyone must care about making together distinctions, watch this post.

The LaTeX command for ħ is hslash. LaTeX go not have a special command because that h due to the fact that it does no make semantic distinctions in between visually identical characters.

If girlfriend look at the source for this page, you’ll view that i deliberately usage the wrong characters for h and ħ. That’s due to the fact that font support for Unicode is poor. I’m betting that much more people will be able to see glyphs because that the Latin letters 보다 for the exactly symbols.

Epi and also mono arrows

Mathematicians occasionally decorate arrows in diagrams to encode much more information about what the arrow represents. These decorations aren’t entirely standard, therefore it’s difficult (at the very least for me) to remember what they mean.

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


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


I don’t know whether the complying with pattern was intentional, yet it’s at least a mnemonic. If you know the an interpretation of the arrows above, you have the right to use this to remember the group theory definitions. Or if you have the right to remember the group theory definitions, you deserve to use this come remember the meaning of the arrows.

In group theory, monomorphisms are characterized using an chart that has two arrows ~ above the left. You might imagine these turning into the 2 lines in the tail of one arrow.


Epimorphisms are identified using a diagram through arrows ~ above the right. You might imagine putting these two arrows on height of each other then sliding one horizontally so the you have actually 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 point in Unicode is U+2203. The TeX command to produce the price is exists.

There are a couple common sport on the symbol. One is to monitor the prize by an exclamation point to denote that the point whose visibility is being asserted is unique, i.e. You can read the pair of symbols as “there exists a unique.”

Another sport is to draw a slash with the price to denote that something does not exist. The Unicode code allude for this sport is U+2204 and also the TeX command is exists.

Long S

Gottfried Leibniz provided the early ‘s’ that summa, the Latin for sum, to denote integration. He provided a “long S,” a type of the letter frequently used in his day.


A famous instance of the lengthy S is the word “Congress” created atop the bill of Rights.


The lengthy S fell out of usage in publish shortly after ~ the bill of legal rights was written, though it persisted in handwriting a few decades longer.

The Unicode code suggest for the long S is U+017F. The code point for the integral sign is U+222B.


It’s difficult to imagine mathematics without an same sign, yet our symbol for equality to be invented fairly recently loved one to the background of mathematics. Robert Recorde developed the symbol in 1557 and it was not widely offered until the 1700’s. Recorde used parallel currently to indicate equality since “noe 2 thynges deserve to be moare equalle.”

The definition behind an same sign deserve to be subtle. We have to be careful around in what sense we take into consideration the things on either next to it is in equal. Barry Mazur composed a sophisticated article ~ above this through the deceptively simple title when is one thing equal to some various other thing?

Equality deserve to be ethereal in programming languages together well. The declare a = b have the right to mean various things in different languages. In C it way to place the value b in the attend to a. In Python it way assign the name a come the worth b. In C++, the assignment operator deserve to be overloaded, and also so a = b can reason arbitrary password to execute. In a sensible programming language, a = b method to entrust the value of b to a permanently. In other languages, it method that a has or refers to the value b for now.


The only Hebrew letter generally used in math is aleph, א, the first letter that the Hebrew alphabet. It is provided with subscripts to denote the cardinality of various boundless sets. The most common use is v a subscript 0, pronounce “aleph naught” or “aleph null,” to denote the cardinality of the integers. The TeX command for aleph is, unsurprisingly, aleph.

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

א is often transliterated aleph on the end; Wikipedia, because that example, complies with this convention. The Unicode standard, however, supplies alef because that the letter and many sports on it.


I’ve never used a Hebrew letter in HTML before writing this post and also apparently there space things I have to learn about HTML and around WordPress. Here’s a tiny HTML paper I created, beginning the letter aleph 3 ways: together an HTML entity, making use of the Unicode value, and directly start the letter:


Here’s how this rendered in mine browser:


I’m surprised the entering HTML to escape sequences for aleph produces various output than straight entering the aleph character. And also I’m surprised the א0 puts the subscript shows up on the left yet ℵ0 puts the subscript top top the right.

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


Now the escaped Unicode value and directly beginning the character create the same output, however the HTML reality ℵ produces a bigger symbol. Also the subscript on the critical item relocated from the ideal side to the left side.

(The screenshots above were taken indigenous Firefox running on Ubuntu. I get similar results ~ above Windows utilizing Firefox or web Explorer other than that א and ℵ render in various fonts.)

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

See more: What Is An Upside Down Triangle Called, What Is The Name Of Upside Down Triangle

Update: I had assumed the the HTML entity ℵ to be an abbreviation for the Hebrew letter aleph, but it is one abbreviation for the mathematical prize aleph. This defines the various subscripting behavior. Also, believed that the default character collection for HTML was UTF-8. But when ns explicitly include to my HTML file I get the results I had expected, i.e. א and א job-related the same.