The Mathematical Symbol "Left-Pointing Angle Bracket (⟨)"

The "Left-Pointing Angle Bracket" Symbol (⟨): Bridging Mathematical and Computational Domains

Throughout the annals of mathematical notation, certain symbols transcend their original purpose, gaining prominence across diverse fields. One such symbol is the ⟨, commonly known as the "Left-Pointing Angle Bracket." Originally rooted in set theory and logic, its usage has expanded into various mathematical and computational domains. This article aims to illuminate the different contexts in which this symbol finds its utility.

Origins of the ⟨ Symbol

The ⟨ symbol, paired with its counterpart, the "Right-Pointing Angle Bracket," is often used to denote ordered pairs, tuples, or inner products in different mathematical contexts.

Example 1: Ordered Pairs and Tuples

In set theory, the ordered pair (a, b) can be represented as ⟨a, b⟩. This notation emphasizes the sequence of elements, distinguishing it from sets where order doesn't matter.

Example 2: Inner Products in Quantum Mechanics

In the realm of quantum mechanics, the ⟨ symbol is part of the bra-ket notation, introduced by physicist Paul Dirac. In this context, an inner product between two quantum states, ⟨ψ|φ⟩, is represented using the angle brackets.

Broadened Applications

Over time, the ⟨ symbol has found applications in various fields:

  • Computer Science: Used in angle-bracket notation for generics in certain programming languages.
  • Linguistics: Employed to indicate the phonetic transcription of a word.
  • XML and HTML: Angle brackets are fundamental in marking the beginning and end of tags.

It's essential to note that while the ⟨ symbol has diverse applications, its meaning is context-dependent. For instance, in quantum mechanics, its interpretation is distinct from its use in computer science or linguistics.

In summary, the ⟨ symbol is a testament to the evolving nature of mathematical notation. From delineating ordered pairs to representing intricate quantum states, its versatility is a reflection of the interconnectedness of modern scientific and mathematical disciplines.

Mathematical symbol 'Left-Pointing Angle Bracket'

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Codes for the 〈 Symbol

The Symbol〈
Alt CodeAlt 9001
HTML Code〈
HTML Entity⟨
CSS Code\2329
Hex Code〈
UnicodeU+2329

How To Insert the 〈 Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the 〈 symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 9001. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML TextOutput
<b>My symbol: &#9001;</b>My symbol: 〈

(Method 4) Use the HTML Entity Code (for webpages).

HTML TextOutput
<b>My symbol: &lang;</b>My symbol: ⟨

(Method 5) Use the CSS Code (for webpages).

CSS and HTML TextOutput
<style>
span:after {
content: "\2329";}
</style>
<span>My symbol:</span> 		My symbol: 〈

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML TextOutput
<b>My symbol: &#x2329;</b>My symbol: 〈
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2329 to place the 〈 symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);
Output

〈

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for 〈 is U+2329. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:
TypeOutput
2329
[Hold down Alt]
[Press x]		〈
(The 2329 turns into 〈. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2329. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2329"
document.write("My symbol: " + str)		My symbol: 〈