The Mathematical Symbol "Ring Operator (∘)"

The "Ring Operator" Symbol (∘)

The ∘ symbol, commonly referred to as the "Ring Operator" or more often as the "Function Composition", is an essential notation used predominantly in mathematics and computer science. It denotes the act of composing two functions, meaning the output of one function becomes the input of another.

Usage of ∘

Function composition is a fundamental concept, especially in higher mathematics and functional programming. The symbol ∘ signifies that two functions are being combined in a specific sequence.

Example 1: Mathematical Functions

Let's say we have two functions \( f(x) = x^2 \) and \( g(x) = x + 3 \). The composition of \( f \) with \( g \), written as \( f ∘ g \), is a new function defined as \( f(g(x)) \). For \( x = 2 \), \( f(g(2)) = f(5) = 25 \).

Example 2: Functional Programming

In functional programming, functions are often treated as first-class citizens, meaning they can be passed as arguments, returned from other functions, and assigned to variables. Here, the ∘ operator might be used to denote function chaining or pipelining, where the output of one function directly feeds into the next.

Significance in Mathematics and Computer Science

The concept of function composition is foundational in several areas:

  • Calculus: When studying the chain rule, function composition plays a pivotal role.
  • Functional Programming: Languages like Haskell and Lisp lean heavily on the idea of chaining functions together.
  • Discrete Mathematics: In the study of relations and functions, composition helps define new relationships from existing ones.
  • Category Theory: A branch of abstract mathematics where morphism composition is a central theme, typically denoted by the ∘ symbol or similar notations.

In summary, the ∘ symbol plays a central role in representing the operation of function composition across various disciplines. It encapsulates a foundational idea of building complex operations from simpler ones, making it a cornerstone concept in both mathematics and computer science.

Mathematical symbol 'Ring Operator'

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Codes for the ∘ Symbol

The Symbol
Alt CodeAlt 8728
HTML Code∘
HTML Entity∘
CSS Code\2218
Hex Code∘

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 8728. 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: &#8728;</b>My symbol: ∘

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

HTML TextOutput
<b>My symbol: &compfn;</b>My symbol: ∘

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

CSS and HTML TextOutput
span:after {
content: "\2218";}
<span>My symbol:</span>
My symbol: ∘

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

HTML TextOutput
<b>My symbol: &#x2218;</b>My symbol: ∘
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2218 to place the ∘ symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);

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

The Unicode for ∘ is U+2218. 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:
[Hold down Alt]
[Press x]

(The 2218 turns into ∘. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2218. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2218"
document.write("My symbol: " + str)
My symbol: ∘