## The Mathematical Symbol "Circled Dot Operator (⊙)"

The ⊙ Symbol in Mathematics: Circled Dot Operator or Tensor Product

In the vast landscape of mathematical notations, the ⊙ or "Circled Dot Operator" plays a pivotal role in various branches, especially in linear algebra and quantum mechanics. This article aims to shed light on its usage and provide examples for a better grasp of its significance.

## Usage

The ⊙ symbol typically denotes the tensor product of two algebraic structures, such as vector spaces or modules. The tensor product allows for the combination of two algebraic systems into a new one in a bilinear manner.

## Examples

• Example 1: Tensor product of vector spaces:
Given two vector spaces $$V$$ and $$W$$ over the same field $$F$$, their tensor product, denoted $$V ⊙ W$$, consists of all formal linear combinations of simple tensors of the form $$v \otimes w$$ where $$v$$ is from $$V$$ and $$w$$ is from $$W$$.
• Example 2: Quantum mechanics:
In quantum mechanics, the tensor product allows for the description of composite systems. If $$| \psi \rangle$$ and $$| \phi \rangle$$ are state vectors of two quantum systems, the combined state of the two systems can be represented as $$| \psi \rangle ⊙ | \phi \rangle$$.

In summary, the ⊙ symbol is crucial for representing bilinear combinations across multiple mathematical contexts. Understanding its applications is key to deciphering advanced algebraic structures and quantum systems. ## Are You Good at Mathematical Symbols?

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## Codes for the ⊙ Symbol

 The Symbol ⊙ Alt Code Alt 8857 HTML Code ⊙ HTML Entity ⊙ CSS Code \2299 Hex Code ⊙ Unicode U+2299

## 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 8857. 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: &#8857;</b>My symbol: ⊙

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

HTML TextOutput
<b>My symbol: &odot;</b>My symbol: ⊙

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

CSS and HTML TextOutput
<style>
span:after {
content: "\2299";}
</style>
<span>My symbol:</span>
My symbol: ⊙

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

HTML TextOutput
<b>My symbol: &#x2299;</b>My symbol: ⊙
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2299 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+2299. 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
2299
[Hold down Alt]
[Press x]

(The 2299 turns into ⊙. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2299. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2299"
document.write("My symbol: " + str)
My symbol: ⊙