The Mathematical Symbol "Circled Times (⊗)"
The ⊗ Symbol in Mathematics: Circled Times
In the tapestry of mathematical symbols, the ⊗ or "Circled Times" stands out for its specialized applications, especially in the domains of linear algebra and functional analysis. This article seeks to demystify its usage and provide clear examples for context.
Usage
The ⊗ symbol is primarily associated with the tensor product of vectors, vector spaces, and modules. It signifies the creation of a new mathematical structure from two input structures. The resulting structure encapsulates more complexity than the individual inputs, and understanding this operation is vital in areas such as quantum mechanics and more advanced areas of mathematics.
Examples
- Example 1: In vector spaces:
Given two vectors \( v \in V \) and \( w \in W \), their tensor product is represented as:
\( v ⊗ w \)
The result is a new vector in the tensor product space \( V ⊗ W \). - Example 2: In the context of modules:
Given two \( R \)-modules \( M \) and \( N \), their tensor product over \( R \) is represented as:
\( M ⊗_R N \)
This operation produces another \( R \)-module.
To wrap up, the ⊗ symbol is a profound mathematical notation, pivotal in advanced areas like quantum physics and functional analysis. Understanding its significance and application can pave the way for a deeper insight into intricate mathematical constructs.
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Codes for the ⊗ Symbol
| The Symbol | ⊗ | |
| Alt Code | Alt 8855 | |
| HTML Code | ⊗ | |
| HTML Entity | ⊗ | |
| CSS Code | \2297 | |
| Hex Code | ⊗ | |
| Unicode | U+2297 |
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 8855. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)(Method 3) Use the HTML Decimal Code (for webpages).
| HTML Text | Output |
|---|---|
| <b>My symbol: ⊗</b> | My symbol: ⊗ |
(Method 4) Use the HTML Entity Code (for webpages).
| HTML Text | Output |
|---|---|
| <b>My symbol: ⊗</b> | My symbol: ⊗ |
(Method 5) Use the CSS Code (for webpages).
| CSS and HTML Text | Output |
|---|---|
|
<style> span:after { content: "\2297";} </style> <span>My symbol:</span> | My symbol: ⊗ |
(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).
| HTML Text | Output |
|---|---|
| <b>My symbol: ⊗</b> | My symbol: ⊗ |
| 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+2297. 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:| Type | Output |
|---|---|
| 2297 [Hold down Alt] [Press x] | ⊗ (The 2297 turns into ⊗. Note that you can omit any leading zeros.) |
| JavaScript Text | Output |
|---|---|
| let str = "\u2297" document.write("My symbol: " + str) | My symbol: ⊗ |