## The Mathematical Symbol "Circled Plus (⊕)"

The ⊕ Symbol in Mathematics: Circled Plus or Direct Sum

In the diverse world of mathematical symbols, the ⊕ or "Circled Plus" holds particular importance in various branches such as linear algebra, abstract algebra, and digital logic. This article demystifies its usage and offers examples for a clearer understanding.

## Usage

The ⊕ symbol signifies the direct sum of algebraic structures, most commonly vector spaces or groups. It's a way to combine two structures into a new one while keeping them distinct within the new structure.

## Examples

**Example 1:**Direct sum in vector spaces:

If \( V \) and \( W \) are vector spaces, their direct sum, denoted \( V ⊕ W \), is the set of ordered pairs of vectors with vector addition and scalar multiplication defined componentwise.**Example 2:**Direct sum in digital logic:

In digital logic, the ⊕ symbol represents the XOR (exclusive or) operation. For two binary values \( A \) and \( B \), \( A ⊕ B \) will be true if either \( A \) or \( B \) is true, but not both.

To conclude, the ⊕ symbol provides a compact way to represent the direct sum operation across various mathematical domains. A solid grasp of its meaning and applications can greatly aid in understanding advanced algebraic concepts and logic operations.

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## Codes for the ⊕ Symbol

The Symbol | ⊕ | |

Alt Code | Alt 8853 | |

HTML Code | ⊕ | |

HTML Entity | ⊕ | |

CSS Code | \2295 | |

Hex Code | ⊕ | |

Unicode | U+2295 |

## 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 8853. 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: "\2295";} </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: ⊕ |

**0x2295**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+2295**. 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 |
---|---|

2295 [Hold down Alt] [Press x] | ⊕ (The 2295 turns into ⊕. Note that you can omit any leading zeros.) |

JavaScript Text | Output |
---|---|

let str = "\u2295" document.write("My symbol: " + str) | My symbol: ⊕ |