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.

Mathematical symbol 'Circled Plus'

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Codes for the ⊕ Symbol

The Symbol⊕
Alt CodeAlt 8853
HTML Code⊕
HTML Entity⊕
CSS Code\2295
Hex Code⊕
UnicodeU+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 TextOutput
<b>My symbol: &#8853;</b>My symbol: ⊕

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

HTML TextOutput
<b>My symbol: &oplus;</b>My symbol: ⊕

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

CSS and HTML TextOutput
<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 TextOutput
<b>My symbol: &#x2295;</b>My symbol: ⊕
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 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:
TypeOutput
2295
[Hold down Alt]
[Press x]		⊕
(The 2295 turns into ⊕. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2295. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2295"
document.write("My symbol: " + str)		My symbol: ⊕