The Mathematical Symbol "Therefore (∴)"

The ∴ Symbol: "Therefore"

The ∴ symbol, often pronounced as "therefore", is widely used in both mathematics and logic. This symbol serves as a shorthand notation indicating a conclusion or a resultant deduction from previously stated facts or premises. In this article, we'll explore its meaning and use, with a couple of illustrative examples.

Usage

In mathematical proofs and logical arguments, the ∴ symbol signifies the conclusion drawn from the given information. It often helps in streamlining arguments by reducing verbosity and providing a clear transition from premises to conclusions.

Examples

  • Example 1: Logical Deduction:
    Given: If it rains, the ground will be wet.
    It is raining.
    ∴ The ground is wet.
    In this example, the ∴ symbol indicates the natural deduction based on the given conditions.
  • Example 2: Mathematical Proof:
    Given: For any integer \( n \), if \( n^2 \) is even, then \( n \) is even.
    \( 4^2 = 16 \) which is even.
    ∴ 4 is even.
    Here, the ∴ symbol helps derive a specific case's conclusion based on a general statement.

In summary, the ∴ symbol is a concise and efficient way to indicate the result or conclusion derived from certain facts or premises in logical and mathematical arguments. Its widespread adoption underlines the importance of brevity and clarity in such contexts.

Mathematical symbol 'Therefore'

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Codes for the ∴ Symbol

The Symbol∴
Alt CodeAlt 8756
HTML Code∴
HTML Entity∴
CSS Code\2234
Hex Code∴
UnicodeU+2234

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 8756. 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: &#8756;</b>My symbol: ∴

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

HTML TextOutput
<b>My symbol: &there4;</b>My symbol: ∴

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

CSS and HTML TextOutput
<style>
span:after {
content: "\2234";}
</style>
<span>My symbol:</span> 		My symbol: ∴

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

HTML TextOutput
<b>My symbol: &#x2234;</b>My symbol: ∴
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x2234 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+2234. 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
2234
[Hold down Alt]
[Press x]		∴
(The 2234 turns into ∴. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u2234. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u2234"
document.write("My symbol: " + str)		My symbol: ∴