The Mathematical Symbol "Asymptotically Equal To (≃)"
Unraveling the ≃ Symbol in Mathematics
The mathematical symbol ≃ denotes "Asymptotically Equal To". It is used predominantly in calculus, number theory, and analysis to signify that two functions become arbitrarily close to each other as they move towards a certain point, typically infinity. This article will delve into two primary contexts of this symbol, providing two examples for each use.
1. Analysis and Calculus
In calculus and analysis, ≃ is employed to express the behavior of functions as they approach a certain value, typically infinity.
- Example 1: The function \(f(x) = x^2 + x\) and \(g(x) = x^2\) are asymptotically equal as \(x\) approaches infinity, which can be written as \(f(x) ≃ g(x)\) as \(x \to \infty\).
- Example 2: The sine function oscillates between -1 and 1, so \(sin(x) ≃ 0\) as \(x \to \infty\), although it doesn't converge to any value.
2. Number Theory
In number theory, ≃ is often used to convey the behavior of sequences or series as they grow larger.
- Example 1: Consider the prime-counting function \( \pi(x) \) which represents the number of primes less than or equal to \( x \). It can be shown that \( \pi(x) ≃ \frac{x}{\log(x)} \) as \( x \to \infty \).
- Example 2: The harmonic series, \( \sum_{n=1}^{\infty} \frac{1}{n} \), grows logarithmically, which means that its sum grows similarly to \( \log(x) \) as \( x \to \infty \).
In summary, the ≃ symbol plays a pivotal role in expressing the behavior of mathematical entities as they approach specific values. It offers a nuanced way to convey closeness or similarity without asserting equality.
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Codes for the ≃ Symbol
| The Symbol | ≃ | |
| Alt Code | Alt 8771 | |
| HTML Code | ≃ | |
| HTML Entity | ≃ | |
| CSS Code | \2243 | |
| Hex Code | ≃ | |
| Unicode | U+2243 |
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 8771. 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: "\2243";} </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+2243. 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 |
|---|---|
| 2243 [Hold down Alt] [Press x] | ≃ (The 2243 turns into ≃. Note that you can omit any leading zeros.) |
| JavaScript Text | Output |
|---|---|
| let str = "\u2243" document.write("My symbol: " + str) | My symbol: ≃ |