Infinity Symbol (∞): Meaning, Uses, and How to Type It

The infinity symbol (∞) represents a quantity without any bound: something larger than every number, or a process that never ends. It is called a lemniscate, and you can type it with Alt + 236 on Windows, Option + 5 on a Mac, ∞ in HTML, or \infty in LaTeX.

This guide explains where the sideways-eight shape came from, what ∞ means in limits, intervals, set theory, and physics, what it symbolizes in culture, and how to type it on every device.

Copy and Paste the Infinity Symbol

• ∞ — infinity (U+221E)
• +∞ and −∞ — positive and negative infinity, used on the extended real line
• ℵ₀ — aleph-null, the smallest infinite cardinal (U+2135 plus subscript zero)

The Infinity Symbol Explained

The English mathematician John Wallis introduced ∞ in 1655 in De sectionibus conicis and used it again in Arithmetica Infinitorum the same year. He never explained his choice; the leading theories are that he adapted the Roman numeral CIↃ (a form of M for 1,000, sometimes used loosely to mean "many") or the Greek letter omega (ω).

The curve's name, lemniscate, comes from the Latin lemniscus, a ribbon. Jacob Bernoulli used it in 1694 for the figure-eight curve (x² + y²)² = 2a²(x² − y²), and the name attached to the symbol as well. The Unicode name is simply INFINITY.

The key thing to understand is that ∞ is not a number. It cannot be added, multiplied, or divided like 5 or π. Expressions such as ∞ − ∞ and ∞ / ∞ are undefined, and the arithmetic of limits exists precisely to handle them carefully.

Meaning and Uses of Infinity in Mathematics

Limits

Infinity appears most often in calculus, where it describes what happens as a variable grows without bound or a function blows up:

• lim (x → ∞) 1/x = 0 — the reciprocal shrinks toward zero as x grows
• lim (x → 0⁺) 1/x = ∞ — the reciprocal grows without bound as x approaches zero from the right
• lim (n → ∞) (1 + 1/n)ⁿ = e — the definition of Euler's number

Writing "x → ∞" means "x increases without limit," not "x reaches infinity." Infinite series use the same idea: Σ (n = 1 to ∞) 1/2ⁿ = 1, even though no finite number of terms ever adds up to exactly 1.

Interval Notation

Unbounded intervals use ∞ as a placeholder endpoint, always with a round bracket because infinity is never included:

• (−∞, ∞) — all real numbers
• [0, ∞) — all non-negative numbers
• (−∞, 3) — all numbers less than 3

The Extended Real Line

Analysts sometimes adjoin two new points, +∞ and −∞, to the real numbers to form the extended real line [−∞, +∞]. This makes every monotone sequence converge and lets measure theory assign the "value" ∞ to sets of infinite size. In this system a few operations are defined (∞ + 5 = ∞, 3 × ∞ = ∞), but ∞ − ∞ and 0 × ∞ remain undefined.

Sizes of Infinity

Georg Cantor showed in the 1870s that infinite sets come in different sizes. The natural numbers ℕ, the integers ℤ, and even the rationals ℚ can all be matched one-to-one with each other; their shared size is the cardinal number ℵ₀ (aleph-null). The real numbers ℝ cannot be listed this way (Cantor's diagonal argument), so their cardinality, written 𝔠 or 2^ℵ₀ and called the cardinality of the continuum, is strictly larger. The symbol ∞ is deliberately vague about which infinity is meant; the aleph numbers are precise.

Infinity in Physics and Engineering

• Optics — a lens "focused at infinity" brings parallel rays to a focus at its focal length; camera lenses have an ∞ mark on the focus ring.
• Electrostatics — the potential of a point charge is defined to be zero at infinite distance, V(∞) = 0, so that V(r) = kq / r.
• Circuits — an open circuit is modeled as infinite resistance, and an ideal voltmeter has infinite input impedance.
• Singularities — general relativity predicts infinite density at the center of a black hole, which is usually read as a sign the theory breaks down there.
• Computing — IEEE 754 floating point reserves bit patterns for +∞ and −∞; in Python, float('inf') creates one and 1e308 * 10 overflows to it.

Infinity in Culture

Outside mathematics the lemniscate stands for eternity, endless love, and limitless possibility, which is why it is among the most popular tattoo designs. It appears on the flag of the Métis nation of Canada, in the Toy Story catchphrase "to infinity and beyond," and in countless logos. The closed double loop also suggests balance and continuous return.

How to Type the Infinity Symbol

Windows: Hold Alt and type 236 on the numeric keypad. The code comes from the original IBM PC character set, so it works in Notepad, browsers, Word, and almost everything else. Word users can also type 221E and press Alt + X, or use \infty inside an equation box (Alt + =).

Mac: Press Option + 5 on a US keyboard. The Character Viewer (Control + Command + Space, search "infinity") works on any layout.

iPhone and iPad: The built-in keyboard has no infinity key. Copy the symbol from the block above, or set up a text replacement under Settings > General > Keyboard > Text Replacement so that typing "inf" becomes ∞. On Android, Gboard reveals ∞ when you long-press the = key on the symbols layer.

Infinity in LaTeX

The command is \infty, used in math mode. Do not confuse it with \inf, which typesets the operator "inf" for the infimum of a set.

```latex
\documentclass{article}
\usepackage{amsmath, amssymb}
\begin{document}
\[ \lim_{x \to \infty} \frac{1}{x} = 0, \qquad \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \]
\[ \sum_{n=1}^{\infty} \frac{1}{2^n} = 1, \qquad \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi} \]
\[ x \in (-\infty, \infty), \qquad |\mathbb{N}| = \aleph_0, \qquad |\mathbb{R}| = 2^{\aleph_0} \]
\end{document}
```

\aleph is available without any package, and \mathbb for ℕ and ℝ needs amssymb or amsfonts. In display math the limits of \lim, \sum, and \int are placed automatically; in inline math they move to the side, and \displaystyle restores the stacked form.

Related Symbols

Is Infinity a Number?

No. Infinity describes a quantity that exceeds every real number, but it is not itself on the number line and does not obey the normal rules of arithmetic. Extended number systems such as the extended real line and the cardinal numbers give particular kinds of infinity precise meanings, each with its own rules.

What Is Infinity Minus Infinity?

Undefined. Depending on how the two infinite quantities arise, the difference can approach any value: lim (x → ∞) (x − x) = 0, but lim (x → ∞) (2x − x) = ∞. Calculus treats ∞ − ∞ as an indeterminate form that must be resolved by further analysis.

Why Is the Infinity Symbol a Sideways Eight?

John Wallis chose the shape in 1655 without saying why; the leading theories are a Roman numeral for 1,000 that also meant "a great many," or the lowercase omega. The endless loop has stuck because it suggests a path with no beginning or end.

Are There Different Sizes of Infinity?

Yes. Cantor proved that the set of real numbers is strictly larger than the set of natural numbers, even though both are infinite. The smallest infinity, that of the counting numbers, is written ℵ₀; the infinity of the real numbers is 2^ℵ₀.

Writing About Limits and Infinite Sums

Limits, series, and improper integrals all lean on \infty, and the difference between \infty and \inf or between inline and display limits trips up many first-time LaTeX users. Octree, the AI LaTeX editor at useoctree.com, generates correct calculus notation from a plain-language description and shows the compiled PDF as you type. For every limit, integral, and series symbol with its LaTeX command, see https://tools.useoctree.com/symbols/calculus.