**Infinity** is an abstract mathematical concept that refers to something endless or boundless. While it’s important in math, you’ll also see it in computing, art, physics, cosmology, and popular culture. Here is the definition of infinity, a look at its symbol, infinity examples, and the mathematical rules for using it.

### What Is Infinity?

Infinity is anything endless. It refers to unending time, a series of numbers that continues forever, or a perpetual series of operations.

### The Infinity Symbol and Early History

English clergyman and mathematician John Wallis introduced the infinity symbol ∞ in 1655. The symbol is called the lemniscate.

The word “leminscate” comes from the Latin word *lemniscus*, which means “ribbon.” The word “infinity” comes from the Latin word *infinitas*, meaning “boundless.” Wallis may have based the lemniscate on the Roman numeral for 1000 (M), which Romans used to mean “countless” as well as the actual number. Another possibility is that the leminscate is a form of the Greek letter omega (Ω or ω), which is the last letter of the Greek alphabet.

But, the concept of infinity has been around long before its symbol. The Greek philosopher Anaximander (c. 610 – c. 546 BC) described the concept of *apeiron*, which means “unbounded.” Aristotle (350 BC) distinguished between different types of infinity. Euclid’s theorems referenced the concept.

Meanwhile, Jain mathematicians in India also developed the concept. Surya Prajnapti (c. 4th-3rd century BCE) described numbers as either enumerable, innumerable, or infinite.

### Examples of Infinity

You may think of the number of grains of sand on the beach or number of stars in the sky as infinite, but they are actually extremely large finite numbers. Infinity goes on forever. Here are some infinity examples:

- The sequence of natural numbers is infinite. {1, 2, 3, …}
- A line or even a line segment consists of infinite points.
- Similarly, a circle consists of infinite points.
- The number pi (
*π*) goes on forever. (3.14159…) - Certain fractions are finite, but they are infinite when written as decimal numbers. (1/3 is 0.333…)
- The number of prime numbers is infinite.
- The number phi (Φ) is the golden ratio, (1 + √5)/2, which is an infinite decimal number 1.618…
- While astronomers can see the edge of the Universe formed by the Big Bang, it’s unknown whether it will expand forever (infinitely) or stop and contract again (finite).
- Fractals are structures that can be magnified infinitely without losing their structure.
- In complex number theory, dividing 1 by 0 is an infinity that doesn’t collapse. (On a calculator, dividing any number by zero is just an error code.)
- If you cross a room, going half the remaining distance with each step, it will take you infinite time or infinite number of steps to reach your destination.
- There are many examples of infinite series in math. For example, 1 + 1/2 + 1/3 + … is an infinite series.

### Different Sizes of Infinity

Mathematicians deal with different sizes of infinity.

- The sets of positive whole numbers (numbers greater than 0) and negative whole numbers (numbers less than 0) are infinite sets of the same size. But, if you combine the two sets you get a new infinite set that is twice as large.
- You can add a number to infinity to make it larger. For example, ∞ + 1 > ∞.
- The set of whole numbers is a smaller infinite set than the set of real numbers.

### Positive and Negative Infinity

In math, there is negative infinity and there is positive infinity (which is just called infinity):

**-∞ < x < ∞**

In other words, negative infinite is less than any real number, while infinity is greater than any real number.

### Is Infinity Divided by Infinity Equal to 1?

While infinity is like an ordinary number in some ways, it differs in others. For example, if you divide a number by itself (e.g., 2/2 or -3/-3) you get 1. But, ∞/∞ isn’t equal to 1. It’s “undefined.” The reason for this goes back to the different sizes of infinities.

In a way, ∞/∞ = (∞+∞)/∞. But, it doesn’t work the same as 1/1 = 2/1 because different infinities may be different size. Confusing, right?

### Undefined Operations

Dividing infinity by itself isn’t the only undefined operation.

Undefined Operations Using Infinity |

0 × ∞ |

0 × -∞ |

∞ + -∞ |

∞ – ∞ |

∞ / ∞ |

∞^{0} |

1^{∞} |

### Special Properties of Infinity in Math

Infinity has special properties in mathematics.

Infinity Special Properties |

∞ + ∞ = ∞ |

-∞ + -∞ = -∞ |

∞ × ∞ = ∞ |

-∞ × -∞ = ∞ |

-∞ × ∞ = -∞ |

x + ∞ = ∞ |

x + (-∞) = -∞ |

x – ∞ = -∞ |

x – (-∞) = ∞ |

For x>0 : x × ∞ = ∞ |

For x>0 : x × (-∞) = -∞ |

For x<0 : x × ∞ = -∞ |

For x<0 : x × (-∞) = ∞ |

### References

- Cajori, Florian (1993) [1928 & 1929].
*A History of Mathematical Notations*. Dover. ISBN 978-0-486-67766-8. - Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008).
*The Princeton Companion to Mathematics*. Princeton University Press. p. 616. - Kline, Morris (1972).
*Mathematical Thought From Ancient to Modern Times*. New York: Oxford University Press. ISBN 978-0-19-506135-2. - Rucker, Rudy (1995).
*Infinity and the Mind: The Science and Philosophy of the Infinite*. Princeton University Press. ISBN 978-0-691-00172-2. - Scott, Joseph Frederick (1981),
*The Mathematical Work of John Wallis, D.D., F.R.S.*, (1616–1703) (2nd ed.), American Mathematical Society. p. 24.