# Infinity

Infinity is the concept of something boundless, something that has no end. In the context of mathematics it may be referred to as a "number," but infinity is not a real number. It is used to represent a value that is immeasurably large, and cannot be assigned any kind of actual numerical value. Infinity is represented using the symbol ∞.

Infinity is larger than the largest conceivable number, has no end, and does not grow in any way. It is not like an exponential value.

Although we can use infinity in mathematics, it does not behave like a real number would.

Example

1 + ∞ = ∞

100 + ∞ = ∞

10^{10000} + ∞ = ∞

It doesn't matter how large the number we add to infinity is, the value will still always be infinity, and even though we know that 10^{10000} is much larger than 1, adding infinity to either of this value still results in the same, unchanged, value of infinity. The same is true of negative numbers and negative infinity (-∞).

We can therefore say that

-∞ < x < ∞

where x is a real number. This says that -∞ will always be smaller than any real value of x, which will always be smaller than ∞.

### Properties of infinity

Although infinity does not act like a real number, it acts fairly similarly with respect to negative and positive values. Below are some of the properties of infinity.

∞ + ∞ = ∞

-∞ + -∞ = -∞

∞ × ∞ = ∞

-∞ × -∞ = ∞

-∞ × ∞ = -∞

x + ∞ = ∞

x - ∞ = -∞

x - (-∞) = ∞

If x > 0

x × ∞ = ∞

x × -∞ = -∞

If x < 0

x × ∞ = -∞

x × ∞ = ∞

### Operations that cannot be defined

Due to its nature, there are some operations that cannot be performed with infinity, because they cannot be defined. Below are said operations:

0 × ±∞

∞ - ∞

∞ ÷ ∞

∞^{0}

1^{∞}