Infinity: Symbol, Value, Types, Properties

Infinity, in mathematics, describes something larger than the natural number. It usually describes something without a limit. This concept is useful not only in mathematics but also in physics. The concept of infinity can be useful in extended number systems, sets, calculus, etc. There are various properties of infinity. Mathematical operations like addition, subtractions, multiplications, exponents can be performed with infinity.

Keywords: Infinity, limits, relations, Sets, Natural numbers, whole numbers, subsets, number series, rational numbers

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What is Infinity?

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Infinity is a concept that tries to express that something has no end or exists without any boundary or limitation. In mathematics, a set of numbers is infinite if there is a one to one correspondence between the set and its subset. For example, x + 3 = x, is only feasible if the number x is an infinite number. The addition of 3 is not going to change the result of this equation.

An infinite number can be represented in another way and like, 1/x , where x→0 . Infinity can be both negative or positive and in terms of a real number x, we can represent it like:

-∞ < x < ∞

Discover about the Chapter video:

Continuity and Differentiability Detailed Video Explanation:

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Symbol of Infinity

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The infinity is represented by a sign ∞ . The infinity sign is also called a lemniscate sometimes. It was first proposed by English mathematician John Wallis in 1657. The philosophical concept of infinity was under debate since the time of the Greeks. In the 17th century, when the infinite symbol and infinitesimal calculus were invented, mathematicians began studying infinite series. The L’Hospital’s Rule is frequently deliberated with infinity as it states that when we have an indeterminate form like ∞ , then we can differentiate the numerator and the denominator and take a limit.

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Examples of Infinity

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• A set of whole numbers of natural numbers is an infinite sequence since it is not stated where the set will end.
  Natural Numbers = {1, 2, 3, 4, 5, 6, 7, 8, 9, ....}
  Whole Numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ....}
• When we divide 10 by 3, we get the value 3.3333... which doesn’t end.
• A line segment is made up of an infinite number of points.

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Value of Infinity

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Infinity is not a real number; it has no value. The reason is, a value needs to be defined and must be specific which is impossible in the case of infinity. The reality is infinity is not specific.

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Types of Infinity

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There are three types of infinity: mathematical, physical, and metaphysical. In mathematics, the infinity symbol is used directly to compare the sizes of sets. In mathematics, infinity exists in the number of points on a line segment or in the size of the never-ending counting numbers, like, 1,2, 3, 4, 5, ... In physics, temporal and spatial concepts of infinity exist, like the infinite universe.

Alternatively, infinity can also be classified as:

• Potential infinity
• Actual infinity

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Properties of Infinity

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Addition with Infinity

• Infinity Plus a Number

If infinity is added to or subtracted from a number, the result is infinity.

x + ∞ = ∞

• Infinity Plus Infinity

Adding infinity to infinity we get infinity.

∞ + ∞ = ∞

Subtraction with Infinity

• Infinity Minus Infinity

Subtracting infinity from infinity, we will get an indeterminate form:

∞ – ∞ = Indeterminate form

Multiplication with Infinity

• Infinity by a Number

Multiplying a non-zero number with infinity, we get infinity:

∞ × k = ∞ , if k≠0

• Infinity by Infinity

Multiplying infinity by infinity, we get infinity.

∞ × ∞ = ∞

• Infinity by Zero

If infinity is multiplied by zero, we will get an indeterminate form:

0 × ∞ = Indeterminate form

Division with Infinity and Zero

• Zero Over a Number

If any number is divided by zero, in other words, the numerator is zero and the denominator is any number, we get zero.

0 / k = 0, where k is a non-zero number

• A Number Over Infinity

If a number is divided by infinity, i.e., the numerator is number and denominator is infinity, the result is zero:

k / ∞ = 0

• Zero over Infinity

If zero is divided by infinity, we get 0.

0 / ∞ = 0

• Infinity over Zero

If infinity is divided by zero, the result is infinity:

∞ / 0 = ∞

• Zero over Zero

If zero is divided by zero, we get an indeterminate form:

0 / 0 = Indeterminate form

• Infinity over Infinity

If infinity is divided by infinity, i.e., infinity in the numerator and infinity in the denominator, an indeterminate form is obtained:

∞ / ∞ = Indeterminate form

Powers with Infinity and Zero

• A Number to the Zero Power

A number to the power zero results in 1.

k⁰ = 1 , where k is a non-zero number

• Zero to the Power Zero

Zero to the power zero gives us an indeterminate form

0⁰ = Indeterminate form

• Infinity to the Power Zero

Infinity to the power zero results in an indeterminate form:

∞⁰ = Indeterminate form

• Zero to the Power of a Number

If the power of zero is greater than zero, then what we get is zero:

0^k = 0, where k is greater than zero

• A Number to the Power of Infinity

A number to the power infinity has two situations:

If the number is greater than one, we get infinity:

k^∞ = ∞, where k is greater than 1

If the number is greater than zero but less than one, we get zero:

k^∞ = 0, 0<k<1

• Zero to the Power of Infinity

Zero to the power infinity results in zero:

0^∞ = 0

• Infinity to the Power of Infinity

Infinity to the power infinity gives infinity:

∞^∞ = ∞

• One to the Power of Infinity

One to the power infinity gives an indeterminate form:

1^∞ = Indeterminate form

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Things to Remember

• In Calculus, Leibniz was the first to derive and use infinite numbers in mathematics.
• Set made up of all points in a line segment is an infinite set.
• As infinity is not a number, it is not an integer, neither rational nor irrational.
• Infinity is neither odd nor even.
• Infinity is not a number; it can never be a real number and if it is not a real number it cannot be represented as a complex number either.

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