In this article, we will discuss what is infinity, how to represent it, and what are its examples, types, and different properties of infinity. We will especially discuss different properties of infinity in detail as they are quite helpful in solving various questions in mathematics and calculus. Let us start with the introduction of infinity.

Introduction

In mathematics, the concept of infinity describes something  larger than the natural number. It usually describes something without a limit. This concept is not only used in the mathematics but also in physics. In mathematics and physics, this concept is utilized in a wide range of fields. The term infinity can also be employed for an extended number system.

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What is Infinity?

Infinity is a concept that tells us that something has no end or it exists without any limit or boundary. It indicates a state of endlessness or having no boundaries in terms of space, time, or other quantities. In mathematics, a set of numbers can be referred to as infinite if there is a one to one correspondence between the set and its subset. For instance, y + 2 = y, is only possible if the number y is an infinite number. The addition of 2 will not change the result of this equation.

We can represent an infinite number in another way and that is \frac{1}{x}, where x \rightarrow 0. We can have negative or positive infinity and in terms of a real number x, we can depict it mathematically like this:

- \infty < x < \infty

 

Symbol of Infinity

The infinity symbol is \infty. The infinity symbol is also referred to as a lemniscate sometimes. It was first proposed by English mathematician John Wallis in 1657. The philosophical nature of infinity was under discussion since the time of the Greeks. In the 17th century, when the infinite symbol and infinitesimal calculus were discovered,  mathematicians began working on infinite series.  The L’Hospital’s Rule is often discussed with infinity as it states that when we have an indeterminate form \frac{\infty}{\infty} or \frac{0}{0}, then we can differentiate the numerator and the denominator and take a limit.

Examples of Infinity

You may be wondering what are examples of infinity in mathematics. Well, we have compiled a list of examples related to infinity:

  • A set of whole numbers of natural numbers is an infinite sequence because it is not specified where the set will end. We can represent these sets as:

Natural Numbers = {1, 2, 3, 4, 5, .....}

Whole Numbers = {0, 1, 2, 3, 4, 5, .....}

  • When you divide 10 by 3 you will get the value 3.33... Here the number three represents infinitely or indefinitely
  • A line is composed of an infinite number of points

 

Types of Infinity

The three types of infinity are mathematical, physical, and metaphysical. In mathematics, we use the infinity symbol directly to compare the sizes of sets. In mathematics, infinities occur as the number of points on a continuous line or as the size of the never-ending counting numbers, for instance, 1,2, 3, 4, 5, .... Temporal and spatial concepts of infinity occur in physics when one if one wonders if there are infinitely many stars in the universe.

In the next section, we will discuss some of the properties of infinity.

Properties of Infinity

Addition with Infinity

Infinity Plus a Number

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

\infty \pm k = \infty

Infinity Plus Infinity

Adding infinity to infinity results in infinity.

\infty + \infty = \infty

 

Infinity Minus Infinity

Subtracting infinity from an infinity will result in an indeterminate form:

\infty - \infty = Indeterminate form

Multiplication with Infinity

Infinity by a Number

Multiplying infinity by a non-zero number results in infinity:

\infty \cdot (\pm k) = \pm \infty, if k \neq 0

Infinity by Infinity

Multiplying infinity by infinity will result in infinity.

\infty \cdot \infty = \infty

Infinity by Zero

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

0 \cdot \infty = Indeterminate form

Division with Infinity and Zero

Zero Over a Number

If a number is divided by zero which means that the numerator is zero and the denominator is the number, then the result is zero.

\frac{0}{k} = 0, where k is a non-zero number

 

A Number Over Zero

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

\frac{k}{0} = \pm \infty, where k is a non zero number

 

A Number Over Infinity

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

\frac{k}{\infty} = 0

 

Infinity Over a Number

If a number is divided by infinity, the result is infinity:

\frac{\infty}{k} = \infty, where k is a non-zero number

 

Zero over Infinity

If zero is divided by infinity, the result is 0.

\frac{0}{\infty} = 0

 

Infinity over Zero

If infinity is divided by zero, we will get infinity:

\frac{\infty}{0} = \infty

 

Zero over Zero

Zero divided by zero results in an indeterminate form:

\frac{0}{0} = Indeterminate form

 

Infinity over Infinity

Infinity divided by the infinity results in the indeterminate form:

\frac{\infty}{\infty} = Indeterminate form

 

Powers with Infinity and Zero

A Number to the Zero Power

A number to the power zero is equal to 1.

k ^ 0 = 1, where k is a non-zero number

 

Zero to the Power Zero

Zero to the power zero results in the indeterminate form:

0^0 = Indeterminate form

 

Infinity to the Power Zero

Infinity to the power zero is an indeterminate form:

\infty ^ {0} = Indeterminate form

 

Zero to the Power of a Number

If the power of zero is greater than zero, then the result is zero:

0 ^ k = 0, where k is greater than zero

If the power of zero is greater than infinity, then the result is infinity:

0 ^ k = \infty, where k is less than zero

 

A Number to the Power of Infinity

A number to the power infinity has two scenarios:

If the number is greater than one, then the result is infinity:

k ^ {\infty} = \infty, where k is greater than 1

If the number is greater than zero but less than one, then the result is zero:

k ^{\infty} = 0, 0 < k < 1

 

Zero to the Power of Infinity

Zero to the power infinity is equal to zero:

0 ^ {\infty} = 0

 

Infinity to the Power of Infinity

Infinity to the power infinity is equal to infinity:

\infty ^ {\infty} = \infty

 

One to the Power of Infinity

One to the power infinity results in an indeterminate form:

1 ^ {\infty} = Indeterminate form

 

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Rafia Shabbir