Math Doubts

$1^{\infty}$ form

Form

$1^{\infty}$

Introduction

The number $1$ is a real number and it involves in multiplication in some cases. When the number $1$ is multiplied by itself, the product of them is equal to one.

$1$ $\times$ $1$ $\,=\,$ $1$

In this case, the number of factors is two. Hence, the product of ones can be written as the one raised to the power of two as per exponentiation.

$\therefore\,\,\,\,$ $1^2$ $\,=\,$ $1$ $\times$ $1$ $\,=\,$ $1$

Finite times

Observe the following three examples

$(1).\,\,$ $1^3$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\,=\,$ $1$

$(2).\,\,$ $1^4$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\,=\,$ $1$

$(3).\,\,$ $1^5$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\,=\,$ $1$

The number $1$ can be multiplied by itself as many times as we want but their product is always equal to $1$ in all the cases.

Infinite times

Let us assume that the number $1$ is multiplied by itself infinite times.

$1^\infty$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $\cdots$ $\times$ $1$

According to the above examples, we can think that the one raised to the power of infinity is also equal to one. Logically, it can be true but it is incorrect practically.

The exponent is infinite in this case. It means, the number of times the number $1$ should be multiplied by itself is indeterminate. So, it is not possible to perform the multiplication infinite times.

$\therefore\,\,$ $1^\infty$ $\,=\,$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $1$ $\times$ $\cdots$ $\times$ $1$ $\,\ne\,$ $1$

The value of $1$ raised to the power of infinite is indeterminate and the $1$ raised to the power of infinity is called an indeterminate form in mathematics.

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved