In mathematics, the concept of infinity describes something larger than the natural number. It usually describes something without a limit. You might have done operations with large numbers. Have you ever come across an expression, where infinity is divided by infinity? Is infinity divided by infinity equal to 1?
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 such as natural numbers, whole numbers, integers, and real numbers are referred to as infinite.
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: – $\infty < x < \infty$.
Is Infinity Divided By Infinity Equal To 1?
At first, you may think that infinity divided by infinity equals one. After all, any number divided by itself is equal to one, however, infinity is not a real or rational number.
Let’s find whether $ \frac {\infty}{\infty}$ is equal to $1$ or not.
Method 1:
To do so, let’s assume that infinity divided by infinity is equal to one, i.e., $ \frac {\infty}{\infty} = 1$.
$=> \frac {\infty + \infty}{\infty} = 1$ (Since, $ \infty + \infty = \infty $)
$=> \frac {\infty}{\infty} + \frac {\infty}{\infty} = 1$
$=> 1 + 1 = 1 => 2 = 1$
This equation is obviously incorrect. Therefore, infinity divided by infinity is NOT equal to one. Instead, we can get any real number to equal to one when we assume infinity divided by infinity is equal to one, so infinity divided by infinity is undefined.
Method 2:
Let’s again start with an assumption that infinity divided by infinity is equal to one, i.e., $ \frac {\infty}{\infty}= 1$.
Next, splitting this fraction into two parts, we get
$ \infty \times \frac {1}{\infty} = 1$
First of all, let’s solve the fraction $\frac {1}{\infty}$.
At first, you would think $1$ divided by $\infty$ is equal to $0$, but that is not correct because that would mean $0 \times \infty$ would equal to $1$.
And you can see that it is not true.
However, $1$ divided by $\infty$ does equal a limit approaching $0$. In other words, $1$ divided by $\infty$ does not equal a number or is undefined.
$=> \infty \times$ undefined = $1$
As a result, we reached a dead end. Therefore, infinity divided by infinity is undefined.
What is 1 Divided by Infinity?
Solving $\frac{1}{\infty}$ is the same as solving for the limit of $\frac{1}{x}$ as $x$ approaches infinity, so using the definition of limit, $1$ divided by infinity is equal to 0. Now, we want to know the answer when we divide $1$ by infinity, denoted as $\frac{1}{\infty}$, which we know does not exist since there exists no number that is largest among all. However, if we will use the definition of a limit of a function and evaluate the function $\frac{1}{x}$, where $x$ becomes larger and larger, we will see that the function $\frac{1}{x}$ approaches a particular number.
The following table shows the value of $\frac{1}{x}$, as $x$ gets larger and larger.
| $x$ | $\frac{1}{x}$ |
| $10$ | $0.1$ |
| $100$ | $0.001$ |
| $1,000$ | $0.0001$ |
| $1,000,000$ | $0.0000001$ |
| $1,000,000,000$ | $0.0000000001$ |
| $1,000,000,000,000$ | $0.0000000000001$ |
The above table shows that as $x$ gets larger and larger or as $x$ gets closer and closer to infinity, $\frac{1}{x}$ becomes closer to the value of $0$.
FAQs
What happens if you divide 1 by infinity?
Infinity is not a real number and is only used as a representation of an extremely large real number. Dividing 1 by infinity is equal to zero. In general, any real number divided by infinity is zero, and the quotient of nonzero real numbers that divide infinity is infinity.
What is something divided by infinity?
Something divided by infinity can be written as $x \times \frac{1}{\infty}$, where $x$ is something. Since $\frac{1}{\infty}$ is equal to zero, therefore, something divided by infinity becomes zero.
Is $\infty$ a number?
No. Infinity is not a number. Instead, it’s a kind of number. You need infinite numbers to talk about and compare amounts that are unending, but some unending amounts – some infinities – are literally bigger than others.
Conclusion
Since infinity is just a concept and is not a fixed number, hence the operations like addition, subtraction, multiplication, or division do not give similar results as obtained by other numbers when operated by these operations. Infinity over infinity is Undefined.
