Question

Prove that $\dfrac{1}{0}$ = infinity.

Answer 1

Hint: Here, in the given question, we need to prove that $\dfrac{1}{0}$ is equal to infinity. As we can see, here $1$ is divided by $0$. Let us first understand division using an example of $\dfrac{9}{3}$. We know that $\dfrac{9}{3} = 3$, but what does it mean? $\dfrac{9}{3} = 3$ means how many times we can take $3$ out of $9$. We can also write it as: $ 9 - 3 = 6 , 6 - 3 = 3, 3 - 3 = 0 $
as we can see, we can take $3$ out of $9$, $3$ times. That is why, $\dfrac{9}{3} = 3$. Division also means repeated subtraction. Now, we will apply the same concept to prove that $\dfrac{1}{0}$ is equal to infinity.

Complete step-by-step answer:
Given, $\dfrac{1}{0}$
We need to find how many times we can take $0$ out of $1$. Let us find this by using the same concept we used above.
\[\left\{
  1 - 0 = 1 \\
  1 - 0 = 1 \\
  . \\
  . \\
  . \\
  \right.\]
As we can see this subtraction will never end because $0$ is less than $1$. We can do this infinite time. Therefore, $\dfrac{1}{0}$ is equal to infinity.

Note: There is another method also to prove that $\dfrac{1}{0}$ = infinity
Let us prove this using another method also, $\dfrac{1}{0}$ =?
Let us approach zero by decreasing the value of the denominator.
$\dfrac{1}{1} = 1$
$\dfrac{1}{{0.1}} = 10$
$\dfrac{1}{{0.01}} = 100$
$\dfrac{1}{{0.001}} = 1000$
.
.
$\dfrac{1}{{0.0000000001}} = 10000000000$
As we can see, we are still far away from zero but as we are approaching zero by decreasing the value of the denominator the answer of division is increasing or the value of quotient is increasing. Even if we reach zero, we will get answers in large numbers.
From this, we can conclude that $\dfrac{1}{0}$ = infinity.
Remember that $\dfrac{1}{0}$ is also undefined.