Question

Which of the following statements are true?
$0\div 0=0$
A. True
B. False

Answer 1

Hint: We need to tell if $0\div 0$ is equal to 0 or not. Here, we will make use of the fact that any defined number multiplied by 0 is also 0 and any defined number divided by 0 is not defined. Here, we will write $0\div 0$ as $\dfrac{0}{0}$ and then $\dfrac{0}{0}$as $0\left( \dfrac{1}{0} \right)$ . Then we will determine if this is definite or not. If this number will be not defined, then the statement will be false as 0 is defined and if the number will be defined then we will check if it will or will not be equal to 0.

Complete step-by-step solution
We know that any defined number multiplied by 0 is 0 and any defined number divided by 0 is not defined. Here, we have that 0 is divided by 0. Thus we can write it as:
$\dfrac{0}{0}$
Now, we have to tell if this is equal to 0 or not.
Now, we mentioned above that any defined number divided by 0 is not defined and any defined number divided by 0 is not defined.
Now, that means that if any undefined number is multiplied by 0, the result is also not defined not 0.
Now, we can write $\dfrac{0}{0}$ as:
\[\begin{align}
  & \dfrac{0}{0} \\
 & \Rightarrow 0\left( \dfrac{1}{0} \right) \\
\end{align}\]
Now, 1 is defined but $\dfrac{1}{0}$ is not defined. Thus, $0\left( \dfrac{1}{0} \right)$ ,i.e. $\dfrac{0}{0}$ is also not defined as $\dfrac{1}{0}$ is not defined and 0 multiplied by any not defined number is also not defined.
Thus, $\dfrac{0}{0}\ne 0$
Hence, the given statement is false.
Thus, option (B) is the correct option.

Note: Here, we have both the numerator and the denominator exactly equal to 0. That’s why the concept of not defined and defined numbers is used. If both the numerator and the denominator were tending to be 0, then the answer would be 1. Thus, take care if there’s a limit or it is exactly equal to 0.