Question

For a non-zero integer ‘a’ which is the following is not defined?
A. \[a \div 0\]
B. \[0 \div a\]
C. \[a \div 1\]
D. \[1 \div a\]

Answer 1

Hint: Here, to solve this, we need to know the difference between whole numbers and integers. We are given a non-zero integer and we need to find which is not defined. Thus, by solving them, we will get the final output.

Complete step by step answer:
Given that, ‘a’ is a non-zero integer and we need to find out which is not defined from the given options.

Let’s see what whole numbers and integers are and how they are different from each other.
Before knowing this, we should know that the Natural numbers are all numbers 1, 2, 3, 4… and so on, count and continue on into infinity.

Next, whole numbers are numbers including natural numbers (begin from 1) along with zero.
Example: 0, 1, 2…

Then, the integers are numbers that include all positive and negative numbers along with zero.
Example: -3, 2, 0, 1, -5…

Thus, both integers and whole numbers include zero.

We will solve all the cases given below:
a) \[a \div 0\]
This means that, \[\dfrac{a}{0}\] which is not possible. Because, when a number is divided by zero then the division process does not happen, so the result is not defined.
We can take a real life example for this.
For example: If we want to share one apple to two people, We cut the apple into two parts. Each part of the apple will be equal to $\dfrac{1}{2}$.
Now in our case, we want to cut the apple into zero parts which is practically and theoretically not possible. So, this value can’t be defined.

b) \[0 \div a\]
This means that, \[\dfrac{0}{a} = 0\] which is possible. Because, zero divided by any other number is zero.
c) \[a \div 1\]
This means that, \[\dfrac{a}{1} = a\] which is possible. Because, any number divided by 1 equals to the number itself.

d) \[1 \div a\]
This means that, \[\dfrac{1}{a}\] is equal to the decimal number, which is possible.

Hence, for a given non-zero integer ‘a’, \[a \div 0\] is not defined.

Note:
Here, the nonzero integers are rational integers other than zero. Thus, they have positive absolute value; they may be positive or negative numbers. Integers count numbers in the number line including negative, zero and positive numbers.