Last updated on July 22nd, 2025
Dividing any number by zero is undefined because no number can be multiplied by zero to get the original number. “Division” refers to breaking something into equal parts, while zero represents the absence of quantity. Zero is a unique number since it is even and lies between negative and positive numbers, yet is neither positive nor negative. Let’s learn more about division by zero.
Division by zero is undefined in mathematics. When zero appears as the denominator, written as a/0, where ‘a’ is any number. Because no number multiplied by zero results in ‘a’, if a ≠ 0. Dividing zero by a non-zero number (like 0 ÷ 5) equals zero, but the reverse isn't true. For example, 8 ÷ 0 = x or x × 0 = 8. Here, x has no value that can make this equation true. Therefore, division by zero is undefined.
Here are some facts about division and zero:
Any number divided by zero is undefined. For example, 1 ÷ 0 does not have an answer. If we divide 1 by a number close to zero, like 0.1, we get a large number, but division by zero itself is not possible, as it does not yield a valid result.
When you divide 1 by smaller and smaller positive numbers, the result increases and approaches positive infinity, whereas dividing 1 by smaller and smaller negative numbers gives results that decrease toward negative infinity.
When you divide 1 by a small negative number, the result becomes a large negative number and approaches negative infinity. However, dividing by zero itself has no answer, which makes it undefined.
In mathematics, when something is called undefined, it means there is no valid or meaningful answer. When we do a mathematical calculation that defies logic, such as dividing by zero, we maintain that the result is undefined. For example, dividing by zero is undefined because it breaks the rules of arithmetic, where no number can make the equation work.
Some students may get confused while working with division by zero. Here are some tips and tricks that will be helpful for the students:
Understanding the concept of division by zero helps students recognize when a mathematical operation is not possible. However, they often make mistakes when working with division by zero. Here are some common errors and helpful solutions to avoid these mistakes.
A kid has 12 cookies and wants to divide them among 0 friends. How many cookies does each friend get?
Undefined
Division by zero is undefined because there is no one to divide or share with; it's like trying to share the cookies when there is no one to receive them.
What is 0 ÷ 12?
0 ÷ 12 = 0
When zero is divided by any non-zero number, the result will always be zero. In this case, since 12 is a non-zero number, dividing 0 by 12 gives 0.
What is 6 ÷ 0?
Undefined
By dividing 6 by 0, you are trying to divide 6 into 0 groups. The answer is pointless because zero groups are impossible. For this explanation, 6 ÷ 0 is not defined.
A teacher creates 30 homework tasks that a group of students should solve. However, if there are 0 groups formed, how many tasks would each group receive?
Undefined
Since there are no groups, it is impossible to divide the homework evenly. Since the solution is undefined, the division by zero is also undefined.
Annie made 24 caps and wanted to share them equally among 0 kids. How many caps does each kid get?
Undefined
There are no kids to receive the caps. The answer is undefined, since it is impossible to divide 24 caps by 0 kids.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.