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Last updated on July 22nd, 2025

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Division By Zero

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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 for UK Students
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What is 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:

 

  • Dividing any number by 1 always results in the original number, since dividing by 1 doesn’t affect its value. For example, 20 ÷ 1 = 20.

 

  • The result of any number divided by zero is undefined. For example, 50 ÷ 0 = undefined (but 0 ÷ 50 = 0).

 

  • Zero can be considered a whole number, an integer, a real number, and a rational number.

 

  • Dividing any number by itself (except zero) equals one. For example, 62 ÷ 62 = 

 

  • Zero is neither positive nor negative, so it is simply written as +0 or -0. 
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One Divided By 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.

 

  • 1 ÷ 0.1 = 10
     
  • 1 ÷ 0.01 = 100
     
  • 1 ÷ 0.000001 = 1,000,000

 

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.

 

  • 1 ÷ -0.1 = -10
     
  • 1 ÷ -0.01 = -100
     
  • 1 ÷ -0.000001 = -1,000,000

 

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.

Professor Greenline from BrightChamps

What is 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.

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Tips and tricks for Division by Zero

Some students may get confused while working with division by zero. Here are some tips and tricks that will be helpful for the students:

 

  • Type any number ÷ 0 into a calculator, and it shows an error because division by zero is impossible.

 

  • Dividing any number by zero, like 4 ÷ 0, has no value because you cannot break something into zero parts.

 

  • If zero is divided by a non-zero number, like 0 ÷ 4 = 0. For example, zero cookies shared among 4 friends still means nothing for everyone.

 

  • Every division problem can be checked by multiplication. For example, 10 ÷ 2 = 5 because 5 × 2 = 10. But for 10 ÷ 0 = x, no number x can be multiplied by 0 to make 10, so the division is undefined.
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Common Mistakes and How to Avoid Them in Division By Zero

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.

Mistake 1

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Students might think that dividing zero by a very small number (like 0 ÷ 0.001) is the same as dividing by zero.

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These are not the same. Dividing zero by a small non-zero number is valid and equals zero, but dividing by zero (like 0 ÷ 0) is undefined.

 

For example: 0 ÷ 0.001 = 0 → This is valid. 0 ÷ 0 = undefined → This has no valid answer because division by zero is not defined.

Mistake 2

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Some students may assume that 0 ÷ 0 = 0, thinking that dividing zero by anything always results in zero.

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0 ÷ 0 is undefined. Since it is impossible to divide by zero, division by zero remains undefined even when zero is the numerator.

 

For example, if 0 ÷ 0 = x, then x × 0 = 0, which is true for any x. Since many values could work, there's no single answer, so the result is undefined.

Mistake 3

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Students may believe 1 ÷ 0 equals infinity.

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1 ÷ 0 is undefined. Division by zero is an impossible operation; it does not produce infinity.

 

For example, if we assume 1 ÷ 0 = x, multiplying both sides provides x × 0 = 1.

 

However, no matter what x is, x × 0 always equals 0, not 1. This indicates that 1 ÷ 0 is undefined, since no value satisfies the equation.

Mistake 4

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Assuming that dividing by a value extremely close to zero, such as 1 ÷ 0.0001, is similar to dividing by zero.

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Division by a non-zero number, no matter how small, is mathematically valid. However, the result becomes very large as the divisor gets closer to zero. Dividing by zero itself is undefined and not allowed.

 

For example: 1 ÷ 0.0000001 = 10,000,000 — a valid operation with a very large result. 1 ÷ 0 = undefined — not a valid mathematical operation.

Mistake 5

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Thinking a calculator’s error message during division by zero means the answer is a number like infinity.

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The error means the operation is not valid. It doesn’t result in a real number.

 

For example, if you calculate 5 ÷ 0, the calculator might show “Error” or “Undefined.” That doesn’t mean the answer is infinity; it means the operation cannot be done.

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Solved Examples on Division by Zero

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Problem 1

A kid has 12 cookies and wants to divide them among 0 friends. How many cookies does each friend get?

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Undefined

Explanation

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.

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Problem 2

What is 0 ÷ 12?

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0 ÷ 12 = 0

Explanation

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.

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Problem 3

What is 6 ÷ 0?

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Undefined

Explanation

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.

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Problem 4

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?

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Undefined

Explanation

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.

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Problem 5

Annie made 24 caps and wanted to share them equally among 0 kids. How many caps does each kid get?

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Undefined

Explanation

There are no kids to receive the caps. The answer is undefined, since it is impossible to divide 24 caps by 0 kids.

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FAQs on Division By Zero

1.What does division by zero mean?

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2.Is it possible to divide a number by zero?

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3.Why is dividing by zero impossible?

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4.What is division by zero referred to as?

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5.What is the quotient of 0 divided by 9?

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6.How can children in United Kingdom use numbers in everyday life to understand Division By Zero ?

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7.What are some fun ways kids in United Kingdom can practice Division By Zero with numbers?

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8.What role do numbers and Division By Zero play in helping children in United Kingdom develop problem-solving skills?

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9.How can families in United Kingdom create number-rich environments to improve Division By Zero skills?

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Hiralee Lalitkumar Makwana

About the Author

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.

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Fun Fact

: She loves to read number jokes and games.

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