What is 1 divided by 0?

In Mathematics, 1 divided by 0 is undefined. Division involves splitting something into equal parts, but dividing by 0 means no parts exist to divide into. Thus, it is conceptually impossible, to break the rules of arithmetic and logic.

Complete Answer:

Let’s carefully go step by step to understand what 1 divided by 0 means and why it’s undefined.

Step 1: Understanding Division

In any division, we deal with:

• Dividend: The number to be divided (here, it’s 1).
• Divisor: The number by which we divide (here, it’s 0).
• Quotient: The result of the division.

The goal of division is to find a number such that:

Step 2: Setting Up the Equation

Let’s assume the result of 1 ÷ 0 is some real number x. Then:

Step 3: Solving the Equation

Key Property of Zero: Any number multiplied by 0 is always 0. For example:

From the equation 1 = 0 × x, we are trying to find a number x such that multiplying it by 0 gives 1. This is impossible, because no real number satisfies this condition.

 

Step 4: Special Cases to Consider

1. If x = 0:

   The equation becomes 1 = 0 × 0 = 0, which is not true. So x ≠ 0.

2. If x is any non-zero real number:

   Any number multiplied by 0 is still 0, not 1. Hence, no real number x works.

3. What if x could be “infinity”?

   In theory, dividing 1 by smaller and smaller numbers (like 0.1, 0.01, 0.001) results in larger and larger values. However, in pure mathematics, infinity is not a real number—it’s a concept. Therefore, division by 0 is still undefined.

 

Step 5: Summary

Why is 1 ÷ 0 Undefined?

Division requires finding a number x such that multiplying it by 0 equals 1. This is not possible in the real number system, as 0 × x = 0 for all values of x.

Therefore, 1 ÷ 0 is undefined in the set of real numbers.

Step 6: A Note on Limits

In calculus, when dividing by numbers very close to 0 (but not exactly 0), the result becomes arbitrarily large. For example:

As the divisor approaches 0, the result approaches infinity. However, this is only valid in the concept of limits, where we say:

In normal arithmetic (without limits), 1 ÷ 0 is still undefined.

 

Final Answer:

Common Maths Questions:

Find the Mean of the First Five Whole Numbers	Explain Zero Factorial
Find the Mode of 10, 12, 11, 10, 15, 20, 19, 21, 11, 9, 10	Degree of the Zero Polynomial is