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Chapter 1: Problem 128

Why is division by zero undefined?

Short Answer

Expert verified
Division by zero is undefined because it leads to results that are contradictory and doesn't make sense in a real-world context. Additionally, considering the indeterminate form, it gives ambiguous results. Thus, in mathematics, division by zero is generally considered undefined.

Step by step solution

01

Understanding of Division

Division is the operation where we distribute a certain quantity into equal parts. The divisor refers to the number of equal parts the initial amount should be split into. Here, we can clearly see why division by zero wouldn't make any sense because we can't possibly split a certain quantity into no parts.
02

Potential Outcomes of Division by Zero

To further illustrate the dilemma, let's consider a numeric example. If 2 is divided by zero, there are two potential outcomes: if we follow the belief that any number divided by zero is zero, we would have 0, but there's no way 0 multiplied by 0 can give us the original 2. Conversely, if we follow the belief that anything divided by zero is infinite, we would have infinity, but infinity multiplied by 0 gives us an indeterminate form, not our original 2.
03

Indeterminate Form of Zero

The term 'indeterminate form' refers to expressions involving two functions whose limit cannot be determined solely from the limits of the individual functions. In the context of division by zero, we talk about the indeterminate form when we have a zero divided by a zero or an infinity multiplied by a zero. This only gives ambiguity and confusion, not a numerical value; hence it is termed 'indeterminate'.

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