Why is division by zero undefined?

In summary, the conversation discusses the question of why division by zero is undefined and the concept of absolute value when approaching zero. It also mentions that division by zero is a one-to-many operation and does not make sense. The conversation includes an explanation by a mentor and a reference to a sticky thread on the topic.
  • #1
Beer w/Straw
49
0
This question seems to befall everyone at one point or another. So much so I begin to get deliberately silly when it is asked http://www.wolframalpha.com/input/?i=Abs[1/0]

Anyway, I'm wondering if there is a sticky present on these forums that addresses it specifically. Something besides mathworld or wiki.

Thanks.
 
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  • #2
Here's a quick rundown: You can't divide by zero because if you re-arrange the following:
[tex]
\frac{x}{y} = w
[/tex]
Multiplying each side of the equation by y:
[tex]
y\cdot w = x
[/tex]

And we are looking at the case where y=0 and x=anything, then:

[tex]
\frac{3}{0} = w
[/tex]
Which we put as:
[tex]
0\cdot w = 3
[/tex]

So here we are asking, "what number, w, times 0 (zero), will give 3?" (or anything non-zero)

But, any number times 0 must be zero (by definition). So it doesn't make sense to divide by zero. Moreover, this is called a one-to-many operation, because instead of 3 we could have chose any other number, and we would still be in the same boat. So it's not hard to see why we leave division by zero undefined, most of the time it simply does not make much sense.

Here's another explanation by one of our mentors:
https://www.physicsforums.com/showthread.php?t=530207 [Broken]
 
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  • #3
Thanks, I just found that sticky.

My bad for not doing a little searching first.
 
  • #4
Beer w/Straw said:
This question seems to befall everyone at one point or another. So much so I begin to get deliberately silly when it is asked http://www.wolframalpha.com/input/?i=Abs[1/0]
There is not a simple question of 1/0 being undefined. There is no ambiguity to abs(1/0).

The reason 1/0 is undefined is because [itex]\lim_{x\to0} \frac 1 x[/itex] does not exist. It's either +∞ or -∞, depending on the direction from which x approaches zero. Extending to the complex domain, [itex]\lim_{z\to0} \frac 1 z[/itex], doesn't help. Now you get a number with infinite magnitude but unknown direction.

On the other hand, [itex]\lim_{z\to 0} \left|\frac 1 z\right|[/itex] does exist. It is +∞, and this is exactly what Mathematica reports.
 
  • #5


Division by zero is undefined because it violates the fundamental principles of mathematics. In mathematics, division is defined as the inverse operation of multiplication. This means that when we divide a number by another number, we are essentially asking "what number multiplied by the divisor will give us the dividend?"

However, when we try to divide by zero, we run into a problem because there is no number that can be multiplied by zero to give us any non-zero number. In other words, there is no solution to the equation "x * 0 = a" where a is any non-zero number.

This leads to a contradiction and breaks the rules of mathematics. Therefore, division by zero is considered undefined and cannot be performed.

It is important to remember that mathematics is a logical and precise system, and division by zero simply does not fit into this system. It is not just a matter of not having a specific answer, but rather a violation of the fundamental principles that govern mathematical operations.
 

1. What is division by zero?

Division by zero is a mathematical operation in which a number is divided by zero. It is considered undefined and results in an error or infinity.

2. Why is division by zero undefined?

Division by zero is undefined because it violates the fundamental rules of arithmetic. In division, the result is the number that when multiplied by the divisor, gives the dividend. However, there is no number that when multiplied by 0 will give a non-zero result.

3. Can division by zero ever be defined?

No, division by zero cannot be defined. It is a fundamental mathematical concept and changing its definition would lead to inconsistencies and errors in calculations.

4. What happens when you divide a number by a very small number close to zero?

When dividing a number by a very small number close to zero, the result will be a very large number. This is because as the divisor gets smaller, the result gets larger. However, it is still considered undefined and should not be used in calculations.

5. Can division by zero ever be useful?

No, division by zero is never useful. It can lead to incorrect or meaningless results, and should always be avoided in mathematical calculations.

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