Division by 0: A Puzzle That Has Baffled Us for Years

In summary: As you subtract smaller and smaller amounts, you get closer and closer to zero, but never reach it. In this sense, it can be thought of as infinitely far away. However, this is just a concept and not an actual number, which is why division by zero is undefined in most mathematical systems.In summary, division by 0 is undefined because the process of dividing by 0 never reaches a specific number or value. While in certain mathematical systems, such as the Riemann Sphere, it is possible to define division by 0 as infinity, this can result in complications and break desirable properties in other operations. Therefore, in most mathematical systems, division by 0 is left undefined.
  • #1
Nick_85
3
0
Hello
This is question is bugging me for years.
Why is division by 0 undefined and not multiplication by 0 undefined?
If any number multiplied by 0 is 0 then the logical answer for any number divided by 0 would be ∞ (for me at least).

If you "define" N/0=∞ and N*0=undefined then there are no contradictions. Did the mathematicians had to choose between n/0=undefined and n*0=undefined?

What about this:
n*0=0
0/0=n special rule
n/0=∞
∞*0=n special rule

Sorry for my poor english, I am not a native speaker.
 
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  • #2
If you allow that multiplication by 0 gives 0 this produces no difficulties of any kind.

If you allow division by 0 then you can arrive at mathmatically impossiblities (1=0, for example)
 
  • #3
phinds said:
If you allow that multiplication by 0 gives 0 this produces no difficulties of any kind.

If you allow division by 0 then you can arrive at mathmatically impossiblities (1=0, for example)

My question is :if you allow division by 0 (=∞) and not allow multiplication by 0, what then? do you arrive at mathmatically impossiblities like your example?
 
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  • #4
You can define division by zero any way you want. But most definitions of division by zero turn out to be fairly useless and cause more problems than they solve.

In complex analysis, one sometimes studies the set ##\mathbb{C}^*=\mathbb{C}\cup\{\infty\}##. In this setting, functions like ##f(z)=\frac{1}{z}## are defined everywhere - with ##f(0)=\infty## and ##f(\infty)=0## - and it looks like we're permitting division by ##0##. But then the algebraic structures involved - in regards to ##\mathbb{C}^*## and in the functions mapping ##\mathbb{C}^*## into itself - start to get a little more complicated. For instance you need to invent special rules for handling multiplication by ##\infty## now. The topological structures are also more complicated. Everything gets more complicated just so a relatively small class of "kinda bad" functions can get lumped in with the "good" ones.

As far as division between reals/rationals/integers (and complex numbers for that matter) goes, those divisions are typically defined in terms of multiplication; i.e. ##x\div y=z \Leftrightarrow x=z\times y## is a(n) (incomplete) definition for division rather than a rule derived from a definition. So from a strictly algebraic standpoint, which I'm guessing is where you're coming from, there is no way to define division by ##0## in a consistent way that actually looks like the way division is defined for the other numbers.
 
  • #5
This FAQ may help(It was very helpful to me)
https://www.physicsforums.com/showthread.php?t=530207 [Broken]
 
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  • #6
editing
 
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  • #7
Look very simply at what DIVISION means. Start with a number, and then subtract another number (divisor) from it, adjusting the Start number after each subtraction, until no more of the divisor can be subtracted. If your divisor is zero, then the process will never be finished. The result is not a number; it is not a variable; it is not any value.
 
  • #8
It is undefined in the reals. There is no real number x that satisfies 1/0 = x within that arena. Your idea that 1/0 = inf is actually used in many other arenas.

This is completely well-behaved, provided the other definitions of operations regarding infinity are followed.

Take a look at the Riemann Sphere.
http://en.m.wikipedia.org/wiki/Riemann_sphere

There is no contradiction that can be found by saying that 1/0 = inf in the Riemann Sphere. Other operations on infinity do seem impossible to make work (while maintaining valuable properties of operations on the set, that is). For example, I personally know of no example where inf-inf is defined, because this would break many desirable properties.

Basically, the results of arithmetic depend upon the "world" we are working in, and the world is our creation. Although 1/0 is not infinity in the reals with addition and multiplication in the normal way, there is nothing wrong with thinking about other "worlds" where infinity is a number, and 1/0 = inf, and seeing what happens.

symbolipoint said:
If your divisor is zero, then the process will never be finished.

This only supports his intuition regarding infinity.
 

What is division by 0 and why is it considered a puzzle?

Division by 0 is a mathematical operation in which a number is divided by 0. This operation is undefined and has no real solution. It is considered a puzzle because it challenges our understanding of basic mathematical principles and has no clear explanation.

Why is division by 0 impossible?

Division by 0 is impossible because it violates the fundamental rule of division, which states that any number divided by 0 is undefined. In other words, there is no number that can be multiplied by 0 to give a specific result. Therefore, division by 0 has no solution.

Can division by 0 ever have a real solution?

No, division by 0 can never have a real solution. This is because it would require a number to be both 0 and not 0 at the same time, which is mathematically impossible. Division by 0 will always result in an undefined or infinite value.

Are there any exceptions to the rule of division by 0?

No, there are no exceptions to the rule of division by 0. This rule is a fundamental principle in mathematics and applies to all numbers, regardless of their value or context.

What are the practical implications of division by 0?

The practical implications of division by 0 are that it can lead to errors and incorrect calculations in mathematical equations. It is also used in various fields, such as physics and engineering, to represent undefined or infinite values. Additionally, understanding the concept of division by 0 can help us better understand and solve complex mathematical problems.

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