# Division of a complex number by zero

• I
• dyn
In summary, the conversation discusses the concept of dividing by zero in complex analysis. It is stated that in ordinary sense, dividing by zero is not allowed in complex numbers. However, in certain contexts, such as non-standard analysis or Möbius transformations, it is possible to extend the definition of a function in a smooth way where the denominator is zero. The use of the Riemann sphere and the projective complex line is also mentioned. Overall, it is concluded that dividing by zero in complex numbers should be approached with caution and not considered as a general rule.
dyn
Hi
I know that division of a real number by zero is not defined. I just came across the following in a textbook on Complex Analysis by Priestley , " we are allowed to divide a complex number by zero as long as the complex number ≠ 0 "
Is this correct ? What happens if the complex number is purely real ?
Thanks

This is nonsense. Complex numbers are a field. A field consists of two groups, one for addition and one for multiplication. They are related by the distributive law. Zero is no element of the multiplicative group, so the question doesn't even arise.

More context would be helpful here. E.g. if we consider complex functions such as ##f(z)=\dfrac{1}{(z-a)^n}## then we can investigate the kind of the singularity.

If it is a book about non standard analysis, then certain constructions with infinities are possible, but I don't know the details.

In any ordinary sense this statement is false. We are allowed to divide zero by a complex number which isn't zero. This is because of the distributive law:
$$\dfrac{0}{z} = (1-1)\cdot z^{-1} = 1\cdot z^{-1} - 1\cdot z^{-1} = z^{-1} -z^{-1} =0$$

FactChecker
Division by zero is not allowed in complex numbers. If the behavior is good enough, it may be possible to extend the definition of a function in a smooth way where the denominator is 0, but that is not the same as dividing by zero. For instance, we can smoothly extend ##f(z) = (z-1)(z-2)/(z-1)## to equal -1 at ##z=1##. That is actually defining a new function and is not the same as doing an actual division by zero.

Thanks for your replies. They totally make sense. But I found it another book , Complex Analysis by Howie where it states , the convention is c/0 = ∞ where c is a finite complex number.
In both books I found the information in a section on Mobius Transformations and the extended complex plane

This is rather sloppy and should not be used. It is un-mathematical. The same can be done with the reals, but it is not precise.

The Möbius transformation has a singularity, and the values tend to infinity the closer we get to that singularity. However, in complex calculus such singularities are classified and they have different consequences if we e.g. integrate around them. To throw all of them in a container labeled infinity is a bit too easy and creates more problems than it solves.

dyn said:
In both books I found the information in a section on Mobius Transformations and the extended complex plane
We should be careful here. The extended complex plane (aka Riemann sphere ) is not really the same. It has a point for ##\infty##, but that point does not have the usual algebraic properties of complex numbers. For instance, ##0 * \infty## is undefined.

That being said, the Riemann sphere can be useful. Riemann was not a dummy.

FactChecker said:
That being said, the Riemann sphere can be useful. Riemann was not a dummy.
Yes, but we need the projective complex line not the complex plane for it! This makes a big difference.

FactChecker
fresh_42 said:
Yes, but we need the projective complex line not the complex plane for it! This makes a big difference.
I agree. But in the right context that (apparently) the books had, it is not sloppy or non-mathematical. @dyn just needs to be aware that those statements were in a very specific context and not for general use.

Last edited:

## 1. What is the result of dividing a complex number by zero?

The result of dividing a complex number by zero is undefined. This is because division by zero is not a valid mathematical operation and does not have a real solution.

## 2. Can a complex number be divided by zero?

No, a complex number cannot be divided by zero. Division by zero is undefined and not allowed in mathematics.

## 3. Why is division by zero not allowed in mathematics?

Division by zero is not allowed in mathematics because it leads to contradictions and inconsistencies in the mathematical system. It also violates the fundamental concept of division, which is the process of finding the number of times one quantity is contained in another.

## 4. Is there any situation where division of a complex number by zero is defined?

No, there is no situation where division of a complex number by zero is defined. Division by zero is always undefined and not allowed in mathematics.

## 5. What happens when a complex number is divided by a very small number close to zero?

When a complex number is divided by a very small number close to zero, the result will be a very large number, approaching infinity. This is because the smaller the divisor, the larger the quotient will be. However, this is not the same as dividing by zero, which is undefined.

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