Dividing by Zero=undefined or complex infinite?

In summary, evaluating cot(pi) results in undefined because you cannot divide by zero. However, in complex analysis, the answer is considered to be complex infinity, represented by a point on the Riemann sphere. This concept is often covered in advanced math courses and can be confusing for those who have not studied complex variables. Ultimately, the answer is undefined and it is important to understand the concept behind complex infinity before using it as an answer.
  • #1
physicsdreams
57
0
A few days ago, I had a problem that looked like this:

evaluate cot(pi)

I know that on the unit circle, cot(pi) ends up as -1/0. In my precalc class, we say that this is undefined because you can't divide by zero.
I decided to plug the problem into wolfram and it tells me that there is in fact an answer, that being complex infinity.
I'm not sure what to make of this as I've never heard of complex infinity.
Am I wrong to say that cot(pi), or any other number divided by zero is undefined, or is the correct answer complex infinite?

Thanks!
 
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  • #2
physicsdreams said:
A few days ago, I had a problem that looked like this:

evaluate cot(pi)

I know that on the unit circle, cot(pi) ends up as -1/0. In my precalc class, we say that this is undefined because you can't divide by zero.
I decided to plug the problem into wolfram and it tells me that there is in fact an answer, that being complex infinity.
I'm not sure what to make of this as I've never heard of complex infinity.
Am I wrong to say that cot(pi), or any other number divided by zero is undefined, or is the correct answer complex infinite?

Thanks!

The correct answer is undefined. When you take complex analysis, the correct answer is complex infinity :-)

Here's a page that reveals all.

http://en.wikipedia.org/wiki/Riemann_sphere
 
  • #3
SteveL27 said:
The correct answer is undefined. When you take complex analysis, the correct answer is complex infinity :-)

Here's a page that reveals all.

http://en.wikipedia.org/wiki/Riemann_sphere

Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus.

Thank you
 
  • #4
physicsdreams said:
Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus.

Thank you


Hey physicsdreams.

A complex number is written in the form of z = a + bi where a and b are just real numbers.

The infinite-complex number is just a number that has an infinite 'length'. We define the 'length' of a complex number to be SQRT(a^2 + b^2).

Basically if you look at the Riemann-Sphere wiki that was posted above, this 'infinite' complex number is at the point where the 'north pole' is, and the complex number that is 'zero' (i.e. z = 0 + 0i = 0) is at the south pole.
 
  • #5
physicsdreams said:
Would it be possible for you to explain this in laymen's terms, seeing as I am only in precalculus.

Thank you

You can visualize the complex numbers as a sphere with zero at one pole and "complex infinity," a symbolic extra point, at the other pole. When you do this, you can make sense of saying that the function 1/z takes the value complex infinity at z = 0. That's where Wolfram is getting its answer from. You're not really dividing by zero, but rather taking the limit of a complex function as the function's value approaches the north pole in the complex sphere.

This is somewhat advanced math, typically taken by undergrad math majors after a couple of years of calculus and a class in real analysis. It would never be accurate to say you can divide by zero. Perhaps Wolfram should do a better job of explaining what they're doing so as not to confuse people who haven't taken a course in complex variables.
 
  • #6
Thank you all for your explanations.

Hopefully I'll gain a better understanding of this advance concept in the future.
 
  • #7
It's undefined when working with the reals. I don't think there's any reason to get into anything more complex at this point (certainly not complex analysis).
 
  • #8
all it means is that as t-->pi, then cot(t)--> infinity.
 

FAQ: Dividing by Zero=undefined or complex infinite?

1. What does it mean to divide by zero?

Dividing by zero means attempting to divide a number by zero, which is mathematically undefined or impossible.

2. Why is dividing by zero undefined?

Dividing by zero is undefined because it violates the fundamental principles of mathematics, including the commutative, associative, and distributive properties. It also leads to contradictions and inconsistencies in mathematical equations.

3. Can dividing by zero ever be defined?

No, dividing by zero cannot be defined because it would require redefining the entire field of mathematics and fundamentally changing the way we understand numbers and operations.

4. What is the result of dividing a number by a very small number close to zero?

The result of dividing a number by a very small number close to zero is a very large number. As the divisor gets closer to zero, the quotient approaches infinity. This is known as an infinite limit and is different from dividing by zero.

5. Are there any real-world applications of dividing by zero?

No, there are no real-world applications of dividing by zero. In real-life scenarios, dividing by zero is impossible and does not have a meaningful interpretation. It only exists as a mathematical concept used to prove theorems and highlight the importance of defining operations.

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