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There is no way to find the square root of a negative number. It just doesn't work. So the answer to the impossible question, "What number squared equals a negative number?" is just said to be $i$, an imaginary number.

So now let's look at a different problem. What is one divided by zero? Of course, you can't answer that question. It doesn't make any sense. Splitting up a chocolate bar to a group of 0 friends isn't possible. It's like being told to walk north when standing on the north pole. So, what if just like $i$, we just say $1/0$ is an imaginary number, referenced by the letter $o$?

What applications would this have? Does it even work? The number $i$ has real life applications, but it can also be used to create abstract designs like the Mandelbrot Set. Could $o$ even be used for an abstract purpose?

NoRt9001
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    i don't think it is possible . Even if we assume, 1/0 = j, some number, then 2/0=4/0 would also be j and j will be inconsistent. – Kiran Oct 16 '16 at 04:09
  • You can do something sort of like this with hyperreals. Conceptually, you construct for every real number $a$, $a + \epsilon$ is a hyperreal that is "infintesmally close to" $a$, and $\epsilon$ is infintesmally close to 0. You can then extend the usual arithmetic to work with these infintesmals. Then for every infintesmal $\epsilon$, $\frac{1}{\epsilon}$ is an infinite number (larger than any finite number). Application wise, you can use infintesmals to prove theorems in introductory calculus. –  Oct 16 '16 at 04:15
  • See the point at infinity on the Projectively extended real line. Note however that even in that context $\infty$ does not follow all the usual algebraic rules, for example $\infty+\infty, 0 \cdot \infty, \frac{0}{0}$ are undefined. – dxiv Oct 16 '16 at 04:17
  • See also this. –  Oct 16 '16 at 04:23
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    It's worth noting that $i$ has LOTS of real life applications in electricity and magnetism. – A. Thomas Yerger Oct 16 '16 at 04:23
  • @AlfredYerger I wasn't aware of that, thanks for bringing that to my attention! – NoRt9001 Oct 16 '16 at 04:24
  • Many algebraic rules break down if you consider naive division by zero. If you focus on the geometric aspects and consider functions formed by operations, however, they turn out to mix perfectly with infinity and even division by zero cen be appropriately interpreted to give an elegant theory. To be more technical, rational functions on the complex plane extends to holomorphic maps on the Riemann sphere $\hat{\Bbb{C}}$. – Sangchul Lee Oct 16 '16 at 04:25
  • If you want an algebraic structure that gives a meaning to "$\frac10$" you can have a look at the bibliography of this wikipedia article on wheel theory. –  Oct 16 '16 at 15:08

2 Answers2

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Fundamentally, complex numbers were used because they arose naturally in polynomial roots. However, division by zero has no such purpose. In fact, if division by zero was defined, then $(0\cdot 1)\frac 10=(0\cdot 2) \frac{1}{0}\implies 1 = 2.$ (Oops). However, if you are willing to forego standard properties, then you get an algebra called a wheel. You can learn more about it here: https://en.wikipedia.org/wiki/Wheel_theory

Teoc
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  • I figured that something like this would come up. I was just wondering about the "rules" in math that we just ignore when they are broken. Thanks for the quick reply! – NoRt9001 Oct 16 '16 at 04:14
  • @NoRt9001 If you found my answer helpful, then please click the check mark on the left side of the answer. This lets people know that this question was answered, and also will remove the question from the unanswered list. – Teoc Oct 16 '16 at 04:17
  • Will do, I asked this question just under 15 minutes ago, so the accepted answer checkmark is still locked. Don't worry, any second now it'll send your delicious reputation. – NoRt9001 Oct 16 '16 at 04:19
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"The number $i$ doesn't have many real life applications" is a very false statement. Indeed, much of electrical engineering relies on complex numbers. The solution to differential equations (both ordinary and partial) also often involves complex numbers, of which we find applications for in nearly every area of science and engineering.

That's besides the point though. It's an interesting thought you raise. It seems that we get some very nice structure (quite beautiful to be honest) and interesting properties out of this set that we call the complex numbers. Naturally, you might be compelled to ask - or even expect - to get the same thing out of the other famous "undefined operation' - division by zero. As it turns out, however, there's no number system in which division by zero is defined. The process breaks down almost immediately. There's a quite popular area that deals with stuff like this called "mathematical fallacy" (see https://en.wikipedia.org/wiki/Mathematical_fallacy), but here's a quick way of showing why division by zero leads to nonsensical answers: $$\text{Let } a=1=b$$ $$\Rightarrow a^2=ab$$ $$\Rightarrow a^2 - b^2 = ab - b^2$$ $$\Rightarrow (a-b)(a+b) = b(a-b)$$ $$\Rightarrow a+b = b \text{ (observe that we divided by } (a-b)=(1-1)=0 \text{)}$$ $$\Rightarrow 1 + 1 = 1$$ Once you have this it becomes hard to make a case for any kind of benefit for defining such an operation. It should also be noted that in more serious mathematics, the complex numbers tend to arise quite naturally, with the definition $\sqrt{-1}=i$ being completely done away with. This post is already quite long, but here's a starting point for research if you want an example of how to do this: Can the complex numbers be realized as a quotient ring?. Hope that helps!

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AlkaKadri
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  • Thank you for explaining things better. I now realize i's importance in real, physical applications. This kind of math has always interested me. It really makes you wonder if humans are inventing math, keeping what works and throwing out (or just ignoring) inconsistencies, or discovering laws and patterns just created by nature. – NoRt9001 Oct 16 '16 at 04:35