Imaginary Numbers – Defining i with the Property i^2 = -1

abstract-algebra

So I'm a computer science graduate and I'm trying to understand imaginary numbers.

What I found is that it is often said: $i$ is the number defined such as $i^2=-1$, but this confuses me, as multiplication was defined as operation on real numbers, so how is it used now with imaginary or complex numbers?

My rough understanding is something along the lines: multiplication is an abstract operation,so we will just allow it for some abstract object we invent and call it $i$, and define $i*i$ as $-1$.

They say stuff like this is enlarging the real number system, but I don't fully understand this process.

I have studied calculus(single and multi-variable), linear algebra, probability and discrete mathematics in college. I just want to understand all these stuff of enlarging a number system, regarding multiplication as abstract operation performed on abstract objects, instead of thinking of it as repeated addition, etc..

I have searched the table of contents of many abstract algebra books, but failed to find what I want, may I just don't know what to search for and where.

Best Answer

It seems to me your problem is not with the imaginary numbers per se, but with the stipulation that $i$ is such an abstract object that $i^2=-1$.

And yes, as mentioned in the comments, you can just define $\mathbb C:=\mathbb R^2$ with peculiarly defined addition/multiplication, then set $i:=(0,1)$ and verify that it satisfies $i^2=-1$, but this is a different approach and it still does not make it clear why can we take some abstract $i$ out of thin air and impose $i^2=-1$ on it.

The proper way to explain that particular wording is based only on the addition/multiplication operations.

Let's start simple with $\mathbb N_0=\{0,1,2,\ldots\}$ with the addition (+) operation. As you have a CS background, you can imagine it in terms of OOP as a class with a member function "plus".

What is a negative one (and $\mathbb Z$), then? It's such an entity that sums with $1$ to zero. We can take an arbitrary object $\xi$ (denoted by $(-1)$) outside $\mathbb N_0$ and try to define a new set (class) $\mathbb N_0 \cup \{ \xi \}$ with a new overloaded (in OOP terms) function "plus", which acts just like $\mathbb N_0$'s "plus" on $\mathbb N_0$, but for $\xi + a$ returns the element previous to $a$.
You should now immediately notice that this "function" is undefined at $\xi+\xi$, because that should sum to 0 when added to 2, while no such element exists in the set (class) we're working - $\mathbb N_0\cup\{\xi\}$. To make it work, we have to add another artificial element from thin air - denoted $(-2)$ - to the set and define $(-1)+(-1):=-2$. But that still does not work, because now $(-2)+(-1)$ is undefined on $\mathbb N_0\cup\{-1,-2\}$. As you notice, we can go like this for arbitrarily long time and never get a properly defined "plus" function, which extends $\mathbb N_0$'s "plus" (i.e. acts like the normal plus on the $\mathbb N_0$ subset) and satisfies $(-1)+1=0$.

So what we want is an extension of $\mathbb N_0$ which supports a total function "plus" in the sense described above, but nothing more! There are a lot of these, but there's a unique -- up to isomorphism (of monoids, but let's not go there) -- "smallest" one, that we denote by $\mathbb Z$. I have no space to delve into details here, but I hope you can agree that the set you intuitively know as $\{\ldots,-2,-1,0,1,2,\ldots\}$ works here - it's "plus" function works like $\mathbb N_0$'s "plus", satisfies $(-1)+1=0$, and you cannot take any elements from $\mathbb Z$ without breaking these properties.

Now let's return to $i$. By analogy with above, here we have a set $\mathbb R$, but now the key operations include not only "plus", but "multiply" too.

Again, take an arbitray object outside $\mathbb R$ (formally, you can even take the set $\mathbb R$ itself!) and denote it by $i$. Construct the set $\mathbb R\cup\{i\}$ and try to define "plus" and "multiply" on it so that they coincide with the respective functions on $\mathbb R$, but also satisfy $i\cdot i=-1$.

You'll immediately notice that you need to add an element $2\cdot i$ to the set in an attempt to make the function "multiply" total, just like you had to add $(-1)+(-1)$ to $\mathbb N_0$ to get to $\mathbb Z$. Similarly, you'll have to add $-1$, because the multiplication function has to be defined on $i$ and $(-1)$, too.

The "smallest" set containing $\mathbb R$, on which "plus" and "multiply" are total functions restricting to the standard real addition and multiplication, with the additional condition that $i\cdot i=-1$, exists, and is denoted by $\mathbb C$.

To continue the OOP analogy, this is now only an "interface" - it just asserts that the object has methods satisfying some properties. To get an "implementation" of it, you have multiple options:

  1. Probably the easiest one for you is to use $\mathbb C:=\mathbb R \times \mathbb R$ and define $(+),(\cdot)$ in the familiar way.
  2. Another option (as mentioned in another answer) is to start with the polynomials $\mathbb R[x]=\{a_0+a_1x+\cdots+a_nx^n:n\in\mathbb N,a_i\in\mathbb R\}$ and consider any two of them equal if they differ by a factor of $x^2+1$.

But you didn't ask about the implementation, I feel, you asked about the interface.

Related Question