For complex number z = 3 - 2i, why is Im(z) -2 and not -2i?

Question

The general part of the mathematical area of complex numbers is having me stumped, and I feel like as soon as I understand this, it may broaden my understanding of every consequent part of this topic (that I'm currently studying at A-Level).

For a given complex number z, where z = 3 - 2i (for example), why do we express Im(z) as -2 as opposed to -2i? And if we did express Im(z) as -2i, what sort of effect would this have on calculations?

Answer 1

It's the definition. It can be what we want. And what we want when we say "Re(3-2i)" and "$Im(3-2i)$" is not the "real component" or "complex component" which would be "the real number $3$" or "$2\sqrt{-1}$". If we wanted those we could have them and call them anything we want.

But we don't want them. We want something else that answers this entirely different question: A complex number is composed be affixing a real number component and another real number times $\sqrt{-1}$--- what are the two real numbers that we use to create the complex number?

Your question is a bit like asking why when someone ask what your last name is why we say "Ford" and not "Mr. Ford". After all, we don't want people calling us "hey, Ford!"-- we want people calling us "hey, Mr. Ford". Well.... fine... but that wasn't the question. We didn't ask "how do we refer to you by your last name"; we asked "What is your last name".

If we actually go into how the complex numbers are defined it might make more sense.

Saying $z = 3 - 2\sqrt{-1}$ is a bit of a handwaving cheat; not so much because it is wrong but because it isn't really defined. We can't wave our hands in the air and say "yesterday there was no such thing as $\sqrt{-1}$ and today there is. Poof. We call it $i$."

The real definition is:

A complex number is a two dimensional pair of real numbers $(a,b)$ with a multiplication operation defined as such $(a,b)\times(c,d) = (ac - bd, bc + ad)$ (and the addition operation is the more intuitive $(a,b) + (c,d) = (a+b, b+d)$.

Off hand neither number of the pair is more "real" or "imaginary" than the other. The only difference is $a$ is the first of the pair, and $b$ is the second of the pair. And we call the first of the pair $Re(z)$ and we call the second of the pair $Im(z)$. And that's all they mean.

Now, the student should be shouting: Wait, WHY do we call them Real and Imaginary parts if we only mean "first" and "second" parts.

Well, because if we take the set of all complex numbers where the second part is strictly $0$ then we get that algebraically that set is completely equivalent to the real numbers. That is $(a,0) + (c,0) = (a + c, 0)$ and $(a,0)\times (b,0) = (ab - 0, a*0 + b*0)= (ab,0)$. so if we identify $b \equiv (b,0)$ we, in essence have the real numbers as a subset of the complex numbers.

So the complex numbers is a set of pairs of real numbers. The first of the pair is called the "real" part because it by itself (if the other part is $0$) acts as a real number. We call the second of the pair the "imaginary" part because... well, the opposite of real is imaginary.

In notation we have $(a,0) \mapsto a \in \mathbb R$ and we have $(0,b)^2 =(0, b)\times (0,b) = (0*0 - b*b,0*b + b*0) = (-b^2,0)\mapsto -b^2$ we realize we can claim $(0,b)\mapsto \sqrt {-b^2} = b\sqrt{-1}$. We call $(0,1) \equiv \sqrt{-1} = i$ and as $i$ and $1$ can't mix (like oil and water) we can use the notation $(a,b) \mapsto a + bi$ without any inconsistencies.

This allows us to not just say that $(a,0)$ is equivalent to $a$ and $\{(a,0)|a \in \mathbb R \}$ is equivalent to $\mathbb R$. We can say $(a,0) = a + 0*i = b$ IS $a$. And $\{(a,0)|a \in \mathbb R \}$ IS $\mathbb R$.

And although the imaginary component of $3-2i$ is $2i$ the imaginary part is the real value we put in the component; $-2$.

And we note that if we take the set of all the complex number where the second number in the pair is

Answer 2

If $z=x+iy$ when $x,y\in\mathbb{R}$ then by definition $Re(z)=x, Im(z)=y$. The reason? Because we need $y$ much more than $yi$, that's it. Using the real and imaginary part as I just defined them we can easily define norm, argument and so on.

Answer 3

By definition, when we write a complax number $z$ as $a+bi$, with $a,b\in\mathbb R$, then the real part and the imaginary part of $z$ are $a$ and $b$ respectively. Then we defined the absolute value $\lvert z\rvert$ of $z$ as $\sqrt{a^2+b^2}$. If the imaginary part was $b$, this formula would have to become $\sqrt{a^2-b^2}$ in order to make sense. And the imaginary part function would not be a function from $\mathbb C$ into $\mathbb R$ anymore.