Is "$a + 0i$" in every way equal to just "$a$"?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$).

I say this is practically the case, so in every calculation you just assume that "$a + 0i$" is equal to the real number "$a$" and you always get the right results (this might not even be true, but I am no expert and as far as I can see this is the case). But then again, I believe that just cutting away "$+ 0i$" makes this number is not the same kind of number anymore, it's changed its structure and is not totally equivalent anymore. This is where he says I am absolutely wrong.

So, is it mathematically (strictly speaking!) perfectly fine to say that every complex number with an imaginary part of $0$ is just a real number? Or does this change the mathematical structure so much that it cannot be mathematically-valid but just something that happens to work (which is what I believe)?

I must say that I believe a complex number is one entity, whilst he believes that it is just a conglomerate of other entitites. So, as one said here, "$(a + bi) - bi = a$", I believe is barely a good argument. But then again: is it correct to view $a + bi$ as a single entity (like "$5$" or "$2$", just consisting of multiple symbols)? I'm (as a non-mathematician) not sure there either.

Solution 1:

Well, yes and no. You are both right.

What does it mean to be the number 1? This is a fascinating philosophical point, worth careful consideration. The usual attitude of modern mathematics is that something is the number 1 if it belongs to a system in which it behaves like the number 1 and does the things we want the number 1 to do. For example, the system should have a notion of multiplication, and it should have $1\cdot x = x$ for each $x$. We would also like there to be something recognizable as 2, and we should have $1+1=2$. There may be other properties we require of 1.

But there are many systems that can serve as models of the natural numbers. In Peano arithmetic, we take the numbers $0,1,2\ldots$ to be sequences of symbols: $\mathbf{0}, \mathbf{S0}, \mathbf{SS0}, \ldots$. In another system we take them to be the sets $\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}, \ldots$. In each system we are obliged to define addition and multiplication in a way that makes the numbers do what they are supposed to, so that in the first case we ought to have $\mathbf{S0}+\mathbf{S0} = \mathbf{SS0}$ and in the second case we should have $\{\varnothing\}+ \{\varnothing\} = \{\varnothing, \{\varnothing\}\}$. If we can't do this, we have no right to say we are dealing with the number 1. If we can do it, we consider ourselves mathematically satisfied, and the question of whether $\mathbf{S0}$ or $\{\varnothing\}$ is the number 1 is no longer a mathematical but a philosophical one. But notice the strange situation we are in: Each of $\mathbf{S0}$ and $\{\varnothing\}$ is the number 1, but they are completely different objects: one is a set and one is a sequence of symbols. And yet each has an equally good claim to being ‘the’ number 1.

Similarly there are many ways to construct the complex numbers. A typical construction starts with the set of pairs $\def\c#1#2{\langle #1,#2\rangle}\c ab$ of real numbers and identifies $i$ as the pair $\c01$. In this construction, the complex number $a+bi$ is the pair $\c a b$, and the real number $x$ makes an appearance as the pair $\c x0$. This latter pair is not the same as the real number $x$, because the former is a pair of real numbers and the latter is a single real number.

But on the other hand, a pair of the form $\c x0$ behaves just like the real number $x$, and the set of such pairs behaves like $\Bbb R$. Its elements correspond exactly with the real numbers, and do all the things we expect the real numbers to do. So just as both $\mathbf{S0}$ and $\{\varnothing\}$ had equally good claims to being ‘the’ number 1, there is no reason to prefer the original $1$ over the pair $\c 10$ as ‘the’ real number 1. The set of $\c x0$ is $\Bbb R$ for all practical purposes. So much so that, having made this identification, we might then discard our original real numbers and consider thenceforth only the set of pairs of the form $\c x0$.

We do this same thing when we construct the real numbers in the first place. We start with the rationals, and then construct the reals as certain structures of rationals. Having done this, we identify certain reals as being morally equivalent to the rationals we started with. For example, we might observe that the pair $\left\langle \left\{x\in\Bbb Q \mid x\le \frac12\right\}, \left\{x\in\Bbb Q \mid x\gt \frac12\right\}\right\rangle$ behaves, in this new system, just the way we expect the rational number $\frac12$ to behave. These new rational numbers aren't the same objects as the rationals we started with: those were single rationals, and these new ones are pairs of sets of rationals. But the new rationals behave like the original rationals.

(Barry Mazur's paper “When is one thing equal to some other thing” is about this exact question, and discusses it at greater length. It is very readable, and I recommend it highly. Sample pullquote: “The heart and soul of much mathematics consists of the fact that the ‘same’ object can be presented to us in different ways.”)

Solution 2:

You first define $\mathbb{R}$, the real numbers, and then you define $\mathbb{C}$, the complex numbers, to be the pairs of numbers $(x,y)$ of real numbers that obey the rules $(x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1+y_2)$ and $(x_1,y_1)\cdot (x_2,y_2) = (x_1x_2 - y_1y_2,x_1y_2+x_2y_1)$. If we use the notation $a+bi$ for the pair $(a,b)$ then it amounts to saying that $i^2=-1$ where $i=(0,1)$. With this notation $\mathbb{R}$ is not really a subset of $\mathbb{C}$, however the set of numbers $(x,0)$ where $x$ is real is now identified as a subset of $\mathbb{C}$ which is essentially the same as the real numbers.

Solution 3:

One place it matters is less purely mathematical and more about what information you can get about the context the number is in.

Consider this: occasionally, you will see a number with its sign attached, even if the sign is positive. What this tells you the reader is that we expect that numbers in this particular context could potentially be negative, but that today it is positive.

Similarly, when you see a number like 23.00, we expect that numbers in this particular context could potentially have two decimal places worth of fractions as part of it, but that today it is whole.

Following this pattern, 2 + 0i says that a number in this particular context could potentially have a complex part, but that today it does not.

Solution 4:

A common construction of $\mathbb C$ is that $\mathbb C=\mathbb R\times\mathbb R$ with some suitable addition and multiplication laws, namely, $(a,b)+(c,d)=(a+c,b+d)$ and $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$ for every $a$, $b$, $c$, $d$ in $\mathbb R$. Then $(1,0)$ is a unit for this multiplication in $\mathbb C$ and one defines $i=(0,1)$. Thus, every $(a,b)$ in $\mathbb C$ is also $(a,0)+(b,0)\cdot i$, abbreviated as $a+b\cdot i$.

In this sense, $2+0\cdot i=(2,0)$ is an element of $\mathbb C$ while $2$ is an element of $\mathbb R$, hence these cannot coincide unless $2$ really means the element $(2,0)$ of $\mathbb C$. Every $(a,0)$ with $a$ in $\mathbb R$ is indeed often abbreviated as $a$. This corresponds to identifying $\mathbb R$ with $J(\mathbb R)$, where $J:\mathbb R\to\mathbb C$ is defined by $J(a)=(a,0)$ for every $a$ in $\mathbb R$.