ΣNΠ

The Reals are just as imaginary

Author: Samuel Peterson

Date Published
2017-09-11 (ISO 8601)
72-09-11 (Post Bomb)

As a student of Mathematics, I have received questions about i more than a few times. By i, I am referring to what is called the imaginary number, whose defining characteristic is: $ i^2 = -1 $. These questions are usually from variations of: "Don't square roots only exist for non-negative numbers?", "Isn't that just made-up?", "Is there any use for i?", and "What's the big idea? You can't just make up numbers!" These are typically not questions one hears about integers, rationals, or reals, and I think they stem from a non-universal acceptance of i as a legitimate mathematical construct, whatever legitimate means.

Based on my experience first as a student of Engineering and then of Physics at Oregon State University, there was no denying i's essential role in these disciplines. However in those first years, there was no time given to a philosophical appreciation of what i was. It was treated in a utilitarian manner, a means to an end; such was the tenor of most "lower-division" mathematics courses -- those courses in the first two years of study which were requirements of one engineering/science degree or another. It was not until my first "upper-division" mathematics course -- those courses intended solely for mathematics students -- that some real thoughtful attention was paid to the number systems which we dealt with. It is from these first upper-division courses that I shall draw upon to defend the legitimacy of $i$ to you, the reader.

Given the length this post grew to (not my original intention), I suppose a cliff-notes defense of $i$'s legitimacy can be given as follows: Apart from the natural numbers and maybe the integers, all numbers are made up. What's important for things like $i$ to exist in the same context as, say, numbers like $3$, is that that the number system in which $i$ lives must be extended from the number system in which $3$ lives in a logically consistent manner. The complex numbers satisfy this requirement.

So if you wish you can consider yourself done. Otherwise, let's get started with a discussion of the number systems which are more universally accepted, and how we can think of them.

1: The Natural Numbers: Because you have to start somewhere.

Since one of the concerns about i which I will attempt to address is its legitimacy as a mathematically rigorous construct, I will ask you to accept one set of numbers as given: the natural numbers. That is to say all non-negative whole numbers $\mathbb{N} = \{1, 2, 3, ... \}$. It should be noted that I omitted 0 from the natural numbers, although some conventions for the natural numbers include it. I omitted $0$ in deference to the common misconception that 0 as we use it today is somehow a relatively new concept¹. Misconception or no, I'll leave it out. I am, after all, striving to lay a foundation of argument which is as free of controversy as I can.

A set of numbers without any other context is rather boring -- otherwise it really would just be a collection of things. I belabor this point because a primary motivation for adding further elements to this set is to enrich its structure. For the natural numbers, I shall formally point out what you no doubt already know about them:

Every number except $1$ has a predecessor and a successor. $1$ only has a successor.
There is an ordering: $x < y$ if y is the successor to x, $x > y$ if $y$ is the predecessor of $x$, $x < z$ if $x < y$ and $y < z$, $x > z$ if $x > y$ and $y > z$
There is an operator called addition ($+$), which works as you would expect: $x + 1 = x_{successor}$, and $x + y = x_{successor} + y_{predecessor}$
There is an operator called multiplication ($\times$), which works as you would expect: $x \times 1 = x$ and $x \times y = x + (x \times y_{predecessor})$
The operators ($+$) and ($\times$) are associative: $x \circ y \circ z = (x \circ y) \circ z = x \circ (y \circ z)$
The operator $\times$ distributes across addition: $x \times (y + z) = x \times y + x \times z$

That's pretty much it. You may notice that I omit any mention of division or subtraction in the list above. This is because I would like to talk of these operators as the inverses of addition and multiplication; that way, we need not worry about pesky irregularities like remainders, or the restrictions on subtraction which demand that the first number in the expression must be greater than the second. However, in order to talk about such inverses, we need larger number systems, which is what the next couple of sections deal with.

2: The Integers: Let there be a group!

To the natural numbers, we may also want to include the notion of $0$. $0$ is a remarkable number which you probably think of daily which arguably is not demonstrable in the real world (unlike, say positive natural numbers). With regards to the structure of the natural numbers above, the property that sets $0$ apart from the other elements is the following: $x + 0 = x$. There is a term for this property: we say that $0$ is the additive identity.

So now we have $\mathbb{N}^+ = \{0, 1, 2, ...\}$, and we have addition $ (+) $ and multiplication $(\times)$. You might be thinking that it would also be nice to have subtraction. To get that in a very general sense, let's discuss the idea of an inverse. We have an additive identity $0$. For a given number $x$, the additive inverse $-x$ is by definition a number such that: $x + -x = 0$. i.e. a number plus its additive inverse is equal to the identity under addition. Unfortunately, if we consider $\{0, 1, 2, ...\}$ the only number that has an inverse in this set is $0$. We could rectify this by "inventing" additive inverses to $1$, $2$, $3$ and so on. In fact this is exactly what negative numbers are. For each natural number $x$ we can define its additive inverse: $-x$ by the following property: $x + -x = 0$. Poof! We now have the integers: $\mathbb{Z} = \{0, 1, -1, 2, -2, ...\}$ just by proclaiming like the LORD: "Let there be additive inverses!" I will omit the details of how addition and multiplication work in this set, as I assume it is common knowledge. I will however point out that we can now think of the operation of subtraction as: $x - y = x + (-y)$.

Note that the inclusion of negatives to the natural numbers in order to get the integers is an extension. We have not in any way changed the structure of the natural numbers. By that, I mean the following:

The natural numbers is a subset of the integers.
When apply the addition and multiplication operators as defined for $\mathbb{Z}$ to the subset, $\mathbb{N}$, we see they behave just as those same operators when defined on $\mathbb{N}$.

These criteria of extension will be a common theme. We will see that the rational numbers are an extension of the Integers, the real numbers are an extension of the rationals, and the complex numbers (of which i is an element) are an extension of the reals. All of these extensions are motivated by a desire to add structure -- in the case of the integers, the extra structure was to introduce additive inverses (formally we would call the integers a group under addition). There is another operator which we could subject to the same consideration: multiplication. That's where the rationals come in.

3: The Rationals: Think of them as Greek-Reals

Just as $0$ is the identity under addition, $1$ is the identity under multiplication, i.e. $y \times 1 = y$. For a given number, $y$, the multiplicative inverse $y^{-1}$ is by definition a number such that: $y \times y^{-1} = 1$. That is, a number times its multiplicative inverse is equal to the identity under multiplication(sound familiar?). Does $\mathbb{Z}$ have multiplicative inverses for each of its elements? No. Only for $1$ which is its own inverse (just like $0$ is its own inverse under addition.) How do we get inverses under multiplication. We invent them in the same way we invented additive inverses to get the integers, along with extensions of the operations of multiplication and addition in the new set.

Unfortunately, it is a bit more laborious to do that with the rationals than it was with the integers. The above idea is more or less correct, it's just that unlike the case with integers, if we were to just include the inverses of the integers, we are not done, because those inverses must have addition and multiplication defined in way consistent with addition and multiplication on the integers. Extending these operators creates additional elements which themselves must have inverses under multiplication and addition (or else what was the point?) and so on... The end result is a set of numbers and a pair of operators which can be described in the following way:

The rational numbers are given by $\mathbb{Q} = \{\frac{n}{p} \text{ where n and p are any mutually-prime integers, and } p \neq 0\}$
If $n$ and $p$ are not mutually prime, we identify $\frac{n}{p}$ with $\frac{n^*}{p^*}$ where $n = q \times n^*$ and $p = q \times p^*$
Addition of two rational numbers is defined by $\frac{n_1}{p_1} + \frac{n_2}{p_2} = \frac{n_1 \times p_2 + n_2 \times p_1}{p_1 \times p_2}$
Multiplication of two rational numbers is defined by $\frac{n_1}{p_1} \times \frac{n_2}{p_2} = \frac{n_1 \times n_2}{p_1 \times p_2}$

Some words of clarification: First, I am over-loading the operators here, I use the rational $+$ and $\times$ operators when they are between items such as $\frac{\cdot}{\cdot}$, and I use the integral $+$ and $\times$ operators when they are above and below the dividing line; second, with this representation of rational numbers, we identify the arbitrary integer $n$ with the rational number $\frac{n}{1}$ -- a quick check will confirm that the rational versions of addition and multiplication when restricted to rational numbers of the form $\frac{n}{1}$, i.e. when restricted to our identification with the integers, behave exactly as the integer versions of addition and multiplication. So the rationals are an extension of the integers.

What about inverses? Observe from the rules above $\frac{1}{1} \times \frac{n}{p} = \frac{n}{p}$ -- i.e. $\frac{1}{1}$ is the multiplicative identity, and $\frac{n}{p} \times \frac{p}{n} = \frac{n \times p}{n \times p} = \frac{1}{1}$ -- the inverse is determined by "flipping" the ratio.

Before we continue, for the sake of simplicity of notation, let it be understood that if we say $q$ is a rational number, then there are some integers associated with q, and if their specific values are not important, we don't have to explicitly write out $\frac{q_{numerator}}{q_{denominator}}$.

The rationals, like the integers and natural numbers, have an ordering to them; there is a notion of greater-than and less-than. Unlike the integers and the natural numbers, the rationals have a fine structure. What do I mean by that? Consider two integers: $n$ and $n + 1$. Clearly, $n+1 > n$, yet there is no integer, $m$ such that $n+1 > m > n$. The same is not true of the rational. For any two rationals $q$ and $p$ such that $q > p $, there is another rational number $r$ such that $q > r > p $ ( in fact, $ (q + p) \times \frac{1}{2} $ would do the trick).

We are getting pretty far from the world of natural numbers by this point. Mind you that after agreeing to agree on the existence of natural numbers we are just making up everything else. I point that out so that when we get around to saying: "oh yeah, and let's say that $\sqrt{-1}$ exists" it will seem like a reasonable thing to do provided we can come up with a set that includes the number systems our extension is based off of and the operators of addition and multiplication which are consistent when restricted to these foundational subsets.

As a final note on the topic of rational numbers. Some ancient schools of greek mathematics liked to think that the rational numbers were all that existed. There is in fact a (most likely apocryphal) story about a Pythogorean who was executed by his fellow occultists for demonstrating that if $\sqrt{2}$ existed then it couldn't be a rational number. In this case, the executed mathematician was indeed right, and greek mathematicians were indeed aware of irrational numbers however from what I've read they were considered to be oddities to them. So for those who view $\sqrt{-1}$ with suspicion, they would do well to remember that back in the day, $\sqrt{2}$ was just as absurd. Nowadays, $\sqrt{2}$ lives in a widely-used number system which is yet another extension, this time of the rationals. We call these numbers the Real numbers.

4: The "Real" numbers

The rational numbers are fine and all, but don't you want a number system that has $\pi$ too? I know I do. Alas the rationals do not have $\pi$, but they get arbitrarily close. In fact, there are lot's of sequences of rational numbers that in some sense get arbitrarily close to quantities which are themselves not rational numbers. We will use this property to construct the real numbers, $\mathbb{R}$.

First off, we have the following notion of distance: The distance between two numbers, $q$ and $p$ is given by $|q - p| = \max(q - p, p - q)$. Another bit of useful information for what follows is that a number is rational if and only if its decimal expansion terminates (e.g. $5.37281$) or if it is eventually repeating (e.g. $5.3728181818181\dots$). With those bits of information in mind, let's consider a sequence of rational numbers: $\{3, 3.1, 3.14, 3.141, 3.1415, \dots\}$. You might see where I'm going with this. I am giving a sequence of rational numbers, where the $n^{th}$ element of the sequence is the value of $\pi$ up to the $n^{th}$ decimal place. These numbers are getting arbitrarily close to something, because we see that if we measure the distance between the $n^{th}$ and the $m^{th}$ element of the sequence, for $m > n$ we get $|a_n - a_m| = 0.0(\text{n zeros})(\text{last m-n elements of a_m})$. Since this sequence is non-repeating, we also see that there is no rational number for which the sequence is getting arbitrarily close -- if we picked any rational number, p, we can say that all elements of the sequence have a distance from p of at least $t > 0$. So if the sequence is getting close to something, but that something isn't rational, what do we do? We invent further numbers so that we can formally discuss what it is that the sequence is converging to.

Loosely speaking, we can get at the idea of real numbers by just saying that the real numbers include all decimal expansions, be they infinitely long or no. This is quite an abstract thing to do if you think about it. There is no way of knowing in any exact sense the value of an irrational number, as infinitely long decimal expansions are physically impossible to express in our finite universe. The irrationals are concepts which can only be numerically approximated with rational numbers to some non perfect degree of precision. What's more, the concept of addition and multiplication is also abstracted by simply applying the addition and multiplication operators learned in grade school arithmetic to the rational approximations of said irrational numbers. By this point I think it should be clear that the term "real number" is quite the misnomer.

What do we get from this extension? Quite a lot actually. First of all, we now have an entirely new order of infinity in the size of our numbers². We also get a nice topological property from this extension: The reals are what we call topologically complete, which just means that sequences that appear to converge to something actually converge to an element in the set. It is a number system which has a rich enough structure to allow for limits, and all sorts of other calculus goodness. There are still some pesky problems with it though: Sometimes polynomials in $\mathbb{R}$ don't have solutions for the $0$.

5: The complex numbers: for those of us who want to be algebraically complete too

Consider an arbitrary $n^{th}$ degree polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 = P_n(x)$ If $x$ and $a_k$ were just real numbers, then it is possible for the equation $P_n(x) = 0$ to have no solution; an example of such a polynomial is $x^2 + 1$. This can sometimes be problematic. In a number of physical settings, solutions to differential equations get needlessly complicated or downright impossible unless we have such solutions to these polynomials. Pretty much anything involving oscillation (which is almost all of physics) is basically pleading for this difficulty to be removed. So let's do what we have done for all the other number systems we have looked at so far. Let's just say there is a solution to $x^2 + 1 = 0$. Let's call it $i$.

Now we need to define the number system in which $i$ fits in, and how addition and multiplication work in this system:

The complex numbers are all numbers of the form $\mathbb{C} = \{ a + i \times b \text{ :where a and b are real numbers}\}$
Addition is defined as component-wise as follows: $ (a_1 + i \times b_1) + (a_2 + i \times b_2) = (a_1 + a_2) + (b_1 + b_2) \times i $
Multiplication is as you would expect -- just distribute like usual and remember that $i \times i = -1$: $ (a_1 + b_1 \times i) \times (a_2 + b_2 \times i) = (a_1 \times a_2 + -1 \times b_1 \times b_2) + i \times (a_1 \times b_2 + a_2 \times b_1) $
We identify the real numbers with the complex numbers of the form: $a + i \times 0 $. This way we consider the complex numbers as an extension of the reals.

A quick check shows that addition and multiplication in the complex number system works just like it does with the Real numbers when we restrict our attention to numbers of the form $a + i \times 0$.

But do we need the complex numbers? That depends... is chocolate a necessary ingredient to happiness? But seriously, the question is not whether we need them, but rather, why on earth would we choose not to use them? The complex numbers have some really nice properties. One of the neatest is that of algebraic completeness: every complex polynomial has at least one root -- $P_n = 0$ is guaranteed to have a solution in this space for arbitrary $P_n$ unless $n = 0$. This property is also called the fundamental theorem of algebra. A nice consequence of this is that all n-degree polynomial can be expressed in the following way: $$ P_n(x) = c \times \displaystyle\prod_{k = 0}^{n}(x - a_k)$$ Another neat thing about the complex numbers is that differentiable functions in the complex numbers are really differentiable -- infinitely differentiable in fact. And at every point, $x_0$ where a function is differentiable, they have a power series expansion in some neighborhood about that point: $$ f(x) = \displaystyle\sum_{n = 0}^{\infty} a_n \times (x - x_0)^n $$ There are a bunch of other cool properties of differentiable complex functions, but that is just outside the scope of this paper.

6: Are we done here? Can we go home?

I guess so. My only real point here is that the invention of the complex numbers is no less legitimate than the invention of the reals or the rationals. The complex numbers, by the way are not where the whole extension-of-number-systems game ends. There's the hyper-reals, the surreals, and others. I do not know much about them, but they are extensions of the reals which have been studied. That being said, I believe it is true that the complex number system is of sufficient richness to be perfectly adequate for most modern natural and social scientific applications.

¹: Although it is true that written numerals in the greco-roman age employed a more archaic system than the arabic numerals which are widely used today, there were devices known throughout Eurasia which demonstrate a virtually identical means of calculation and of number representation as with modern arabic numerals: the abacus. Such devices were already ancient by the time of Pericles.

²: While the natural numbers, integers and rational numbers are all countably infinite in cardinality, the irrational are what is called uncountably infinite. The curious reader can look up in greater detail what is meant by that.