Complex Numbers are Real - Infinity is Really Big

Complex Numbers are Beautiful Too

Real Analysis Study Help for Baby Rudin, Part 1.7

Complex numbers can be represented either using Cartesian coordinates or polar coordinates. The geometric interpretation of complex multiplication is very beautiful using polar coordinates.

Are imaginary numbers real? After all, why are they be called “imaginary”? Doesn’t this name imply that they don’t exist? To some degree, your answers to these questions probably depend on your definition of reality. However, I will say the answer to the first question is “yes”: imaginary numbers are real.

The main thrust of this article is to argue for the truth of a broader fact: that complex numbers are real. Complex numbers include imaginary numbers and more.

The reality of complex numbers will be true in both metaphysical and practical senses. I will illustrate these realities through discussion and examples.

In addition, I will also illustrate how complex numbers are beautiful. As a Christian mathematician, seeing this beauty causes my heart to praise God.

The original impetus for this article is that it is also the seventh in a series of articles about the Real Analysis textbook Baby Rudin. However, my hope is that a broader audience can learn from and enjoy this content. So, I will discuss simpler material before getting into the more complex real analysis at the very end (pun intended).

The Imaginary Unit is Real

Imaginary numbers are usually introduced in the following way. We start by saying that we would like to be able to solve the polynomial equation $x^{2}+1=0$ . This equation is equivalent to $x^{2}=-1$ . Now the square of any real number cannot be negative. This means that there are no real number solutions to the equation $x^{2}+1=0$ .

Ancient and pre-modern people left it at that: there are no solutions to $x^{2}+1=0$ . However, in the 1500’s, some mathematicians decided to “create” solutions. We let $i$ represent a “new” number with the property that $i^{2}=-1$ . In other words, $i^{2}+1=0$ so that $i$ is a solution of the polynomial equation $x^{2}+1=0$ .

This new number is called the imaginary unit. And we might reasonably wonder: is this a figment of our imaginations? Well, yes and no. We need more information.

Actually, as explained in Chapter 6 of William Dunham’s outstanding book “Journey Through Genius: the Great Theorems of Mathematics”, the practical motivation for the creation of the imaginary unit and resulting complex numbers was as a means of solving more complicated polynomial equations called cubic equations. Complex numbers turned out to be a useful tool for such problems!

However, we want to think about other questions here. Namely, is this “creation” of a “new” number that solves this equation allowed? Isn’t this “cheating”? How can we just create a new number out of thin air?

What Are Numbers Anyway?

Before we try to answer these questions, we should think about what constitutes a “number” in the first place.

Natural Numbers are Real

Most people would describe numbers as objects (or symbols) that can be used to represent quantities that can be counted or measured. The counting numbers (natural numbers) are 1, 2, 3, 4, 5, etc. These are used to count, for example, the number of apples in your bag. They can then be used for comparison: my bag has 8 apples and yours has 11; you have more than I do. Furthermore, they can be combined: if we combine the apples from my bag and yours, then we have $8+11=19$ apples total. Also, you have $11-8=3$ more apples than I do.

It is important to realize that these numbers are both abstract (conceptual) and symbolic. As abstract concepts, they are as real as any other “idea” out there, such as the concepts of color and emotion. We can have meaningful discussions about numbers just as we can have about colors and emotions.

As symbols, they a part of a language that can be used to communicate, and they are as real as words in any written language. Even better, they have the benefit of being very precise in their meaning, whereas colors, emotions, and everyday words are less precise. Even better than that, they have the benefit of having “rules” that they follow that allow us to apply them in helpful and sometimes unexpected ways.

Rational Numbers are Real

The rational numbers are also both abstract and symbolic, following similar rules. They are used for approximate measurement of “continuous” quantities, such as time, length, area, and volume.

For example, I can measure a stick to be $\frac{1}{4}$ (one-fourth) of a meter long. This is represented in decimal notation and percent notation as $\frac{1}{4}=0.25=25\%$ . It’s a fraction of a meter.

Of course, some objects are longer than one meter, leading to “improper” fractions and “mixed” numbers, such as $\frac{5}{4}=1\frac{1}{4}=1.25$ meters.

Fractions are a source of difficulty for many people. If you struggle to work with them, keep at it! There are a lot of helps out there for you.

Zero is Real

The desire to represent the lack of quantity or lack of measure leads to introducing the number zero $0$ . The history of this number is quite fascinating (see “Zero: The Biography of a Dangerous Idea”, by Charles Seife). Many ancient (and even some modern) people groups never even thought of the idea in the first place. This might seem strange to most modern people since we are so used to it.

Negative Numbers are Real

How are negative numbers justified? One practical example is as a convenient way to represent net worth, which is assets minus liabilities. If your assets are larger than your liabilities, your net worth is positive. If your assets are smaller than your liabilities, your net worth is negative.

This is convenient if you are, for example, using formulas in a spreadsheet. If you want to find your average net worth from each year over the course of 10 years, you can just add the 10 net worth amounts from each year, including the negative ones, before dividing by 10. You don’t have to use an extra “marker” for the years where your liabilities are larger than your assets and subtract them rather than add them. The negative sign itself is the marker that gets incorporated into the addition.

Another practical application of negative numbers is to represent position. For example, for the up-and-down oscillation of your car as you drive down the road, it is best to let the “equilibrium” (up/down) vertical position be represented by zero. When the car is above equilibrium, its vertical position is positive. And when the car is below equilibrium, its vertical position is negative. This is once again very convenient in many circumstances, including in modeling this position using trigonometric functions.

Irrational Numbers are Real

In Ancient Greece, the views of those in the Pythagorean Society held sway in many philosophical and religious groups. The Pythagoreans believed that the concept of “number” was the essence of all reality. Furthermore, they initially took it on faith that the rational numbers were sufficient for all continuous measurement.

This can also be described as faith in the commensurability of all measurements. What does this mean? It means, for example, that given any two lengths $a$ and $b$ , they believed there was a smaller length $c$ that divided both $a$ and $b$ . In other words, they believed that there existed two natural numbers $m$ and $n$ such that $a=m\cdot c$ and $b=n\cdot c$ .

This faith was dealt a death blow after the Pythagorean Theorem, which they proved was true, lead the Pythagoreans to realize that the diagonal of a square was incommensurable with the sides of that square. The story is apocryphal, but supposedly this lead some of them to sacrifice Hippasus of Metapontum at sea.

If the square has side length 1, then its diagonal has length $\sqrt{1^{2}+1^{2}}=\sqrt{2}$ . This is the unique positive real number whose square is two. Its existence as a length seems non-controversial. Its existence using decimal representations (and no geometry) is less clear. (If you know Real Analysis, see my article “Existence of nth Roots of Positive Reals”.)

A short and beautiful argument that $\sqrt{2}$ , assuming it exists, is not rational (a.k.a., irrational) is found near the beginning of my article “Does the Square Root of Two Exist?”

Also of interest for those who like proofs may be this video I made in 2017.

Prove Square Roots of Non-Perfect Squares are Irrational

Complex Numbers are Real (Arithmetic Can Be Done With Them)

But what about the imaginary unit and complex numbers? Are they real?

By definition, they are certainly not what we technically define to be real numbers. The set of real numbers consists of all rational numbers along with all irrational numbers. They are all the numbers that can be represented on the real number line. Alternatively, they are all numbers that can be represented by decimal expansions, including both repeating decimals (rational numbers) and non-repeating decimals (irrational numbers).

**Complex Numbers are Real Because We Define Them to Be Real**

But complex numbers are real in the sense that we can define them to be abstract and symbolic objects that obey “natural” rules that are shared by other “number systems”. What are these rules? They include rules such as the commutative, associative, and distributive laws.

Let’s get more specific. A complex number is any mathematical expression of the form $a+bi$ , where $a$ and $b$ are real numbers (written $a,b\in \Bbb{R}$ , where $\Bbb{R}$ represents the set of all real numbers), and $i^{2}=-1$ . Why does $i^{2}=-1$ ? Because we say so!

That’s a good enough reason for most mathematicians. The only question is whether it’s a good/useful idea for other people who need to use mathematics. It turns out it is!

For future reference, we call the real number $a$ the real part of the complex number $a+bi$ . And we call the real number $b$ the imaginary part of the complex number $a+bi$ (even though $b\in \Bbb{R}$ ).

Complex Number Addition

How do we add complex numbers? We add their real parts together and their imaginary parts together. For example, $(2+3i)+(7+4i)=(2+7)+(3+4)i=9+7i$ .

In general, $(a+bi)+(c+di)=(a+c)+(b+d)i$ .

This kind of addition satisfies the commutative property. Here is the verification. For any $a,b,c,d\in \Bbb{R}$ , we have:

$(a+bi)+(c+di)=(a+c)+(b+d)i=(c+a)+(d+b)i=(c+di)+(a+bi).$

Notice that we used the fact that real number addition is commutative: $a+c=c+a$ and $b+d=d+b$ .

Also notice that the imaginary unit $i$ is not really “doing” anything with complex number addition, other than being a “placeholder”. It’s as if we are just adding two linear polynomials in the “variable” $i$ .

Complex Number Multiplication

On the other hand, $i$ is “doing” something when we define complex number multiplication. We start by treating the product as the product of two polynomials, and use the so-called FOIL method. For any $a,b,c,d\in \Bbb{R}$ , we have:

$(a+bi)(c+di)=ac+adi+bci+bdi^{2}=ac+(ad+bc)i+bdi^{2}.$

This is not wrong, but it is incomplete. It is not fully simplified. Use the fact that, by definition, $i^{2}=-1$ to do so:

$(a+bi)(c+di)=ac+adi+bci+bdi^{2}=ac+(ad+bc)i+bdi^{2}=(ac-bd)+(ad+bc)i.$

Is this operation commutative? Yes, but the verification is just a bit trickier. For any $a,b,c,d\in \Bbb{R}$ , we have:

$(a+bi)(c+di)=(ac-bd)+(ad+bc)i=(ca-db)+(cb+da)i=(c+di)(a+bi).$

We have used the facts that $ac=ca$ , $bd=db$ , $ad=da$ , $bc=cb$ , and $da+cb=cb+da$ .

Complex Number Subtraction

In any number system, subtraction is the inverse operation of addition. This only works when each number has a so-called additive inverse.

For an arbitrary complex number $a+bi$ , its additive inverse is $-(a+bi)=-a+(-b)i$ . The real number $-a$ is the additive inverse of the real number $a$ because $a+(-a)=0$ , the additive identity in $\Bbb{R}$ . And the real number $-b$ is the additive inverse of the real number $b$ because $b+(-b)=0$ . This gives

$(a+bi)+(-(a+bi))=(a+bi)+(-a+(-b)i)=(a+(-a))+(b+(-b))i=0+0i,$

where $0+0i$ is the additive identity in the set of all complex numbers $\Bbb{C}$ .

Now we define subtraction as: $(a+bi)-(c+di)=(a+bi)+(-(c+di))$ . In other words, $(a+bi)-(c+di)=(a-c)+(b-d)i$ . We subtract the corresponding real parts and the corresponding imaginary parts.

Complex Number Division

Division by a nonzero number is the inverse operation of multiplication. This is less clear. For an arbitrary nonzero real number $a$ , its so-called multiplicative inverse is $\frac{1}{a}$ because $a\cdot \frac{1}{a}=1$ , the multiplicative identity. But what about a nonzero complex number $a+bi\in \Bbb{C}$ ?

First, realize that to say $a+bi\not=0+0i$ means that either $a\not=0$ or $b\not=0$ or both. In fact, in general, we say that $a+bi=c+di$ if and only if both $a=c$ and $b=d$ .

If $a+bi\not=0+0i$ , then what should $\frac{1}{a+bi}$ be?

A “hopeful” calculation involving a “trick” reveals how we should define it. The trick involves using the so-called complex conjugate of $a+bi$ , which is $a+(-bi)=a-bi$ . We hope:

$\frac{1}{a+bi}=\frac{1}{a+bi}\cdot \frac{a-bi}{a-bi}=\frac{a-bi}{a^{2}-abi+abi-b^{2}i^{2}}=\frac{a-bi}{a^{2}+b^{2}}=\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i.$

Note that $a^{2}+b^{2}\not=0$ since either $a\not=0$ or $b\not=0$ or both, so we are not dividing by zero. I will leave it as an exercise for you to check directly using FOIL that $(a+bi)\cdot \left(\frac{a}{a^{2}+b^{2}}+\frac{-b}{a^{2}+b^{2}}i\right)=1+0i$ , the multiplicative identity in $\Bbb{C}$ .

This leads us to define complex number division using complex number multiplication. We define $(a+bi)\div (c+di)=(a+bi)\cdot \frac{1}{c+di}=(a+bi)\cdot \left(\frac{c}{c^{2}+d^{2}}+\frac{-d}{c^{2}+d^{2}}i\right)$ when $c+di\not=0+0i$ . This can, of course, be expanded using FOIL if you want.

In practice, with particular examples, it is typically best just to use a “trick” with the complex conjugate of the denominator rather than the formula above. Here’s an example:

$(3-2i)\div (4+5i)=\frac{3-2i}{4+5i}\cdot \frac{4-5i}{4-5i}=\frac{12-15i-8i+10i^{2}}{16-20i+20i-25i^{2}}=\frac{2-23i}{16+25}=\frac{2}{41}+\frac{-23}{41}i$ .

Complex Numbers Are Real (They Can Be Visualized)

Maybe these calculations are not good enough for you. Maybe you have to “see” complex numbers before you “believe” they exist.

Complex Numbers as Points in a Cartesian Coordinate Plane

This is something that can definitely be done. In fact, Baby Rudin does this by defining complex numbers as “ordered pairs” of real numbers. That is, complex numbers are objects of the form $(a,b)$ , where $a,b\in \Bbb{R}$ . And there is a commonly-used and natural association (Cartesian coordinates, a.k.a. rectangular coordinates) between ordered pairs of real numbers and points in a plane. We can literally draw complex numbers and “see” them on paper (or a screen).

To be more precise, “identify” the complex number $a+bi$ with the ordered pair $(a,b)$ . We treat these as “the same”, even though they are different symbols. In fact, we will even “abuse notation” and write an equal sign $(a,b)=a+bi$ .

The real number first coordinate $a$ is the real part of $(a,b)$ and the real number second coordinate $b$ is the imaginary part of $(a,b)$ .

Instead of using “x” and “y” as the labels for the axes, we use “real” and “imaginary”. It is also traditional to give the plane a new name. We call it the complex plane, or Argand plane, in honor of Jean-Robert Argand.

Here is a picture with three complex numbers $P=(4.2,-2.7)=4.2-2.7i$ , $Q=(6.7,4.5)=6.7+4.5i$ , and $R=(-6.4,8)=-6.4+8i$ to illustrate this.

Complex numbers are real because we can visualize them in the complex plane. — Complex numbers are real, in part, because we can visualize them in the complex plane.

In spite of treating complex numbers as points, we do not want to lose their arithmetic properties. For example, we define multiplication of these ordered pairs to exactly match the multiplication defined above. Since $(a+bi)\cdot (c+di)=(ac-bd)+(ad+bc)i$ , we define $(a,b)\cdot (c,d)=(ac-bd,ad+bc)$ . Likewise, for division, we start by defining multiplicative inverses as $(a,b)^{-1}=(1,0)\div (a,b)=\left(\frac{a}{a^{2}+b^{2}},\frac{-b}{a^{2}+b^{2}}\right)$ when either $a\not=0$ or $b\not=0$ or both.

Complex Numbers in Polar Coordinates (Polar Forms)

We are getting close to the point of not only seeing complex numbers as real, but also seeing them as beautiful. This is where my heart “sings” for joy and thankfulness to God because He has beautifully structured the universe and has given us minds that can understand it and appreciate its beauty. We will ultimately illustrate this beauty by giving a geometric interpretation of complex number multiplication.

Before we can do that, we need to describe how to represent complex numbers in “polar form”. Geometrically, we use a polar coordinate system to do this. Symbolically, it gets a bit confusing because there are various conventions for the polar form of a complex number. In addition, the polar form of a complex number is not unique.

Since any complex number is a point in a rectangular coordinate plane, we can also associate it with a directed line segment (a.k.a. “arrow” or “vector”) from the origin $0+0i$ to the point.

Here is the picture of such vectors corresponding to the picture above. Note that the approximate lengths of each vector are shown, along with the corresponding approximate angles from the positive (rightward) real axis to the vectors. By convention, we are taking these angles to be between $-180^{\circ}$ and $180^{\circ}$ (between $-\pi$ radians and $\pi$ radians).

Complex numbers can be represented using polar coordinates. This is also related to their polar forms. — The approximate lengths and angles of each complex number (as vectors) are shown. These lengths and angles represent the polar coordinates of these complex numbers, which is related to their so-called polar forms.

Complex Number Modulus and Argument

For a given complex number $(a,b)=a+bi$ , its length is called its modulus (or absolute value), and is computed using the Pythagorean Theorem to be $\sqrt{a^{2}+b^{2}}$ . Its angle is called its principal argument and is computed using trigonometry as described in the next paragraph. In the next paragraph, recall that $\pi\mbox{ radians}=180^{\circ}$ .

If $a>0$ , then its argument is $\tan^{-1}\left(\frac{b}{a}\right)=\arctan\left(\frac{b}{a}\right)$ . If $a<0$ , then its argument is $\tan^{-1}\left(\frac{b}{a}\right)+\pi=\arctan\left(\frac{b}{a}\right)+\pi$ radians if $b>0$ and $\tan^{-1}\left(\frac{b}{a}\right)-\pi=\arctan\left(\frac{b}{a}\right)-\pi$ radians if $b<0$ . If $a=0$ and $b>0$ , then its argument is $\frac{\pi}{2}$ radians. And if $a=0$ and $b<0$ , then its argument is $-\frac{\pi}{2}$ radians. The argument of the origin $0+0i$ is undefined.

Non-principal values of the argument of a complex number can be computed by adding integer multiples of $2\pi$ radians ( $360^{\circ}$ ) to the principal value.

Polar Forms

Technically, since the polar coordinates of a complex number are specified by two numbers, the polar form of a nonzero complex number $a+bi$ could be represented by an ordered pair $(r,\theta)$ , where $r=\sqrt{a^{2}+b^{2}}$ and $\theta$ is the angle as described in the section above. However, this would be confusing since we are also using ordered pair notation for rectangular coordinates.

Although such confusion is tolerated when switching between rectangular and polar coordinates for “pure” points in a non-complex plane, it is typically not tolerated when working with the complex plane. (Though, to be truthful, ordered pairs are rarely used to represent complex numbers beyond the introductory stages anyway.)

Because of this, when we write a complex number $(a,b)=a+bi$ in its polar form, involving polar coordinates, other conventions are used.

First of all, by using trigonometry, we find that $a=r\cos(\theta)$ and $b=r\sin(\theta)$ . Therefore, we can write $(a,b)=(r\cos(\theta),r\sin(\theta))$ or $a+bi=r\cos(\theta)+r\sin(\theta)i$ .

These equations still highlight the rectangular form, however. How can we just focus on the polar form?

One convention is to introduce a new “function” $\mbox{cis}(\theta)$ , which is just shorthand for $\cos(\theta)+i\sin(\theta)$ . Then we can write $(a,b)=a+bi=r\mbox{cis}(\theta)$ .

The main convention, however, uses a formula called Euler’s formula: $e^{i\theta}=\cos(\theta)+i\sin(\theta)$ , where $e=2.71828\ldots$ , to write $(a,b)=a+bi=re^{i\theta}.$

While the first (less standard) convention is just shorthand notation, the main convention based on Euler’s formula really needs justification: why should the special number $e$ be involved?

Our purpose here is not to justify Euler’s formula, though there are beautiful methods to do so. Instead, our purpose is to illustrate the geometric beauty of complex number multiplication. We will therefore use the shorthand “cis”.

Complex Numbers are Beautiful

We will now describe the surprising and beautiful geometric interpretation of complex number multiplication. We will not prove the mathematical validity of this interpretation. If you are interested, it can be proved using the Law of Cosines.

Instead, after stating the interpretation, it will be illustrated with an example.

Here is the interpretation:

Geometric Interpretation of Complex Number Multiplication: When multiplying two nonzero complex numbers, multiply their moduli and add their arguments. In particular, if $z=(a,b)=a+bi=r_{1}\mbox{cis}\left(\theta_{1}\right)$ and $w=(c,d)=c+di=r_{2}\mbox{cis}\left(\theta_{2}\right)$ are two complex numbers, then the polar form of the product $zw=(ac-bd,ad+bc)=(ac-bd)+(ad+bc)i$ is $r_{1}r_{2}\mbox{cis}(\theta_{1}+\theta_{2})$ .

Note: in this form, it is possible that $\theta_{1}+\theta_{2}$ is not between $-\pi$ radians and $\pi$ radians (it might not be the “principal” value of the argument of the product $zw$ ).

Think about how surprising this is! If you did not already know this fact, there is probably almost no way you could have predicted it! And it’s so simple! This is one big example of what mathematicians think of when they describe the beauty of mathematics!

An Example to Confirm the Geometric Interpretation of Complex Multiplication

Let’s illustrate this with one example. We will multiply the complex numbers $Q=(6.7,4.5)=6.7+4.5i$ and $R=(-6.4,8)=-6.4+8i$ from the pictures above and see the truth of the interpretation is confirmed.

First, in rectangular coordinates, the product is $PR=(6.7+4.5i)\cdot (-6.4+8i)=(6.7\cdot (-6.4)-4.5\cdot 8)+(6.7\cdot 8-4.5\cdot 6.4)i= -78.88+24.8i$ .

Next, compute the polar forms. Since $\sqrt{6.7^{2}+4.5^{2}}\approx 8.07094$ and $\tan^{-1}\left(\frac{4.5}{6.7}\right)\approx 33.88696^{\circ}$ , we have $P\approx 8.07094\mbox{cis}(33.88696^{\circ})$ . Since $\sqrt{(-6.4)^{2}+8^{2}}\approx 10.24500$ and $\tan^{-1}\left(\frac{8}{-6.4}\right)+180^{\circ}\approx 128.65981^{\circ}$ , we have $R\approx 10.24500\mbox{cis}(128.65981^{\circ})$ . And since $\sqrt{(-78.88)^{2}+24.8^{2}}\approx 82.68672$ and $\tan^{-1}\left(\frac{24.8}{-78.88}\right)+180^{\circ}\approx 162.54677^{\circ}$ , we have $PR\approx 82.68672\mbox{cis}(162.54677^{\circ})$ .

Here’s the punchline. Note that $8.07094\cdot 10.24500\approx 82.68678$ (within rounding error of $82.68672$ ) so that the moduli got multiplied. And note that $33.88696^{\circ}+128.65981^{\circ}=162.54677^{\circ}$ so that the arguments got added.

We have confirmed the truth of the geometric interpretation for this example!

Are Complex Numbers Real for Real-Life Applications?

You may ask: sure, this is all real and beautiful from the perspective of pure mathematics, but what about real life? Do complex numbers have real-world applications?

Advanced Applications

The answer is a (qualified) “yes”. Certainly, complex numbers are useful for real-world applications. If you talk to an electrical engineer, that person will tell you that complex numbers are useful for analyzing circuits. If you talk to a physicist, they will probably point out that complex numbers are ubiquitous in quantum mechanics. And if you talk to anyone who needs to solve differential equations, you will be told that complex numbers are an essential tool in some situations.

So why is this “yes” also “qualified”? Because, technically you could do all the calculations in these applications without using complex numbers. In essence, you could always just work with the real and imaginary parts separately (probably giving them different names).

On the other hand, such calculations are done most “naturally” using complex numbers with real and imaginary parts. In fact, the terms “real part” and “imaginary part” are arbitrary (though logical) labels anyway.

So, if you just worked with these parts “separately”, you would be, in essence, working with complex numbers without knowing or acknowledging it.

A More Basic Application

As a “down-to-earth” application that people taking precalculus can understand, I mention that the geometric interpretation of complex multiplication can be used to more efficiently derive many trigonometric identities (also see De Moivre’s formula, which is related to Euler’s formula and the geometric interpretation of complex number multiplication).

Visual Complex Analysis

Finally, before we briefly explore the relationship of this content to ordered fields in real analysis, I mention that a beautiful exposition of complex numbers and complex analysis can be found in Tristan Needham’s book, “Visual Complex Analysis”. It’s not an easy book, and you’ll need to know calculus well beforehand, but it is very much worthwhile and beautiful if you want to get deeply into the subject.

Relationship to Ordered Fields in Real Analysis

As I mentioned at the beginning, the original impetus of this article was as the seventh article in a series to help people study real analysis from Baby Rudin. Other than the fact that Rudin defines complex numbers as ordered pairs of real numbers, I have not yet delved into this more advanced content. That omission will presently be rectified here at the end of this article for those who are interested. On the other hand, this article is already long, so I will keep this content short.

The Set of Complex Numbers Forms a Field

The definition of a mathematical field is described near the middle of the second article in this series: “Definitions of Ordered Set and Ordered Field”. In that article, I emphasize that the set of rational numbers $\Bbb{Q}$ forms a field as does the set of real numbers $\Bbb{R}$ (both under “ordinary” addition and multiplication). In addition, now I emphasize that the set of complex numbers $\Bbb{C}$ forms a field under the addition and multiplication operations described earlier in this article.

Moreover, in Baby Rudin, other properties of $\Bbb{C}$ are emphasized as well. One important example is the triangle inequality for complex numbers. If $z=a+bi\in \Bbb{C}$ and $w=c+di\in \Bbb{C}$ with moduli $|z|=\sqrt{a^{2}+b^{2}}$ and $|w|=\sqrt{c^{2}+d^{2}}$ , then $|z+w|\leq |z|+|w|$ . This can indeed be interpreted using triangles, but I will not get into details. Another important example is the Cauchy-Schwarz inequality.

The Complex Numbers Do NOT Form an Ordered Field

Though the set of complex numbers $\Bbb{C}$ can, somewhat surprisingly (because of two-dimensional nature of the complex plane), be made into a totally ordered set, it cannot be made into an ordered field. No total order $<$ on $\Bbb{C}$ will be “compatible” with the field operations on $\Bbb{C}$ .

The Dictionary (Lexiographic) Order

One example of an ordering on $\Bbb{C}$ that makes it an ordered set is the so-called dictionary order. Let $z=(a,b)=a+bi$ and $w=(c,d)=c+di$ be arbitrary complex numbers. Define $z<w$ if either $a<c$ or, in the case where $a=c$ , if $b<d$ . Note that, in particular $z=(a,b)=a+bi>0$ ( $z$ is positive) if either $a>0$ or, in the case where $a=0$ , if $b>0$ . On the other hand, $z=(a,b)=a+bi<0$ ( $z$ is negative) if either $a<0$ or, in the case where $a=0$ , if $b<0$ .

However, this order does not make $\Bbb{C}$ an ordered field. For example, $z=1+i$ and $w=1+2i$ are both positive with respect to the dictionary order (their real parts are positive). However, the product $zw=(1-2)+(2+1)i=-1+3i$ is negative with respect to the dictionary order, because its real part is negative. This cannot happen in an ordered field (the product of two positives must be positive in an ordered field).

Could Some Other Total Order Work?

But maybe the dictionary order is the “wrong” total order to use to make $\Bbb{C}$ an ordered field. Could there be some other total order we can define on $\Bbb{C}$ to make it an ordered field?

The answer is no!

Why? Use an argument by contradiction and a bit of ingenuity.

Suppose to the contrary that there is a total order $<$ that can be defined on $\Bbb{C}$ to make it an ordered field (and not just an ordered set). Now the imaginary unit $(0,1)=0+1i=i\in \Bbb{C}$ is either positive or negative with respect to $<$ (it’s definitely not zero). In either case, it is a property of ordered fields that the square of any nonzero element must be positive (see Proposition 1.18(d) on page 8 in Baby Rudin).

Therefore, $i^{2}=-1=-1+0i=(-1,0)>0$ (I realize that looks strange to write). But we can also say that $i^{4}=(i^{2})^{2}=(-1)^{2}=1>0$ as well. On the other hand, $i^{2}+i^{4}=-1+1=0$ . In other words, $i^{2}=-1$ and $i^{4}=1$ are additive inverses of each other ( $-i^{2}=i^{4}$ ) which are both positive. But this cannot happen in an ordered field (see Proposition 1.18(a) on page 8 in Baby Rudin).

This contradiction implies that there is no total order $<$ that can be defined on $\Bbb{C}$ to make it an ordered field. In a nutshell, we can say that $\Bbb{C}$ is not an ordered field.

A Final Interesting Observation

Let’s close this article by comparing this to something we saw in Study Help for Baby Rudin, Part 1.5, “The Archimedean Property”.

At the end of that article, we considered ordered fields of rational functions $F(x)$ , where the coefficients were from an ordered field $F$ . In particular, based on what we discussed there, we can say that the field $\Bbb{R}(x)$ of rational functions with real coefficients is a field extension of the field of real numbers $\Bbb{R}$ that is also an ordered field, just as $\Bbb{R}$ is. There is a natural embedding $\Bbb{R}\rightarrow \Bbb{R}(x)$ mapping a real number $c$ to the constant rational function $c=\frac{c}{1}$ so that we can effectively “imagine” that $\Bbb{R}\subseteq \Bbb{R}(x)$ .

On the other hand, $\Bbb{C}$ can also be considered to be a field extension of $\Bbb{R}$ . There is a natural embedding $\Bbb{R}\rightarrow \Bbb{C}$ mapping a real number $c$ to the complex number $c+0i$ (the “real axis” in the complex plane is actually the “real line”). Hence, we can “imagine” that $\Bbb{R}\subseteq \Bbb{C}$ as well.

However, $\Bbb{R}(x)$ is an ordered field (under a natural total order that extends the one defined on $\Bbb{R}$ ) whereas we have just learned that $\Bbb{C}$ is not an ordered field. This is an interesting observation that is worth remembering for the future if you want to be a pure mathematician! It’s always a good thing for pure mathematicians to be aware of various examples!

On the other hand, keep in mind that, although $\Bbb{R}(x)$ is an ordered field, it is does not satisfy, for example, the Archimedean Property.

A Helpful Video

If you like the content of this last section, you will probably find this video I made interesting and helpful.

Archimedean Property of the Real Numbers R, a Non-Archimedean Ordered Field, and Hyperreal Numbers