Since a number is not itself a geometric object, the phrase “symmetry of numbers” might sound a little mysterious. It doesn’t refer to the possible naive interpretation as symmetries of the arabic numerals, say for instance, the symmetry of 8 about its vertical axis. Arabic numerals give one possible way to represent integers, but there are others. The roman IIX is not symmetric about its central vertical axis. Any symmetry of a number that is truly a feature of the number itself, and not just its representation in some script, should depend upon arithmetic properties of the number. Thus, we’ll need to identify what arithmetic properties of numbers should be preserved by symmetry.
We’re jumping the gun a little bit here, as we’ve been using the word number in a bit of a vague sense. There are lots of different flavors of numbers — integer, rational, real, complex, etc. Of these the integers are the simplest, but they lack one feature. If you take two arbitrary nonzero integers, their quotient won’t necessarily be another integer. One must work with rational numbers to handle quotients of nonzero integers. They are defined as the quotients of all integers, where the denominators must be nonzero. Rational numbers can be added, multiplied and divided:
It’s possible to discuss symmetries of integers alone, but it requires a little more mathematical savvy. It will be simpler for us to allow division as above.
To motivate the next definition I’d like to discuss what is sometimes meant by symmetry in geometry. Consider some geometric object in the plane or in 3-space. A transformation of the plane or 3-space is called distance preserving if the distance between any pair of points does not change after performing the transformation. For example, rotations and reflections preserve distances. A symmetry of an object in the plane (or 3-space, or any inner product space) is a distance preserving transformation of the plane which maps the object to itself. For instance, a rotation of the plane by 90 degrees about the center of a square preserves that square. The point to take away from this is that we have defined symmetry of an object by making reference to a larger ambient space, in this case the entire plane. We’ll do the same thing for numbers, which means in particular that we’ll need to introduce the larger “ambient space of numbers”. We turn to this topic now.
The constant is not a rational number, although it is a real number. Nor is , although saying so could have gotten you killed in the time of the Pythagoreans! Cantor discovered over a century ago that there are vastly more real numbers than rational numbers. Although both are real but not rational, there is an important distinction to be made between the numbers and , and it has to do with equations. Namely is a root of an integer polynomial, while is not. What this means is that if you plug into the polynomial then you get the number . Indeed, this is precisely how is defined! It turns out, and this is not so easy to show, that is not a root of any polynomial with integer coefficients. Numbers like this are said to be transcendental over the rationals. Any number which can be expressed as a root of a polynomial with integer coefficients is said to be algebraic over the rationals. Numbers which are algebraic have many symmetries, while transcendental numbers need not. It is for this reason that we restrict our attention to algebraic numbers. Since algebraic numbers each satisfy some polynomial algebraic relation, this forces certain symmetries upon them.
Let’s recapitulate for a second. Now we know what numbers we’d like to consider in this discussion: numbers which are algebraic over the rationals. And the reason why we want to focus on these numbers is because they have lots of symmetries. The same philosophy holds true in geometry. If you take a really wild looking shape in the plane, studying its symmetries might tell you virtually nothing about the shape at all. Geometric symmetry is only useful for studying nice shapes which possess symmetries, just as arithmetic symmetry is useful for studying algebraic numbers. Other techniques are required to study transcendental numbers.
Let’s get a bit more formal and introduce the notation for the set of rational numbers. Now we’ll sometimes say that a number is algebraic over to mean that it is algebraic over the rational numbers. Note that if is a rational number, denoted , then it can be written as for two integers and . It is thus a root of the polynomial , which has integer coefficients. This verifies that every rational number is algebraic over the rationals.
Next we need to introduce the set of all algebraic numbers over the rationals. For this it is illuminating to first recall the complex numbers and the Fundamental Theorem of Algebra. After all of this we’ll pause for the day.
A complex number is an expression where and are real numbers, and is a square root of . Many people balk at this, as common experience shows that whenever you square a number you get a positive number. This is certainly true for real numbers. So what’s the deal with complex numbers? The point is that they can be added, subtracted, multiplied and divided just as well as real numbers, so why not call them numbers as well? We’re just broadening our definition of the term number. With this broader definition, it is no longer true that the square of every number is positive. This is only true for real numbers, and in fact, it doesn’t really make sense to talk about positive complex numbers.
Recall that complex numbers are added “componentwise”: . Similarly for subtraction. Mutiplication is an iota more complicated, but still easy to figure out if you recall that we want to have :
To define division first note that according to the rule above. The number is called the conjugate of . Since is a real number, nonzero as soon as one of or is nonzero, we can divide by it to deduce that the correct definition for the division of complex numbers is given by
as long as one of or is nonzero. The number is the zero complex number, denoted simply . In fact, every real number is complex if we identify .
Complex arithmetic works just like ordinary arithmetic. Addition and multiplication are commutative and associative, and the distributive laws hold. The complex number is the multiplicative identity, while is the additive identity. The ability to do arithmetic with complex numbers is not sufficient justification for their importance. One possible justification is provided by the following fact: every polynomial with real numbers as coefficients has a complex root. That is, if I write down any polynomial, say , then it has a complex root. This fact is the Fundamental Theorem of Algebra and was first proved by Gauss. It is somewhat surprising since not all polynomials have real roots. The polynomial is an example, and is another. Thus, if one wants to consider solutions to polynomials with real coefficients, then complex numbers arise quite naturally.
The Fundamental Theorem of Algebra implies that if a number is a root of a polynomial over the reals, then it is a complex number. Algebraic numbers are roots of polynomials over the rationals, so they are also complex numbers. Let denote the collection of all complex numbers, and let denote the subset of the complex numbers which are algebraic over the rationals. This set will be called the set of algebraic numbers, and it is the ambient space containing that we are after. Next time we’ll define symmetries of numbers as certain functions . Just as symmetries of geometric objects preserve the geometric property of distance, our symmetries of numbers will be required to preserve arithmetic properties.
Remark: Every polynomial with real coefficients factors completely over the complex numbers. What about polynomials with complex numbers as coefficients? Do you need to introduce some sort of hyper-complex number to factor such polynomials? Thankfully not! Even every polynomial with complex coefficients factors completely over the complex numbers.