# II. Exact Arithmetic

This is part II of the series A Practical Guide to Boolean Operations on Triangle Meshes. Part I on interval arithmetic can be found here.

In our last installment, we presented *interval arithmetic* as part of the solution for the problem of floating point error accumulation.
To review, an arithmetic operation on a CPU has associated floating point error because of limited precision.
Small errors in calculations can balloon into large errors in decision making given the right circumstances; if there's anything that we've collectively learned as programmers, it's that the right (wrong) circumstances are essentially inevitable.
To remedy this, we adopt the following scheme:

- Compute arithmetic expressions using interval arithmetic, yielding rigorous lower and upper bounds for the result of the calculation.
- If we can't make a decision because the resulting interval is ambiguous, repeat the calculation using increased precision or exact arithmetic.

### Number Representations

There are a *lot* of real numbers (aka "infinite decimals").
Almost all of them are "non-computable," meaning that we can't come up with any sort of finite representation for them.
In other words, they will never be used, listed or uniquely described by anything or anyone.
It's not as if we just choose not to use them; we are actually incapable of distinguishing them from the vast swaths of other nearby numbers in a finite amount of time and space.
These numbers are by definition boring.
If there was something uniquely interesting about a particular number, we'd need only articulate the fact and the number would be computable (in an informal way, for any math readers out there).
For example, we care about \(\pi\) for many reasons, one being that it is the sum of the infinite series
\[4\left(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\ldots \right)\,. \]
This is a representation that we can use to generate digits of \(\pi\).
We can't generate *all* the digits of \(\pi\), but we can generate as many as we want given enough time and resources.
Of course, we can't point out any particular non-computable number to you in a concrete and satisfying way.
That would make it computable!

Academics love to start off their papers talking about "computable numbers," and for good reason. It is an impossible goal to attempt to create a representation for arbitrary real numbers on a finite computer. Instead, library designers must restrict themselves to smaller subsets of \(\mathbb{R}\). What subset depends on what you need from your exact arithmetic library. Let's give a few examples.

### Example Number Representation #1: Integers

### Example Number Representation #2: Rationals

### Example Number Representation #3: Algebraic Numbers

Algebraic numbers are exactly the numbers that are roots of polynomials with rational coefficients. Any number that can be written as an expression using rational numbers, basic operations (\(+\), \(-\), \(\times\), \(/\)) and radicals is algebraic, though there are many algebraic numbers that cannot be written like this. Compass and straightedge construction is a classic application of algebraic numbers, with each point coordinate equal to some expression involving addition, subtraction, multiplication, division, and square roots.

The most straightforward (and perhaps intuitive) way to represent radical-like numbers is via their expressions. This is how we typically write them out, using radicals, addition, multiplication and division. For example, \[ \frac{\sqrt{3-2\sqrt{2}}}{\sqrt{5}+(7)^{1/7}} \] is an expression that we might write out to denote a certain number. There are internal representations of expressions on computers that suffice for this kind of thing, notably abstract syntax trees (ASTs). This might look something like this:

One problem with this is that there can be multiple expressions that describe the same number. For example, \[ \frac{\sqrt{3-2\sqrt{2}}}{\sqrt{5}+(7)^{1/7}}=\frac{\sqrt{-2\sqrt{2}+3}}{(7)^{1/7}+\sqrt{5}}=\frac{\sqrt{2}-1}{\sqrt{5}+(7)^{1/7}}\,. \] Some of these transformations are obvious, like \(3-2\sqrt{2}=-2\sqrt{2}+3\). Others are less so: \(\sqrt{3-2\sqrt{2}}=\sqrt{2}-1\). This is not an insurmountable hurdle, but it makes it quite difficult to tell whether two numbers are the same. The bigger problem, depending on the application, is that there are many algebraic numbers which cannot be written in terms of radicals (so-called "unsolvable" algebraic numbers).

Algebraics are exactly the real numbers \(\alpha\) such that \(\mathbb{Q}(\alpha)\) is a finite-dimensional field extension over \(\mathbb{Q}\). If \([\mathbb{Q}(\alpha):\mathbb{Q}]=n\) ("the dimension of the field formed by appending \(\alpha\) to \(\mathbb{Q}\) with respect to the base field \(\mathbb{Q}\)"), that means that we can represent any expression involving rationals, \(\alpha\), \(+\), \(-\), \(\times\), and \(/\) as a list of \(n\) rational vector coordinates in the basis \(\{1,\alpha,\alpha^2,\ldots,\alpha^{n-1}\}\). Any number in \(\mathbb{Q}(\sqrt[3]{5})\) can be written as \(a+b\sqrt[3]{5}+c\sqrt[3]{5}^2\), for example, where \(a,b,c\in\mathbb{Q}\). This basis comes from the necessary linear dependence of \(\{1,\alpha,\alpha^2,\ldots,\alpha^n\}\) yielding the minimal polynomial \(p\): \[ p(\alpha)=c_0+c_1\alpha+c_2\alpha^2+\ldots+c_n\alpha^n=0\,, \] together with the canonical isomorphism \[\mathbb{Q}(\alpha)\cong \mathbb{Q}[X]/(p(X)) \\ \alpha\mapsto \overline{X}\,.\] All this seems is simultaneously extremely opaque to the uninitiated and all too obvious to those familiar with Galois theory or field extensions. The upshot is that roots of rational polynomials are