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Real Numbers

Table of Contents

1. Number Systems

The real numbers are typically taught as the final "target" in a sequence of number systems.

1.1. Natural Numbers

We can begin with the natural numbers.

1.1.1. Peano Arithmetic

An axiomatic way to do so is assert there is some constant Z that's a natural number, and if x is a natural number then we can apply a function/constructor of a single argument S to it to get S x another distinct natural number. By iteratively applying S to Z, we get all the natural numbers.

We identify Z with zero, S Z with one, S (S Z) with two, and so on. Further, S is injective as a mapping of the natural numbers to itself…so for any m and n, we have m=n if and only if S m = S n.

These are the Peano Axioms. We could define the familiar binary operators like addition, which satisfies the properties:

  • a + Z = a
  • a + (S b) = (S a) + b

Multiplication could similarly be defined inductively:

  • a * Z = Z
  • a * (S b) = a + a*b

1.1.2. Axioms

We "know" arithmetic, but what axioms do we "need"? The ones we should hope for include, but are not limited to:

  • Closure under addition and multiplication, i.e., \(a\times b\) is a natural number, and \(a + b\) is another natural number.
  • Addition
    Associativity
    \(a + (b + c) = (a + b) + c\)
    Commutativity
    \(a + b = b + a\)
    Identity Element
    \(0 + a = a\)
  • Multiplication
    Associativity
    \(a\times(b\times c)=(a\times b)\times c\)
    Commutativity
    \(a\times b = b\times a\)
    Identity Element
    \(a\times 1 = a\)
    Distributivity
    \(a\times(b + c) = a\times b + a\times c\)
    No nonzero zero divisors
    if \(a\times b = 0\), then either \(a = 0\) or \(b = 0\)

1.2. Integers

We allow subtraction to natural numbers. Subtraction is a binary operation, written \(a - b\) for integers \(a\) and \(b\), which is such that \((a - b) + b = a\). We generically write \(0 - a\) as \(-a\), and refer to it as the negation of \(a\).

If we wanted to be rigorous, we could use the Grothendieck construction: work with pairs of natural numbers \((a,b)\) and instead of equality, we use an equalivalence relation

\begin{equation} (a,b) \sim (x, y)\iff a+y = b+x. \end{equation}

If \(a > b\), then \((a,b)\) is intuitively the equivalence class corresponding to the number \(a - b > 0\). On the other hand, for \(a < b\), the pair corresponds to the equivalence class of the negative number \(a - b\). We should think of the first component as a surplus, the second as a deficit.

1.3. Rational Numbers

The intuitive construction of the rational numbers is any pair of integers \(a/b\) where \(b\neq0\) is nonzero. Addition is defined as

\begin{equation} \frac{a}{b} + \frac{x}{y} = \frac{ay + bx}{by} \end{equation}

Multiplication is defined as

\begin{equation} \frac{a}{b}\times\frac{x}{y} = \frac{a\times x}{b\times y}. \end{equation}

We could go through another Grothendieck construction to derive the rational numbers formally from the integers, but that's tedious.

1.4. Real Numbers

There are a variety of ways to construct the real numbers, at least three I'm aware of:

  1. using Dedekind cuts,
  2. Bourbaki's topological closure of the rationals
  3. as the smallest ordered field containing the rationals

The latter seems the most direct way to get cooking.

1.4.1. Axioms

  1. \((\mathbb{R}, +, \times)\) form a field
    • Associativity of multiplication and addition: for any \(x,y,z\in\mathbb{R}\) we have \((x + y) + z = x + (y + z)\) and \((x\times y)\times z = x\times(y\times z)\)
    • Commutativity of addition and multiplication: for any \(x,y\in\mathbb{R}\) we have \(x + y = y + x\) and \(x\times y=y\times x\)
    • Distributivity: for any \(x,y,z\in\mathbb{R}\) we have \(x\times(y + z) = (x\times y) + (x\times z)\)
    • For any \(x\in\mathbb{R}\), \(x + 0 = x\)
    • 0 is not equal to 1, and for any \(x\in\mathbb{R}\) we have \(x\times1=x\)
    • For any \(x\in\mathbb{R}\) there exists a unique \(-x\in\mathbb{R}\) such that \(x+(-x)=0\)
    • For any nonzero \(x\in\mathbb{R}\) there exists a unique \(x^{-1}\in\mathbb{R}\) such that \(x\times x^{-1}=1\)
  2. \((\mathbb{R}, \leq)\) form a totally ordered set
    Reflexivity
    For any \(x\in\mathbb{R}\), we have \(x\leq x\)
    Antisymmetry
    For any \(x,y\in\mathbb{R}\), if \(x\leq y\) and \(y\leq x\), then \(x = y\)
    Transitivity
    For any \(x,y,z\in\mathbb{R}\), if \(x\leq y\) and \(y\leq z\), then \(x\leq z\)
    Totality
    For all \(x,y\in\mathbb{R}\), either \(x\leq y\) or \(y\leq x\).
  3. The field operations \(+\) and \(\times\) are compatible with the order \(\leq\) on \(\mathbb{R}\), in particular
    • If \(x,y\in\mathbb{R}\) are such that \(x\leq y\), then for any \(z\in\mathbb{R}\) we have \(x+z\leq y+z\).
    • For any \(x,y\in\mathbb{R}\), if \(0\leq x\) and \(0\leq y\), then \(0\leq x\times y\)
  4. The order \(\leq\) is complete (i.e., every non-empty subset of \(\mathbb{R}\) bounded above has a least upper bound)

Tarski provided another axiomatization in 1936 with four undefined notions. Again, we don't need it for our purposes.

1.5. Transcendental and Algebraic Numbers

We call a number \(a\) Algebraic if there exists some polynomial \(p(x)\) for which \(a\) is a root, \(p(a) = 0\).

Observe: any rational number is algebraic, since \(p(x) = x - r\) for \(r\in\QQ\) is such a polynomial.

The set of algebraic numbers is countable.

We call a number \(t\) Transcendental if it is not algebraic.

(Gelfond–Schneider) If \(a\) and \(b\) are algebraic numbers, \(a\neq 0\) and \(a\neq 1\), with \(b\) irrational, then any value of \(a^{b}\) is a transcendental number.

As a consequence, almost all real numbers are transcendental. Most of them are not really even "used" in practice.

1.6. Definable Real Numbers

We say a real number \(a\) is Definable in the Language of Arithmetic if its Dedekind cut can be defined as a predicate in that language; i.e., if there is a first-order formula \(\varphi\) in the language of arithmetic, with three variables, such that

\begin{equation} \forall m,\forall n,\forall p\left(\varphi(m,n,p)\iff\frac{(-1)^{p}m}{n}\lt a\right). \end{equation}

Our intuition is that \(m\), \(n\), and \(p\) range over all natural numbers. The only operations allowed in Peano arithmetic are addition and multiplication. (And, I suppose, the successor operation, which is just "addition with 1" intuitively.) The only constants allowed in Peano arithmetic are 0 and 1.

An equivalent way to characterize arithmetically definable real numbers is by considering Dedekind cuts of the form

\begin{equation} \{m/n \mid \forall x_{1}\exists x_{2}\dots \forall x_{k-1}\exists x_{k},p(m,n,\vec{x})=0\} \end{equation}

for some fixed polynomial with integer coefficients \(p\), and the \(x\)'s range over integers. Here both \(\E\) and \(\pi\) are definable, as are all our favorite real numbers. A standard reference for this line of thinking is:

  • Reference: Simpson's Subsystems of Second-Order Arithmetic discusses system \(ACA_{0}\), which we've basically defined.

It is invalid to reason, "Well, we can define \(\pi\) as the area of the unit circle, and that's a formula, so it is definable." (Or worse, "We define \(\pi\) as…, therefore since it's defined that way it must be definable.") Please don't do this.

We can consider definable real numbers, is the intersection of transcendental numbers with definable real numbers interesting in some way?

2. Scientific Notation

We can represent any real number using three components:

  1. Its sign (+1 or -1)
  2. Its magnitude (an integer)
  3. Its mantissa (a real number less than the base)

In base 10, we would write \(x = s\times m\times 10^{p}\) where \(s\in\{+1,-1\}\), \(0\lt m\lt 10\), and \(p\in\mathbb{Z}\). Every real number may be written in this manner, and it is called "scientific notation".

The difficulty is, not every real number may be represented with finitely many digits. For example \(\pi\) requires infinitely many, as does \(\sqrt{2}\). We then form approximations to these numbers. Since this is an approximation, we are left with errors, which propagate in our calculations. The entire field of Numerical Analysis studies this matter.

2.1. Binary

When we work in base 2, the mantissa is a sequence of bits with leading bit being 1 (otherwise we could shift the mantissa down until it is 1, and the exponent decreases by the number of shifts). This has the advantage that \(1\lt m\lt 2\), which can speed up Taylor series computations.

For example, the logarithm of \(x\) in this notation is given by

\begin{equation} \log(x) = \log(s) + \log(m) + p\log(2) \end{equation}

which consists of looking up stored values for \(\log(2)\), and computing a series approximation for \(\log(m)\).

We can extract the components for a float in Common Lisp by 

USER => (multiple-value-bind (mantissa expon sign)
(decode-float f)
  (scale-float mantissa expon))
;; equivalent to (abs f)

USER> (multiple-value-bind (mantissa expon sign)
          (decode-float pi)
        (list :sign sign :mantissa mantissa :exponent expon))
;; (:SIGN 1.0d0 :MANTISSA 0.7853981633974483d0 :EXPONENT 2)

I've delegated discussion of this number system in Floating Point Arithmetic.

3. References

  • Every real analysis textbook
  • Edmund Landau,
    Foundations of Analysis.
    A "golden oldie" which provides rigorous construction of number systems from Peano arithmetic to integers to fractions and finally giving us the reals.

Last Updated 2022-01-11 Tue 13:15.