The Timpson Lyceum · Real Analysis (2020)

Lecture 2 (Real Analysis 2020) part 1

Overview

This part finishes building arithmetic purely from the field axioms and then opens the second axiom system, the positivity (order) axioms. Working only from the axioms and previously proved theorems, we establish the multiplicative inverse as a unique number (not yet an operation), define division the same way subtraction was defined, and prove several "obvious" algebraic facts (distributing multiplication and division across a sum or difference) rigorously. The discipline throughout is that nothing may be used unless it is an axiom or already a proved theorem — we do not even know what "$2$" or "to a power" means yet. We close by introducing the positivity axioms, which carve out a set of positive reals and give $\mathbb{R}$ its order.

Key Concepts

Recap — what the field axioms give us. The field axioms (commutativity, associativity, additive and multiplicative identities and inverses, distributivity, and $1 \neq 0$) are assumed. From them we already proved: $0$ is unique; $x \cdot 0 = 0$; if $xy = 0$ then $x = 0$ or $y = 0$; the additive inverse of each $x$ is unique; and the identities $-(-x) = x$, $\;-x = (-1)x$, $\;(-x)y = -(xy)$, $\;(-x)(-y) = xy$, and $(-1)(-1) = 1$.

Subtraction is a definition, not an axiom. Having shown each $x$ has a unique additive inverse, we name it $-x$ (the one and only $y$ with $x + y = 0$) and then define $x - y := x + (-y).$ "$-$" marks a particular number; it is not yet an operation.

Distributing across a difference (Theorem). $x(y - z) = xy - xz$. This is not an axiom — it follows from the definition of subtraction plus the distributive axiom.

Sign rule for a negated difference (Theorem). $-(x - y) = -x + y$, obtained by rewriting each "$-$" as multiplication by $-1$, distributing, and using $(-1)(-1) = 1$.

Multiplicative inverse — unique, and just a label. For every $x \neq 0$ there is one and only one $y$ with $xy = 1$ (proved like the uniqueness of $0$). We name that unique $y$ $x^{-1} := \text{the one and only } y \text{ with } x\cdot y = 1.$ Crucially, $x^{-1}$ is not an operation or a power — there is no definition of "to the power of" yet. It is only a tag for a specific number, and it exists only when $x \neq 0$ (a zero element has no multiplicative inverse).

Inverse identities (Theorems). $(x^{-1})^{-1} = x$, $\;(-x)^{-1} = -(x^{-1})$, and $(xy)^{-1} = x^{-1}y^{-1}$.

Division is a definition. Mirroring subtraction, for all $x$ and all $y \neq 0$, $\frac{x}{y} := x \cdot y^{-1}.$ The numerator $x$ may be anything (including $0$) — $x$ is always just a number. The restriction is on the denominator: $y^{-1}$ is only defined for $y \neq 0$, so dividing by $0$ is undefined.

Caution for later proofs. You may be tempted to multiply both sides of an equation by some $a^{-1}$. This is legal only after you have confirmed $a \neq 0$; otherwise $a^{-1}$ may not exist.

The positivity axioms. The field axioms give arithmetic but no notion of order. The positivity axioms remedy this by postulating a subset $P \subseteq \mathbb{R}$, the positive numbers, satisfying:

• Closure: if $x, y \in P$ then $x + y \in P$ and $x \cdot y \in P$.
• Trichotomy: for each real $x$, exactly one of $x \in P$, $x = 0$, or $-x \in P$ holds.

These let us define order (e.g. $x < y$ when $y - x \in P$), giving every pair of reals a definite before/after relationship. Together with the field axioms and the later completeness axiom, these are the three axiom systems the course is built on.

Worked Example

Claim. For all $x, y$ and all $z \neq 0$, $\;\dfrac{x + y}{z} = \dfrac{x}{z} + \dfrac{y}{z}$.

Apply the definition of division ($a/z = a\cdot z^{-1}$) first, then the distributive axiom:

The hypothesis $z \neq 0$ is essential: every "denominator" here is a multiplicative inverse, and $z^{-1}$ exists only when $z \neq 0$.

Summary

• Everything is built strictly from the field axioms and already-proved theorems — no appeals to outside arithmetic facts, and no numbers like "$2$" or operations like "powers" are assumed.
• Subtraction and division are definitions: $x - y := x + (-y)$ and $x/y := x\cdot y^{-1}$, each mirroring an inverse element.
• $-x$ and $x^{-1}$ are labels for unique numbers (the unique additive / multiplicative inverse), not operations; $x^{-1}$ exists only for $x \neq 0$.
• Familiar rules such as $x(y-z) = xy - xz$, $-(x-y) = -x + y$, and $(x+y)/z = x/z + y/z$ are genuine theorems requiring proof.
• Always verify a number is nonzero before multiplying by its inverse — $a^{-1}$ need not exist.
• The positivity axioms introduce a set of positive reals (closed under $+$ and $\times$, with trichotomy), supplying the order that the field axioms alone do not provide.