The Timpson Lyceum · College Algebra

Lecture 4 (College Algebra)

Overview

This lecture builds the real number system from the ground up as an axiomatic structure: a set $\mathbb{R}$ equipped with two operations, $+$ and $\cdot$ , satisfying ten field axioms. The central idea is that algebra is nothing more than tracing the logical consequences of these ten assumptions — every later result is a theorem derived from them. We restate the axioms, see how the natural numbers and integers can be constructed (and even proved to be distinct) from them, and then prove a sequence of foundational theorems, including uniqueness of the additive identity, the cancellation law, $x \cdot 0 = 0$ , the zero-product property, and the rules governing additive inverses and subtraction. A recurring theme is that mathematics is a self-contained logical game: it does not assume its statements are true, only that the axioms imply the theorems.

Key Concepts

The field axioms of $\mathbb{R}$ . $\mathbb{R}$ is a set closed under two operations $+$ and $\cdot$ (the sum and product of two real numbers is again real) satisfying:

A1 — Commutativity of addition: $x + y = y + x$ .
A2 — Associativity of addition: $(x + y) + z = x + (y + z)$ .
A3 — Additive identity: there exists $0 \in \mathbb{R}$ with $x + 0 = x$ for all $x$ .
A4 — Additive inverses: for each $x$ there exists $y$ with $x + y = 0$ .
A5 — Commutativity of multiplication: $x \cdot y = y \cdot x$ .
A6 — Associativity of multiplication: $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ .
A7 — Multiplicative identity: there exists $1 \in \mathbb{R}$ with $x \cdot 1 = x$ for all $x$ .
A8 — Multiplicative inverses: for each $x \neq 0$ there exists $y$ with $x \cdot y = 1$ .
A9 — Distributive law: $x \cdot (y + z) = x \cdot y + x \cdot z$ .
A10 — Nontriviality: $1 \neq 0$ .

Axioms vs. intuition. The symbols $0$ and $1$ are defined only by the axioms they satisfy — not by prior intuition about counting or distance. Familiar facts (e.g. that multiplication is repeated addition) are at best convenient interpretations; what makes $0$ behave like zero is precisely A3, A4, A10 together with the theorems they force.

Constructing the numbers. From the axioms one proves $1 + 1 \neq 1$ and $1 + 1 \neq 0$ , so $1 + 1$ is a genuinely new element, defined as $2$ . Continuing, $3 := 2 + 1$ , $4 := 3 + 1$ , and so on, generates all the natural numbers; once additive inverses are defined, the negative integers follow. The symbols $0$ and $1$ are not defined this way — they receive their meaning directly from the axioms (otherwise one falls into infinite regress).

Proof by contradiction. To show a statement is false, assume it true and derive a falsehood (e.g. $1 = 0$ , which violates A10). This is the engine behind several uniqueness proofs in this lecture.

Uniqueness of identities and inverses.

The additive identity $0$ is unique (Theorem 1): no other element behaves like $0$ .
The additive inverse of each $x$ is unique — this licenses the definition $-x$ as the one and only solution of $x + y = 0$ . ( $-x$ denotes the additive inverse, which need not be a "negative" number; e.g. the inverse of $-4$ is $4$ .)

Cancellation law (Theorem 2). If $x + z = y + z$ then $x = y$ .

Multiplication by zero (Theorem 3). For every $x$ , $x \cdot 0 = 0$ .

Zero-product property (Theorem 4). If $x \cdot y = 0$ then $x = 0$ or $y = 0$ .

Inverse identities.

$-(-x) = x$ .
$(-1)\cdot x = -x$ .
$-(x \cdot y) = (-x)\cdot y = x \cdot(-y)$ — applying an additive inverse to a product requires negating only one factor.
$(-x)\cdot(-y) = x \cdot y$ , and in particular $(-1)(-1) = 1$ .

Subtraction (definition). $x - y := x + (-y)$ . Subtraction is not a new operation but shorthand for adding the additive inverse. From this: $x(y - z) = xy - xz$ and $-(x - y) = -x + y$ .

Why division by zero is forbidden. Division by $b$ means multiplying by $b$ 's multiplicative inverse, which A8 guarantees only for $b \neq 0$ . If $1/0$ were some number $c$ , then $c \cdot 0 = 1$ ; but $c \cdot 0 = 0$ by Theorem 3, forcing $1 = 0$ , contradicting A10. Admitting division by zero would make the system inconsistent.

Worked Example

Theorem 3: for every real number $x$ , $x \cdot 0 = 0$ .

The key is that $0 = 0 + 0$ (A3), so distributing gives an equation we can cancel. Start from $x \cdot 0$ , rewrite $0$ as $0 + 0$ , distribute via A9, then use the cancellation law (Theorem 2) to peel off one copy of $x \cdot 0$ :

\begin{aligned} x \cdot 0 &= x \cdot (0 + 0) &&\text{(A3: } 0 = 0+0\text{)} \\ &= x \cdot 0 + x \cdot 0 &&\text{(A9: distribute)} \\ x \cdot 0 + 0 &= x \cdot 0 + x \cdot 0 &&\text{(A3 on the left)} \\ 0 + x \cdot 0 &= x \cdot 0 + x \cdot 0 &&\text{(A1: commute)} \\ 0 &= x \cdot 0 &&\text{(Theorem 2: cancel } x\cdot 0\text{)} \end{aligned}

By A5, $x \cdot 0 = 0 \cdot x$ as well, so multiplication by $0$ yields $0$ from either side.

Summary

$\mathbb{R}$ is defined axiomatically: a set with $+$ and $\cdot$ satisfying ten field axioms (A1–A10); all of algebra is the logical fallout of these rules.
$0$ and $1$ get their meaning from the axioms themselves; every other integer is constructed (e.g. $2 := 1+1$ ) and proved distinct.
The additive identity is unique, and each element has a unique additive inverse $-x$ , defined as the sole solution of $x + y = 0$ .
Core early theorems: cancellation ( $x+z = y+z \Rightarrow x=y$ ), $x \cdot 0 = 0$ , and the zero-product property ( $xy = 0 \Rightarrow x=0$ or $y=0$ ).
Inverse rules follow cleanly: $-(-x)=x$ , $(-1)x = -x$ , $(-x)(-y)=xy$ , and $-(xy)$ needs only one factor negated.
Subtraction is defined as $x - y := x + (-y)$ ; division by $0$ is impossible because $0$ has no multiplicative inverse — allowing it forces $1 = 0$ and breaks the system.