The Timpson Lyceum · College Algebra

Lecture 5 (College Algebra)

Overview

This lecture closes out the foundational, axiom-based portion of the course and then opens the textbook's "basic review." First it finishes building the multiplicative structure of the real numbers: it proves that every nonzero number has a unique multiplicative inverse, names that inverse $x^{-1}$ , and proves the key facts $(x^{-1})^{-1} = x$ and $(xy)^{-1} = x^{-1}y^{-1}$ , before defining division in terms of inverses. The second half pivots to a higher-level review of numbers themselves: number symbols vs. values, base-10 decimals, the three kinds of decimal expansions, and the precise definition of rational and irrational numbers. The centerpiece of this half is the classic proof that $\sqrt{2}$ is irrational. The throughline is that familiar arithmetic (reciprocals, fractions, decimals) all rests on a small set of field axioms.

Key Concepts

Multiplicative inverse (existence and uniqueness). For every real number $x \neq 0$ , there exists one and only one real number $y$ such that $xy = 1$ . This unique $y$ is called the multiplicative inverse of $x$ , written $x^{-1}$ . So by definition, $x \cdot x^{-1} = 1$ for every $x \neq 0$ . (Zero has no multiplicative inverse, since $0$ times anything is $0$ , never $1$ .) This parallels the additive inverse: $-x$ is the unique number with $x + (-x) = 0$ .

Inverse notation vs. reciprocal notation. Writing $7^{-1}$ and writing $\tfrac{1}{7}$ say exactly the same thing — $7^{-1}$ is "the one number you multiply $7$ by to get $1$ ." The "move a negative exponent to the denominator" rule you may have memorized comes directly from this definition.

Inverse of an inverse. $(x^{-1})^{-1} = x$ for every $x \neq 0$ . Taking the multiplicative inverse twice returns the original number, mirroring $-(-x) = x$ for additive inverses.

Inverse of a product. $(xy)^{-1} = x^{-1}y^{-1}$ for $x \neq 0$ and $y \neq 0$ . Note the contrast with the additive case, where $-(xy) = (-x)y$ applies the inverse to only one factor. For multiplicative inverses, the inverse distributes over both factors.

Definition of division. Just as subtraction is defined by $a - b = a + (-b)$ , division is defined through the multiplicative inverse: for $y \neq 0$ , $\frac{x}{y} = x \cdot y^{-1}.$ Division by zero is undefined precisely because $0$ has no multiplicative inverse.

Splitting a fraction over a sum. Using division-as-inverse plus the distributive axiom, $\dfrac{x + y}{z} = \dfrac{x}{z} + \dfrac{y}{z}$ .

Numbers vs. number symbols. A number is an abstract element with no physical form. A number symbol is a representation of a number, and the number it represents is its value. Many different symbols — $5$ , $3+2$ , $\sqrt{25}$ , "five" — can all share one value.

Base-10 and decimal expansions. Our system uses ten digits ( $0$ – $9$ ). A decimal number can be expanded as a sum of digit-times-power-of-ten terms; for example $361.24 = 3\cdot 10^{2} + 6\cdot 10^{1} + 1 + 2\cdot 10^{-1} + 4\cdot 10^{-2}$ . A negative exponent denotes a repeated multiplicative inverse: $2^{-5} = (2^{-1})^{5} = \tfrac{1}{2^{5}}$ .

Three kinds of decimals.

Terminating — the expansion ends, e.g. $\tfrac{5}{4} = 1.25$ .
Repeating — a block of digits repeats forever, e.g. $\tfrac{1}{3} = 0.\overline{3}$ or $\tfrac{1}{7} = 0.\overline{142857}$ .
Non-repeating — goes on forever with no repeating block, e.g. $\pi$ and $\sqrt{2}$ .

Every terminating decimal is also a repeating decimal (append repeating zeros). Every real number can be written as an endless decimal, repeating or not.

Rational numbers. A real number $x$ is rational if there exist integers $p$ and $q$ with $q \neq 0$ such that $x = \tfrac{p}{q}$ . The set is written $\mathbb{Q}$ . A number is rational exactly when its decimal expansion terminates or repeats. Judge by value, not appearance: $\tfrac{3\pi}{2\pi} = \tfrac{3}{2}$ is rational, and $\sqrt{2}\cdot\sqrt{2} = 2$ is rational even though each factor is not.

Irrational numbers. A real number that is not rational is irrational — it cannot be written as a quotient of two integers, equivalently its decimal expansion neither terminates nor repeats. Every real number is exactly one of: rational or irrational.

Number sets. The natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$ (this course's author starts at $1$ ); the integers $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$ — all whole numbers, positive, negative, and zero.

Converting a repeating decimal to a fraction. Set $x$ equal to the repeating decimal, multiply by $10^{k}$ where $k$ is the length of the repeating block, subtract the original $x$ to cancel the infinite tail, and solve. This always yields an integer over an integer, proving every repeating decimal is rational.

Additive inverse facts (review). $-x = (-1)\cdot x$ , and $-(-x) = x$ . "Positive/negative" (the sign) is unrelated to "even/odd": $x$ is even if $x = 2a$ for some integer $a$ , and odd if $x = 2a + 1$ ; by this definition $0$ is even, and $0$ is neither positive nor negative.

Worked Example

Proving $\sqrt{2}$ is irrational (by contradiction). Suppose, for contradiction, that $\sqrt{2}$ is rational. Then it can be written as a fraction of integers, and any such fraction can be reduced until at least one of the numerator or denominator is odd:

\begin{aligned} \sqrt{2} &= \frac{p}{q}, \quad p,q \in \mathbb{Z},\ q \neq 0,\ \text{($p$ or $q$ odd)} \\ 2 &= \frac{p^{2}}{q^{2}} \quad\Longrightarrow\quad 2q^{2} = p^{2} \\ p^{2} &= 2q^{2} \ \text{is even} \quad\Longrightarrow\quad p \ \text{is even} \\ p &= 2a \quad\Longrightarrow\quad 2q^{2} = (2a)^{2} = 4a^{2} \\ q^{2} &= 2a^{2} \ \text{is even} \quad\Longrightarrow\quad q \ \text{is even} \end{aligned}

Now both $p$ and $q$ are even, contradicting the requirement that one of them is odd. The assumption fails, so $\sqrt{2}$ is irrational. (The steps use the fact that the square of an odd integer is odd, so an even square forces an even base.)

Converting a repeating decimal. Show $0.\overline{9}$ is rational:

\begin{aligned} x &= 0.\overline{9} \\ 10x &= 9.\overline{9} \\ 10x - x = 9x &= 9 \\ x &= \frac{9}{9} = 1 \end{aligned}

Summary

Every nonzero real number $x$ has a unique multiplicative inverse $x^{-1}$ with $x\cdot x^{-1} = 1$ ; this is just $\tfrac{1}{x}$ in cleaner notation.
Key inverse identities: $(x^{-1})^{-1} = x$ and $(xy)^{-1} = x^{-1}y^{-1}$ (the multiplicative inverse distributes over both factors, unlike the additive inverse).
Division is defined as $\tfrac{x}{y} = x\cdot y^{-1}$ for $y \neq 0$ ; dividing by zero is undefined because $0$ has no inverse.
Decimals come in three types — terminating, repeating, and non-repeating — and every real number is an endless decimal.
A number is rational ( $x = \tfrac{p}{q}$ with integers $p,q$ , $q\neq 0$ ) exactly when its decimal terminates or repeats; irrational numbers are everything else, with non-terminating, non-repeating expansions.
$\sqrt{2}$ is irrational (proof by contradiction via even/odd parity), while every repeating decimal can be turned into an integer fraction and is therefore rational.