The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 1 (Intermediate Algebra) Integers, Rationals, Order of Operations, and Properties of Algebra

Integers, Rationals, Order of Operations, and Properties of Algebra – Lecture 1

Overview

This opening lecture builds the number systems that the rest of the course rests on and reviews the arithmetic you will use constantly. It introduces set notation for the empty set, natural numbers, integers, and rationals, then drills the rules for adding, subtracting, multiplying, and dividing both integers and fractions. It also reviews the order of operations (PEMDAS), including how absolute-value bars and division-as-a-fraction fit in. Finally, it lists the axioms (properties) of algebra and shows how to evaluate expressions by substitution and how to combine like terms. The goal is a solid, automatic command of the basic operations so later algebra is just bookkeeping.

Key Concepts

Number systems and notation

Empty set $\varnothing$ — the set containing nothing.
Natural numbers $\mathbb{N}$ — the whole numbers $0, 1, 2, 3, \dots$ continuing without end.
Integers $\mathbb{Z}$ — the negatives, zero, and positives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$
Rational numbers $\mathbb{Q}$ — every quotient of integers $\frac{a}{b}$ where $a, b \in \mathbb{Z}$ and $b \neq 0$ . Every fraction of integers (e.g. $\tfrac12$ , $\tfrac{17}{97}$ ) is rational.

Integer arithmetic

Adding behaves as expected; think of debts and payments. Example: $-7 + (-3) = -10$ .
Subtracting a negative is the same as adding a positive: $a - (-b) = a + b$ . So $-7 - (-3) = -7 + 3 = -4$ .
Sign rules for multiplication: a negative times a negative is positive; a negative times a positive is negative. With repeated factors the sign alternates — an even number of negative factors gives a positive result, an odd number gives a negative result.
Division follows the same sign rules as multiplication, because by definition $a \div b = a \cdot \frac{1}{b}$ . So $\frac{-15}{-3} = 5$ .

Rational-number arithmetic

Reducing: cancel common factors using $\frac{x}{x} = 1$ (for $x \neq 0$ ) and $a \cdot 1 = a$ . Example: $\frac{64}{12} = \frac{16 \cdot 4}{3 \cdot 4} = \frac{16}{3}$ .
Adding/subtracting: rewrite each fraction over a common denominator (ideally the least common multiple), then add or subtract numerators. Brute-force formula: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ .
Multiplying: multiply straight across, $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ . Cross-cancel common factors before multiplying to keep numbers small.
Dividing: multiply by the reciprocal, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ . (E.g. $1000 \div \tfrac12 = 1000 \cdot 2 = 2000$ — how many half-pizzas fit in 1000 pizzas.)

Order of operations (PEMDAS — "Please Excuse My Dear Aunt Sally")

Parentheses → Exponents → Multiplication/Division → Addition/Subtraction.
Multiplication and division share a tier (division is multiplication by a reciprocal), and addition and subtraction share a tier (since $a - b = a + (-b)$ ); within a tier work left to right.
With nested brackets, work from the innermost outward.
Absolute-value bars group like parentheses: $|x| = x$ if $x \ge 0$ , and $|x| = -x$ if $x < 0$ . A fraction bar groups the entire numerator and denominator, so evaluate each separately before dividing.

Properties (axioms) of algebra

Commutative: $x + y = y + x$ and $x \cdot y = y \cdot x$ .
Associative: $(x + y) + z = x + (y + z)$ and $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ .
Identities: $x + 0 = x$ (additive), $1 \cdot x = x$ (multiplicative).
Inverses: $x + (-x) = 0$ (additive inverse); for $x \neq 0$ , $x \cdot \frac{1}{x} = 1$ (reciprocal).
Distributive: $x(y + z) = xy + xz$ .
Substitution / law of identity: equals may be substituted for equals — replace variables with their values and evaluate.

Combining like terms

Only terms with the same variable part combine; $x^2$ and $x$ are different and cannot be merged. Collect like terms by adding their coefficients: e.g. $5x - 2y - 8x + 7y = -3x + 5y$ .

Worked Example

Evaluate the expression below using PEMDAS, with absolute-value bars treated as grouping symbols.

\begin{aligned} &\frac{5 + 3^2 - \dfrac{24}{6}\cdot 2}{\bigl(5 + 3(-2)\bigr)^2 + \bigl|\,2^2 - 5\,\bigr|} \\ &= \frac{5 + 9 - 4\cdot 2}{(5 - 6)^2 + |\,4 - 5\,|} \\ &= \frac{5 + 9 - 8}{(-1)^2 + |-1|} \\ &= \frac{6}{1 + 1} = \frac{6}{2} = 3 \end{aligned}

Summary

The number systems nest: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$ , written $\varnothing$ , $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{Q}$ .
Subtracting a negative equals adding a positive; for products and quotients, an even count of negatives is positive and an odd count is negative.
Add/subtract fractions over a common denominator; multiply straight across (cross-cancel first); divide by multiplying by the reciprocal — and always reduce the result.
Apply PEMDAS, remembering that ×/÷ tie and +/− tie (work left to right), brackets resolve innermost-first, and absolute-value and fraction bars act as grouping symbols.
The axioms of algebra (commutative, associative, identity, inverse, distributive) justify every rearrangement; evaluate expressions by substituting values and simplifying.
Combine like terms only when their variable parts match — $x^2$ and $x$ stay separate.